Large time intrinsic growth and asymptotic behavior for
the classical Timoshenko system
May 30, 2026
In this paper, we investigate the large time behavior of solutions to the classical Timoshenko system in the whole space \(\mathbb{R}\). Although the system is conservative and its natural energy is conserved in time,
the transversal displacement \(\varphi\) and the rotation angle \(\psi\) exhibit intrinsic polynomial growths. We establish sharp \(L^p-L^q\) estimates for
the solutions and show that the growth mechanism originates from the interaction between quadratic oscillations and singular low-frequency amplitudes of different orders. Furthermore, we prove the optimality of the obtained growth rates under a nontrivial
zeroth-moment condition on the initial data, while additional moment cancellations with a nontrivial first-moment condition lead to lower-order growth regimes. As a consequence, we derive large time asymptotic profiles related to an effective plate-type
dispersive structure hidden in the low-frequency regime of the classical Timoshenko system. We also discuss the relation with the dissipative Timoshenko system through a large time vanishing dissipation limit for time-normalized solutions.
Keywords: Timoshenko system, intrinsic growth, sharp asymptotics, plate-type structure, large time asymptotic profile, vanishing dissipation limit
AMS Classification (2020) 35L52, 35B40, 35Q74
In this paper, we investigate the large time behavior of solutions to the following classical Timoshenko system (or, the so-called conservative Timoshenko system) in the whole space \(\mathbb{R}\): \[\begin{align} \label{Eq-Timoshenko} \begin{cases} \rho\,\varphi_{tt}-K(\varphi_x-\psi)_x=0,&x\in\mathbb{R},\;t>0,\\ I_{\rho}\psi_{tt}-EI\psi_{xx}-K(\varphi_x-\psi)=0,&x\in\mathbb{R},\;t>0,\\ (\varphi,\varphi_t)(0,x)=(\varphi_0,\varphi_1)(x),&x\in\mathbb{R},\\ (\psi,\psi_t)(0,x)=(\psi_0,\psi_1)(x),&x\in\mathbb{R}, \end{cases} \end{align}\tag{1}\] where the unknown functions \(\varphi=\varphi(t,x)\) and \(\psi=\psi(t,x)\) represent the transversal displacement and the rotation angle, respectively. The positive constants \(\rho\), \(I_\rho\), \(E\), \(I\), and \(K\) denote the mass density, the rotary inertia, the Young modulus, the moment of inertia of the cross section, and the shear stiffness (cf. [1]), respectively.
The classical Timoshenko system was established in the early 20th century by [2], [3], and remains one of the cornerstones of structural mechanics, because it simultaneously captures the transversal motion, the shear deformation, and the rotational inertia effects of beams. To describe the propagation and low-frequency structures of the system later, we introduce the following characteristic quantities.
| Characteristic quantity | Notation |
|---|---|
| Shear-wave speed | \(c_{\mathrm{S}}:=\sqrt{\frac{K}{\rho}}\) |
| Rotational-wave speed | \(c_{\mathrm{R}}:=\sqrt{\frac{EI}{I_{\rho}}}\) |
| Shear-rotation oscillation frequency | \(c_{\mathrm{O}}:=\sqrt{\frac{K}{I_{\rho}}}\) |
| Bending dispersion coefficient | \(c_{\mathrm{D}}:=\sqrt{\frac{EI}{\rho}}\) |
The quantities \(c_{\mathrm{O}}\) and \(c_{\mathrm{D}}\) play essential roles in the low-frequency structure and the large time asymptotic behavior of solutions to the classical Timoshenko system 1 . In particular, they characterize the oscillatory and dispersive effects appearing in the present paper.
Most existing studies on the Timoshenko system have focused on dissipative mechanisms, stability structures, and decay properties of damped models. In contrast, the large time asymptotic behavior of solutions to the classical conservative Timoshenko system in the whole space \(\mathbb{R}\) remains far less understood due to the lack of crucial damping mechanisms.
To understand the large time behavior of the classical Timoshenko system 1 , it is important to compare it with the corresponding dissipative Timoshenko system as follows: \[\begin{align} \label{Eq-dissipative-Timoshenko} \begin{cases} \rho\,\varphi_{tt}-K(\varphi_x-\psi)_x=0,&x\in\mathbb{R},\;t>0,\\ I_{\rho}\psi_{tt}-EI\psi_{xx}-K(\varphi_x-\psi)+\gamma\psi_t=0,&x\in\mathbb{R},\;t>0,\\ (\varphi,\varphi_t)(0,x)=(\varphi_0,\varphi_1)(x),&x\in\mathbb{R},\\ (\psi,\psi_t)(0,x)=(\psi_0,\psi_1)(x),&x\in\mathbb{R}, \end{cases} \end{align}\tag{2}\] where \(\gamma>0\) denotes the frictional damping coefficient. The dissipative Timoshenko system has been extensively studied in connection with stability structures, regularity-loss phenomena, asymptotic behavior, and decay properties. In particular, most existing asymptotic theories for the Timoshenko system have been developed in the dissipative setting. In the case of non-equal wave speeds \(c_{\mathrm{S}}\neq c_{\mathrm{R}}\), the dissipative structure exhibits the so-called regularity-loss phenomenon, which was clarified in [4] by means of energy methods in the Fourier space. Subsequent studies established refined decay estimates and asymptotic profiles for the dissipative Timoshenko system and related models (see, for example, [4]–[11] and references therein). Recently, [12] discovered a new large time growth phenomenon for the dissipative Timoshenko system 2 , to be specific, \[\|\varphi(t,\cdot)\|_{L^2}\approx t^{\frac{3}{4}} \;\;and\;\; \|\psi(t,\cdot)\|_{L^2}\approx t^{\frac{1}{4}} \;\;for\;\; t\gg1.\] Moreover, the corresponding large time asymptotic profiles were also identified, which are recalled in Proposition 2 and Proposition 3. These observations naturally lead to the following question.
Question .
What is the mechanism responsible for the large time growth in the dissipative Timoshenko system 2 ?
Since the dissipative system still exhibits polynomial growth of the solutions themselves, it becomes important to clarify whether such growth originates from the damping mechanism or from the intrinsic structure of the Timoshenko system.
The present paper shows that the large time growth phenomenon is already contained in the conservative structure of the classical Timoshenko system itself. In particular, the frictional damping mechanism in the dissipative Timoshenko system 2 is not the origin of the growth behavior. Instead, the growth is generated by the interaction between quadratic oscillations and singular low-frequency amplitudes appearing in the Timoshenko system 1 . Moreover, the coincidence between the growth structures of the conservative and dissipative Timoshenko systems naturally raises the following question.
Question .
Whether the dissipative dynamics converge to the conservative ones in the small damping regime as \(\gamma\to0\)?
Motivated by this observation, we further discuss a large time vanishing dissipation limit for time-normalized solutions in the final section.
It is well-known that the classical Timoshenko system 1 possesses the conserved natural energy \[\begin{align} E(t):= \frac{1}{2}\left( \rho\|\varphi_t(t,\cdot)\|_{L^2}^2 +I_{\rho}\|\psi_t(t,\cdot)\|_{L^2}^2 +EI\|\psi_x(t,\cdot)\|_{L^2}^2 +K\|\varphi_x(t,\cdot)-\psi(t,\cdot)\|_{L^2}^2 \right), \end{align}\] satisfying \[E(t)\equiv E(0) \;\;for any\;\;t>0.\] However, the conservation of energy itself does not provide a detailed description of the large time behavior of the transversal displacement \(\varphi\) and the rotation angle \(\psi\). This naturally leads to the following question.
Question .
Can one describe the precise large time behavior of \(\varphi\) and \(\psi\) themselves?
This problem goes beyond the standard energy theory, since the conserved energy \(E(t)\) does not control the displacement variables \(\|\varphi(t,\cdot)\|_{L^2}\) and \(\|\psi(t,\cdot)\|_{L^2}\) themselves in \(\mathbb{R}\). In particular, their large time behavior cannot be captured by standard energy methods.
One of the purposes of the present paper is to answer this question by establishing sharp \(L^p-L^q\) growth estimates for solutions to 1 . More precisely, for suitable initial data in \(H^{s,p}\) with \(1\leqslant p\leqslant2\leqslant q\leqslant+\infty\), we derive \[\begin{align} \|\varphi(t,\cdot)\|_{L^q} &\lesssim (1+t)^{1-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}, \\ \|\psi(t,\cdot)\|_{L^q} &\lesssim (1+t)^{\frac{1}{2}-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}. \end{align}\] Furthermore, in the case \((p,q)=(1,2)\), we establish the optimal large time growth rates \[\begin{align} \|\varphi(t,\cdot)\|_{L^2} \approx t^{\frac{3}{4}} \;\;and\;\; \|\psi(t,\cdot)\|_{L^2} \approx t^{\frac{1}{4}} \end{align}\] for \(t\gg1\), under the nontrivial moment condition \(\int_{\mathbb{R}} \varphi_1(x)\,\mathrm{d}x \neq0\). Moreover, the additional moment cancellation \(\int_{\mathbb{R}}\varphi_1(x)\,\mathrm{d}x=0\) leads to lower-order growth regimes. Our results reveal that the large time growth is an intrinsic phenomenon generated by the interaction between quadratic oscillations and singular low-frequency amplitudes appearing in the classical Timoshenko system. The above questions, therefore, are answered through the sharp \(L^p-L^q\) growth estimates obtained in Section [sec:Sec-Lq], the optimal asymptotic profiles derived in Section [Section-L2-est], and the large time vanishing dissipation limit established in Section [sec:Sec-Final].
The derivation of the sharp \(L^q\)-growth estimates is highly nontrivial. In the low-frequency region, the analysis requires a delicate treatment of the oscillatory and dispersive structures of the Timoshenko system. In the high-frequency region, the coupled propagation structure must be combined with refined dyadic decomposition arguments to identify the dominant contributions. The proof of the optimal \(L^2\)-growth estimates is even more delicate. In contrast to the dissipative case studied in [12], the classical Timoshenko system does not contain the additional low-frequency regularization generated by the fractional damping structure. Consequently, the Fourier analysis developed in [12] is no longer directly applicable. To overcome this difficulty, we adapt several ideas from the wave and plate equations [13]–[16] to the present strongly coupled system, while avoiding explicit calculations of the characteristic roots. This requires a refined analysis of the coupled oscillatory structure generated by the Timoshenko system.
Notation. Let the generic positive constants \(c\) and \(C\), independent of \(t\), vary from line to line. The notation \(f\lesssim g\) means that there exists a constant \(C>0\) such that \(f\leqslant Cg\). Similarly, \(f\gtrsim g\) means \(g\lesssim f\). We write \(f\approx g\) if both \(f\lesssim g\) and \(g\lesssim f\) hold. We denote by \(f\ast_{(x)}g\) the convolution of \(f\) and \(g\) with respect to the spatial variable \(x\). We denote by \(\widehat{f}=\mathcal{F}_{x\to\xi}(f)\) the Fourier transform of \(f\) in the spatial variable \(x\), and by \(\mathcal{F}^{-1}_{\xi\to x}\) its inverse. Moreover, we introduce the following zones in the Fourier space: \[\begin{align} \mathcal{Z}_{\mathrm{int}}(\varepsilon_0) &:= \{\xi\in\mathbb{R}:\;|\xi|\leqslant\varepsilon_0\}, \\ \mathcal{Z}_{\mathrm{bdd}}(\varepsilon_0,N_0) &:= \{\xi\in\mathbb{R}:\;\varepsilon_0\leqslant|\xi|\leqslant N_0\}, \\ \mathcal{Z}_{\mathrm{ext}}(N_0) &:= \{\xi\in\mathbb{R}:\;|\xi|\geqslant N_0\}, \end{align}\] where \(\varepsilon_0>0\) is sufficiently small and \(N_0>0\) is sufficiently large. Let \(\chi_{\mathrm{int}}(\xi)\), \(\chi_{\mathrm{bdd}}(\xi)\), \(\chi_{\mathrm{ext}}(\xi)\) be smooth cut-off functions supported in \(\mathcal{Z}_{\mathrm{int}}(\varepsilon_0)\), \(\mathcal{Z}_{\mathrm{bdd}}(\frac{\varepsilon_0}{2},2N_0)\), \(\mathcal{Z}_{\mathrm{ext}}(N_0)\), respectively, satisfying \[\chi_{\mathrm{bdd}}(\xi) = 1-\chi_{\mathrm{int}}(\xi)-\chi_{\mathrm{ext}}(\xi) \;\; for all \;\; \xi\in\mathbb{R}.\] For simplicity, we write \(\chi:=\chi_{\mathrm{int}}\) and define \[\begin{align} \|f\|_{L^q_{\chi}} := \big\|\chi(D)f\big\|_{L^q} \;\;and\;\; \|f\|_{L^q_{1-\chi}} := \big\| \big(1-\chi(D)\big)f \big\|_{L^q} \end{align}\] to denote the corresponding localized norms. The Bessel potential space is denoted by \[H^{s,p} := \left\{ f\in\mathcal{S}' :\; (1-\partial_x^2)^{\frac{s}{2}}f\in L^p \right\},\] where \(s\in\mathbb{R}\) and \(1\leqslant p\leqslant+\infty\). Finally, we define the weighted \(L^1\) space by \[\begin{align} L^{1,\sigma} := \left\{ f\in L^1: \;\; \|f\|_{L^{1,\sigma}} := \int_{\mathbb{R}} (1+|x|)^{\sigma}|f(x)|\,\mathrm{d}x <+\infty \right\}, \end{align}\] where \(\sigma\in\mathbb{N}_0\). Note that \(L^{1,0}\equiv L^1\). The zeroth moment for \(f\in L^1\), and first moment for \(f\in L^{1,1}\), respectively, are defined by \[\begin{align} P_f := \int_{\mathbb{R}} f(x)\,\mathrm{d}x \;\;and\;\; M_f := \int_{\mathbb{R}} (-x)f(x)\,\mathrm{d}x. \end{align}\]
For brevity, we introduce two data spaces \[\begin{align} X_{p,q}^{\epsilon} &:= H^{s_{p,q}+\epsilon,p} \times H^{s_{p,q}-1+\epsilon,p} \;\;and\;\; Y_{p,q}^{\epsilon} := H^{s_{p,q}-1+\epsilon,p} \times H^{s_{p,q}-2+\epsilon,p} \end{align}\] with the index \[\begin{align} s_{p,q} := \frac{3}{2} \left( \frac{1}{p}-\frac{1}{q} \right) \;\; for \;\; 1\leqslant p\leqslant2\leqslant q\leqslant+\infty. \end{align}\] Our first result reveals the intrinsic polynomial growth of the displacement variables in the \(L^p-L^q\) framework.
Theorem 1. Suppose that the initial data belong to the corresponding spaces appearing on the right-hand side of the estimates below, with an arbitrarily small constant \(\epsilon>0\). Then, the transversal displacement \(\varphi\) and the rotation angle \(\psi\) to the classical Timoshenko system 1 satisfy the following \(L^p-L^q\) growth estimates:
in the case of non-equal speed \(c_{\mathrm{S}}\neq c_{\mathrm{R}}\), \[\begin{align} \|\varphi(t,\cdot)\|_{L^q} &\lesssim (1+t)^{ 1-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\varphi_0,\varphi_1)\|_{X_{p,q}^{\epsilon}} + (1+t)^{ \frac{1}{2}-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\psi_0,\psi_1)\|_{Y_{p,q}^{\epsilon}}, \\ \|\psi(t,\cdot)\|_{L^q} &\lesssim (1+t)^{ \frac{1}{2}-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\varphi_0,\varphi_1)\|_{Y_{p,q}^{\epsilon}} + (1+t)^{ -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\psi_0,\psi_1)\|_{X_{p,q}^{\epsilon}}; \end{align}\]
in the case of equal speed \(c_{\mathrm{S}}=c_{\mathrm{R}}\), \[\begin{align} \|\varphi(t,\cdot)\|_{L^q} &\lesssim (1+t)^{ 1-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\varphi_0,\varphi_1)\|_{X_{p,q}^{\epsilon}} + (1+t)^{ \frac{1}{2}-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\psi_0,\psi_1)\|_{X_{p,q}^{\epsilon}}, \\ \|\psi(t,\cdot)\|_{L^q} &\lesssim (1+t)^{ \frac{1}{2}-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\varphi_0,\varphi_1)\|_{X_{p,q}^{\epsilon}} + (1+t)^{ -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\psi_0,\psi_1)\|_{X_{p,q}^{\epsilon}}. \end{align}\]
Remark 1. The arbitrarily small loss \(\epsilon>0\) is technical and arises from the dyadic decomposition in the high-frequency analysis. More precisely, it is caused by the embedding \(H^{s+\epsilon,p}\hookrightarrow B^s_{p,1}\).
Remark 2. The difference between the equal speed case and the non-equal speed case in Theorem 1 has a natural interpretation from the viewpoint of propagation mechanisms. When \(c_{\mathrm{S}}\neq c_{\mathrm{R}}\), the shear and rotational waves propagate with genuinely different speeds, which produces a stronger separation of high-frequency modes and stronger high-frequency interactions, which require additional regularity in the corresponding estimates. In contrast, when \(c_{\mathrm{S}}=c_{\mathrm{R}}\), the two principal propagation mechanisms are coherent at the leading order. This propagation coherence weakens the high-frequency loss and explains the improved regularity requirements in the equal speed configuration.
Remark 3. The derived \(L^p-L^q\) estimates can be interpreted through the interaction between quadratic oscillations and singular low-frequency amplitudes. More precisely, the dispersive scaling produces the decay factor \[(1+t)^{ -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)},\] whereas the amplitude singularities generate the polynomial growth factors. Therefore, the polynomial growth is an intrinsic phenomenon of the conservative Timoshenko system. In particular, it should not be interpreted as an instability phenomenon, since the characteristic roots remain purely imaginary and the natural energy is conserved in time.
For later convenience, for \(\sigma\in\{0,1\}\), we introduce \[\begin{align} Z_1^{\sigma} &:= L^2\times(H^{-1}\cap L^{1,\sigma}), \\ Z_2^{\sigma} &:= H^{-1}\times(H^{-2}\cap L^{1,\sigma}), \\ Z_3 &:= L^2\times H^{-1}. \end{align}\] Our second contribution reveals the sharp large time asymptotic behavior of solutions to the classical Timoshenko system. In particular, the leading asymptotic profiles are governed by the zeroth moment of the initial velocity \(\varphi_1\), while additional moment cancellations lead to lower-order growth regimes. These asymptotic characterizations confirm the sharpness of the corresponding growth estimates in the special case \((p,q)=(1,2)\) of Theorem 2.1.
Theorem 2. Let \(\sigma\in\{0,1\}\). Suppose that \((\varphi_0,\varphi_1)\in Z_1^{\sigma}\) and \((\psi_0,\psi_1)\in Z_{\psi,\star}^{\sigma}\) for the classical Timoshenko system 1 , where \[Z_{\psi,\star}^{\sigma} := \begin{cases} Z_2^{\sigma} &if\;\;c_{\mathrm S}\neq c_{\mathrm R}, \\ Z_1^{\sigma} &if\;\;c_{\mathrm S}=c_{\mathrm R}. \end{cases}\] Then, the following statements hold for sufficiently large time.
If \(P_{\varphi_1}\neq0\), then the transversal displacement \(\varphi\) satisfies the following optimal growth estimate: \[\begin{align} t^{\frac{3}{4}}|P_{\varphi_1}| \lesssim \|\varphi(t,\cdot)\|_{L^2} \lesssim t^{\frac{3}{4}} \|(\varphi_0,\varphi_1)\|_{Z_1^0} + t^{\frac{1}{4}} \|(\psi_0,\psi_1)\|_{Z_{\psi,\star}^0}, \end{align}\] and the following asymptotic relation: \[\begin{align} \lim\limits_{t\to+\infty}t^{-\frac{3}{4}}\left\| \varphi(t,\cdot) - \sqrt{t}\, \mathcal{G}_{0} \left( \tfrac{\cdot}{\sqrt t} \right) P_{\varphi_1} \right\|_{L^2} =0, \end{align}\] where the intrinsic plate-type asymptotic profile is given by \[\begin{align} \mathcal{G}_{0}(y) := \mathcal{F}^{-1}_{\eta\to y} \left( \frac{ \sin(c_{\mathrm D}|\eta|^2) }{ c_{\mathrm D}|\eta|^2 } \right). \end{align}\]
If \(P_{\varphi_1}=0\) but \(M_{\varphi_1}-\frac{I_{\rho}}{\rho}P_{\psi_1}\neq0\), then the transversal displacement \(\varphi\) satisfies the following optimal growth estimate: \[\begin{align} t^{\frac{1}{4}} \left| M_{\varphi_1} - \frac{I_{\rho}}{\rho}P_{\psi_1} \right| \lesssim \|\varphi(t,\cdot)\|_{L^2} \lesssim t^{\frac{1}{4}} \|(\varphi_0,\varphi_1)\|_{Z_1^1} + t^{\frac{1}{4}} \|(\psi_0,\psi_1)\|_{Z_{\psi,\star}^1}, \end{align}\] and the following asymptotic relation: \[\begin{align} \lim\limits_{t\to+\infty}t^{-\frac{1}{4}}\left\| \varphi(t,\cdot) - \mathcal{G}_{1} \left( \tfrac{\cdot}{\sqrt{t}} \right) \left( M_{\varphi_1} - \frac{I_{\rho}}{\rho}P_{\psi_1} \right) \right\|_{L^2} =0, \end{align}\] where the derivative of intrinsic plate-type asymptotic profile is given by \[\begin{align} \mathcal{G}_{1}(y) :=\partial_y\mathcal{G}_0(y)= \mathcal{F}^{-1}_{\eta\to y} \left(i\eta \frac{ \sin(c_{\mathrm D}|\eta|^2) }{ c_{\mathrm D}|\eta|^2 } \right). \end{align}\]
Remark 4. When \(P_{\varphi_1}=0\) and \(M_{\varphi_1}-\frac{I_{\rho}}{\rho}P_{\psi_1}=0\), by additionally assuming \(\varphi_1\in L^{1,2}\) and \(\psi_1\in L^{1,1}\), we are able to prove the boundedness \(\|\varphi(t,\cdot)\|_{L^2}\lesssim 1\), which lies beyond the scope of the present paper.
Corollary 1. Suppose that \((\varphi_0,\varphi_1)\in Z_1^{0}\) and \((\psi_0,\psi_1)\in Z_{\psi,\star}^{0}\) such that \(P_{\varphi_1}\neq0\) for the classical Timoshenko system 1 . Assume additionally \[\begin{align} \mathrm{supp}\, (\varphi_0,\varphi_1,\psi_0,\psi_1) \subset[-L,L] \;\;for some\;\; L>0. \end{align}\] Then, the transversal displacement \(\varphi\) satisfies the following \(L^q\)-growth estimate with any \(q\geqslant 2\): \[\begin{align} \|\varphi(t,\cdot)\|_{L^q} \gtrsim t^{\frac{1}{4}+\frac{1}{q}}|P_{\varphi_1}| \end{align}\] for sufficiently large time.
Theorem 3. Suppose that \((\varphi_0,\varphi_1)\in Z_{\varphi,\sharp}^0\) and \((\psi_0,\psi_1)\in Z_{3}\) such that \(P_{\varphi_1}\neq0\) for the classical Timoshenko system 1 , where \[Z^0_{\varphi,\sharp} := \begin{cases} Z_2^0 &if\;\;c_{\mathrm S}\neq c_{\mathrm R}, \\ Z_1^0 &if\;\;c_{\mathrm S}=c_{\mathrm R}. \end{cases}\] Then, the rotation angle \(\psi\) satisfies the following optimal growth estimate: \[\begin{align} t^{\frac{1}{4}}|P_{\varphi_1}| \lesssim \|\psi(t,\cdot)\|_{L^2} \lesssim t^{\frac{1}{4}} \|(\varphi_0,\varphi_1)\|_{Z^0_{\varphi,\sharp}} + \|(\psi_0,\psi_1)\|_{Z_{\psi,3}} \end{align}\] for sufficiently large time, and the following asymptotic relation: \[\begin{align} \lim\limits_{t\to+\infty}t^{-\frac{1}{4}}\left\| \psi(t,\cdot) - \mathcal{G}_{1} \left( \tfrac{\cdot}{\sqrt t} \right) P_{\varphi_1} \right\|_{L^2} =0. \end{align}\]
Remark 5. When \(P_{\varphi_1}=0\), by additionally assuming \(\varphi_1\in L^{1,1}\), we are able to prove the boundedness \(\|\psi(t,\cdot)\|_{L^2}\lesssim 1\), which lies beyond the scope of the present paper.
Remark 6. The polynomial growth derived in this paper is generated entirely by the low-frequency structure of the classical Timoshenko system. In particular, the low-frequency zone \(\mathcal{Z}_{\mathrm{int}}(\varepsilon_0)\) produces the leading asymptotic behavior.
Remark 7. The asymptotic profiles \(\mathcal{G}_0(y)\) and \(\mathcal{G}_1(y)\) are generated by the oscillatory multiplier \[\chi_{\mathop{\mathrm{int}}}(\xi)\frac{\sin(c_{\mathrm D} |\xi|^2 t)}{c_{\mathrm D} |\xi|^2},\] which reveals a hidden plate-type structure in the low-frequency regime of the classical Timoshenko system 1 . Therefore, the leading large time behavior is governed not by the hyperbolic wave propagation itself, but by the associated quadratic oscillations and singular low-frequency amplitudes.
Remark 8. The polynomial growth derived in this paper should not be interpreted as an instability phenomenon of the classical Timoshenko system. Indeed, all characteristic roots remain purely imaginary, and the natural energy is exactly conserved in time. The coincidence of the growth rates and asymptotic profiles with those for the dissipative Timoshenko system 2 in [12] shows that the frictional damping mechanism does not generate the leading large time growth. Instead, the polynomial growth is already contained intrinsically in the conservative structure of the classical Timoshenko system. This relation is further clarified in the final section, where we discuss a large time vanishing dissipation limit between the dissipative and classical Timoshenko systems.
To reveal the hidden plate-type structure of the classical Timoshenko system 1 , we first reduce the coupled system to a fourth-order equation. More precisely, following the reduction procedure in [12] (see also [14], [17] for related thermoelastic systems), we eliminate the coupling terms by applying the Klein-Gordon operator \((I_{\rho}\,\partial_t^2-EI\partial_x^2+K)\) to 1 \(_1\) and the free wave operator \((\rho\,\partial_t^2-K\partial_x^2)\) to 1 \(_2\).
As a consequence, each component \(u\in\{\varphi,\psi\}\) satisfies the following fourth-order equation: \[\begin{align} \begin{cases} \displaystyle{ \frac{I_{\rho}}{K}u_{tttt} -\left( \frac{EI}{K} +\frac{I_{\rho}}{\rho} \right) u_{ttxx} +\frac{EI}{\rho}u_{xxxx} +u_{tt} =0, } &x\in\mathbb{R},\;t>0, \\[0.5em] (u,u_t,u_{tt},u_{ttt})(0,x)=(u_0,u_1,u_2,u_3)(x), &x\in\mathbb{R}. \end{cases} \end{align}\] where the higher-order initial data are given by \[\begin{align} u_2(x) := \begin{cases} \displaystyle{ \frac{K}{\rho} \big( \varphi''_0(x)-\psi'_0(x) \big) } &if\quad u=\varphi, \\[1em] \displaystyle{ \frac{1}{I_{\rho}} \Big[ EI\psi''_0(x) + K\big( \varphi'_0(x)-\psi_0(x) \big) \Big] } &if\quad u=\psi, \end{cases} \end{align}\] and \[\begin{align} u_3(x) := \begin{cases} \displaystyle{ \frac{K}{\rho} \big( \varphi''_1(x)-\psi'_1(x) \big) } &if\quad u=\varphi, \\[1em] \displaystyle{ \frac{1}{I_{\rho}} \Big[ EI\psi''_1(x) + K\big( \varphi'_1(x)-\psi_1(x) \big) \Big] } &if\quad u=\psi. \end{cases} \end{align}\]
Remark 9. The partial differential operator \[\begin{align} \mathcal{L}_{\mathrm{Cla}}(\partial_t,\partial_x) := \frac{I_{\rho}}{K}\partial_t^4 - \left( \frac{EI}{K} + \frac{I_{\rho}}{\rho} \right) \partial_t^2\partial_x^2 + \frac{EI}{\rho}\partial_x^4 + \partial_t^2 \end{align}\] associated with the classical Timoshenko system 1 has the same leading structure as the operator corresponding to the dissipative Timoshenko system 2 (see [12]), namely, \[\begin{align} \mathcal{L}_{\mathrm{Dis}} (\partial_t,\partial_x;\gamma) := \mathcal{L}_{\mathrm{Cla}}(\partial_t,\partial_x) + \frac{\gamma}{K} \partial_t \left( \partial_t^2 - \frac{K}{\rho}\partial_x^2 \right). \end{align}\] This common structure explains why both systems exhibit the same leading asymptotic growth rates. However, when \(\gamma=0\), the dissipative correction disappears, and the low-frequency regularization mechanism is no longer available.
Remark 10. Although both \(\varphi\) and \(\psi\) satisfy the same fourth-order equation, their asymptotic behaviors differ significantly. In particular, the growth rates, the asymptotic profiles, and the required regularities of the initial data are different. This phenomenon originates from the distinct higher-order initial data \(u_2\) and \(u_3\) generated by the coupling structure of the original Timoshenko system. Consequently, the low-frequency singular structures appearing in the Fourier representations of \(\varphi\) and \(\psi\) are also different.
Applying the partial Fourier transform with respect to \(x\), one finds that \(\widehat{u}\) solves \[\begin{align} \begin{cases} \displaystyle{ \frac{I_{\rho}}{K}\widehat{u}_{tttt} +\left[ \left( \frac{EI}{K} + \frac{I_{\rho}}{\rho} \right)|\xi|^2 +1 \right]\widehat{u}_{tt} + \frac{EI}{\rho}|\xi|^4\widehat{u} =0, } &\xi\in\mathbb{R},\;t>0, \\[0.5em] (\widehat{u}, \widehat{u}_t, \widehat{u}_{tt}, \widehat{u}_{ttt})(0,\xi) = (\widehat{u}_0, \widehat{u}_1, \widehat{u}_2, \widehat{u}_3)(\xi), &\xi\in\mathbb{R}. \end{cases} \end{align}\] Its characteristic equation is \[\begin{align} \label{Quartic-Eq} \frac{I_{\rho}}{K}\lambda^4 + \left[ \left( \frac{EI}{K} + \frac{I_{\rho}}{\rho} \right)|\xi|^2 +1 \right]\lambda^2 + \frac{EI}{\rho}|\xi|^4 = 0. \end{align}\tag{3}\] Since the discriminant associated with \(\lambda^2\) is strictly positive, the quartic equation 3 has four purely imaginary roots. We denote them by \[\lambda_{1,2} = \pm i\lambda_{\mathrm{I},1} \;\;and\;\; \lambda_{3,4} = \pm i\lambda_{\mathrm{I},2},\] where \[\begin{align} \lambda_{\mathrm{I},1}^2 &= \frac{K}{2I_{\rho}} \left( A(|\xi|) + \sqrt{ [A(|\xi|)]^2 - \frac{4EI}{\rho K}|\xi|^4 }\; \right), \\[0.5em] \lambda_{\mathrm{I},2}^2 &= \frac{K}{2I_{\rho}} \left( A(|\xi|) - \sqrt{ [A(|\xi|)]^2 - \frac{4EI}{\rho K}|\xi|^4 }\; \right), \end{align}\] with \[A(|\xi|) := \left( \frac{EI}{K} + \frac{I_{\rho}}{\rho} \right)|\xi|^2 +1.\] The low-frequency asymptotic behavior of the Timoshenko system is generated by \(\lambda_{\mathrm{I},2}\), whose quadratic oscillatory structure produces the hidden plate-type behavior appearing throughout the paper.
Let us derive the explicit Fourier representations of the two components \(\widehat{\varphi}\) and \(\widehat{\psi}\). Using the expressions of \(\widehat{u}_2(\xi)\) and \(\widehat{u}_3(\xi)\), and applying Cramer’s rule, we obtain, for \(\xi\neq0\), \[\begin{align} \widehat{\varphi}&= \frac{\frac{K}{\rho}(|\xi|^{2}\widehat{\varphi}_0 + i\xi\widehat{\psi}_0) - \lambda_{\mathrm{I},2}^{2}\,\widehat{\varphi}_0}{\lambda_{\mathrm{I},1}^{2} - \lambda_{\mathrm{I},2}^{2}}\cos(\lambda_{\mathrm{I},1}t)+\frac{\frac{K}{\rho}(|\xi|^{2}\widehat{\varphi}_1 + i\xi\widehat{\psi}_1) - \lambda_{\mathrm{I},2}^{2}\,\widehat{\varphi}_1}{\lambda_{\mathrm{I},1}^{2} - \lambda_{\mathrm{I},2}^{2}}\,\frac{\sin(\lambda_{\mathrm{I},1}t)}{\lambda_{\mathrm{I},1}} \notag\\ &\,\quad- \frac{\frac{K}{\rho}(|\xi|^{2}\widehat{\varphi}_0 + i\xi\widehat{\psi}_0) - \lambda_{\mathrm{I},1}^{2}\,\widehat{\varphi}_0}{\lambda_{\mathrm{I},1}^{2}-\lambda_{\mathrm{I},2}^{2} }\cos(\lambda_{\mathrm{I},2}t)-\frac{\frac{K}{\rho}(|\xi|^{2}\widehat{\varphi}_1 + i\xi\widehat{\psi}_1) - \lambda_{\mathrm{I},1}^{2}\,\widehat{\varphi}_1}{ \lambda_{\mathrm{I},1}^{2}-\lambda_{\mathrm{I},2}^{2}}\,\frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}}\notag\\ &=:\sum\limits_{j\in\{0,1\}}\widehat{K}_{\varphi,j}(t,|\xi|)\widehat{\varphi}_j+\sum\limits_{j\in\{0,1\}}\widehat{K}_{\psi,j}(t,|\xi|)\widehat{\psi}_j,\label{Rep-varphi} \end{align}\tag{4}\] moreover, \[\begin{align} \widehat{\psi}&= -\frac{ \frac{K}{I_{\rho}}i\xi\widehat{\varphi}_0 +\big( \lambda_{\mathrm{I},2}^{2} - \frac{E I|\xi|^{2} + K}{I_{\rho}} \big)\widehat{\psi}_0 }{ \lambda_{\mathrm{I},1}^{2}-\lambda_{\mathrm{I},2}^{2} } \cos(\lambda_{\mathrm{I},1}t)- \frac{ \frac{K}{I_{\rho}}i\xi\widehat{\varphi}_1 + \big( \lambda_{\mathrm{I},2}^{2} - \frac{E I|\xi|^{2} + K}{I_{\rho}} \big)\widehat{\psi}_1 }{ \lambda_{\mathrm{I},1}^{2}-\lambda_{\mathrm{I},2}^{2} }\, \frac{\sin(\lambda_{\mathrm{I},1}t)}{\lambda_{\mathrm{I},1}}\notag\\ &\,\quad+ \frac{ \frac{K}{I_{\rho}}i\xi\widehat{\varphi}_0 + \big( \lambda_{\mathrm{I},1}^{2} - \frac{E I|\xi|^{2} + K}{I_{\rho}} \big)\widehat{\psi}_0 }{ \lambda_{\mathrm{I},1}^{2} - \lambda_{\mathrm{I},2}^{2} } \cos(\lambda_{\mathrm{I},2}t)+ \frac{ \frac{K}{I_{\rho}}i\xi\widehat{\varphi}_1 + \big( \lambda_{\mathrm{I},1}^{2} - \frac{E I|\xi|^{2} + K}{I_{\rho}} \big)\widehat{\psi}_1 }{ \lambda_{\mathrm{I},1}^{2} - \lambda_{\mathrm{I},2}^{2} }\, \frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}}\notag\\ &=:\sum\limits_{j\in\{0,1\}}\widehat{G}_{\varphi,j}(t,|\xi|)\widehat{\varphi}_j+\sum\limits_{j\in\{0,1\}}\widehat{G}_{\psi,j}(t,|\xi|)\widehat{\psi}_j.\label{Rep-psi} \end{align}\tag{5}\] Here and in what follows, the quotient \(\frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}}\) at \(\xi=0\) is understood in the limiting sense. The Fourier multipliers introduced in 4 and 5 will be analyzed separately in the low-frequency and high-frequency regions.
The characteristic roots exhibit different asymptotic structures in the low-frequency and high-frequency zones.
A direct expansion yields \[\begin{align} \lambda_{\mathrm{I},1} &= c_{\mathrm O} + \frac{1}{2c_{\mathrm O}} \left( \frac{EI}{I_\rho} + \frac{K}{\rho} \right)|\xi|^2 + O(|\xi|^4), \\[0.5em] \lambda_{\mathrm{I},2} &= c_{\mathrm D}|\xi|^2 + O(|\xi|^6). \end{align}\]
In particular, the quadratic oscillatory structure generated by \(\lambda_{\mathrm{I},2}\) produces the hidden plate-type behavior appearing in the large time asymptotics.
Moreover, \[\begin{align} \lambda_{\mathrm{I},1}^{2} - \frac{EI|\xi|^2+K}{I_\rho} = \frac{K}{\rho}|\xi|^2 + O(|\xi|^4), \end{align}\] where the cancellation of the zeroth-order terms plays an important role in reducing the singular structure of the Fourier amplitudes associated with \(\widehat{\psi}\).
If \(c_{\mathrm S}\neq c_{\mathrm R}\), then \[\begin{align} \lambda_{\mathrm{I},1} &= \max\{c_{\mathrm S},c_{\mathrm R}\}|\xi| + O(|\xi|^{-1}), \\[0.5em] \lambda_{\mathrm{I},2} &= \min\{c_{\mathrm S},c_{\mathrm R}\}|\xi| + O(|\xi|^{-1}). \end{align}\]
If \(c_{\mathrm S}=c_{\mathrm R}\), then \[\begin{align} \lambda_{\mathrm{I},1} = c_{\mathrm S}|\xi| + \frac{c_{\mathrm O}}{2} + O(|\xi|^{-1}), \\[0.3em] \lambda_{\mathrm{I},2} = c_{\mathrm S}|\xi| - \frac{c_{\mathrm O}}{2} + O(|\xi|^{-1}). \end{align}\]
Furthermore, one has the asymptotic relation \[\begin{align} \label{Eq-01} \lambda_{\mathrm{I},1}^2-\lambda_{\mathrm{I},2}^2\approx \begin{cases} |\xi|^2 &if\quad c_{\mathrm{S}}\neq c_{\mathrm{R}},\\[0.3em] |\xi| &if\quad c_{\mathrm{S}}=c_{\mathrm{R}}. \end{cases} \end{align}\tag{6}\] due to an additional cancellation of the leading high-frequency terms. This distinction reflects the propagation coherence in the equal speed configuration and explains the improved regularity requirements in the equal speed case.
As a basic tool for estimating oscillatory integrals soon afterwards, we recall the classical van der Corput lemma (cf. [18]).
Lemma 1.
Let \(I\subset\mathbb{R}\) be an interval. Suppose that \(\Psi\in \mathcal{C}^2(I)\) is real-valued and satisfies \[|\Psi''(\xi)| \geqslant \mu >0 \;\; for all \;\; \xi\in I,\] and let \(b\in \mathcal{C}^1(I)\) be complex-valued. Then, \[\begin{align} \left| \int_I \mathrm{e}^{i\Psi(\xi)}\, b(\xi) \,\mathrm d\xi \right| \lesssim \mu^{-1/2} \left( \|b\|_{L^\infty(I)} + \|b'\|_{L^1(I)} \right), \end{align}\] where the implicit constant is independent of \(\mu\).
We also recall the following estimate for weighted \(L^1\) data (cf. [19]).
Lemma 2. Suppose that \(f\in L^{1,1}\). Then, \[\begin{align} |\widehat{f}(\xi)| \lesssim |P_f| + |\xi|\,\|f\|_{L^{1,1}}. \end{align}\]
The representations 4 and 5 show that the leading low-frequency contributions are generated by \[\begin{align} the oscillatory multiplier \;\;\frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}}, \end{align}\] whose low-frequency behavior is governed by (see Lemma 7 later) \[\begin{align} the quadratic oscillation\;\;\frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2}. \end{align}\] The estimate below reveals the precise balance between the dispersive decay generated by the oscillatory phase and the polynomial growth caused by the amplitude singularity.
Lemma 3.
Suppose that \(f\in L^p\) and \(\ell\in\{0,1,2\}\). Then, the Fourier multiplier satisfies the following \(L^p-L^q\) estimate: \[\begin{align} \left\| \mathcal{F}^{-1}_{\xi\to x} \left( |\xi|^\ell\, \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right) \ast_{(x)} f(\cdot) \right\|_{L^q_{\chi}} \lesssim (1+t)^{ 1-\frac{\ell}{2} -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|f\|_{L^p} \end{align}\] for \(1\leqslant p\leqslant2\leqslant q\leqslant+\infty\).
Proof. Let us rewrite the Fourier multiplier as \[\begin{align} \mathcal{K}_{1,\ell}(t,x) := \chi_{\mathop{\mathrm{int}}}(D) \mathcal{F}^{-1}_{\xi\to x} \left( |\xi|^{\ell}\, \frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}} \right) & = \int_{\mathbb{R}} \chi_{\mathop{\mathrm{int}}}(\xi)\, \mathrm{e}^{ix\xi}\, |\xi|^{\ell}\, \frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}} \,\mathrm{d}\xi \\ &= \int_{\mathbb{R}} \chi_{\mathop{\mathrm{int}}}(\xi)\, \mathrm{e}^{ix\xi}\, |\xi|^{\ell}\, \frac{ \sin\big(|\xi|^2g_0(\xi)t\big) }{ |\xi|^2g_0(\xi) } \,\mathrm{d}\xi, \end{align}\] where we denoted \(\lambda_{\mathrm{I},2} = |\xi|^2g_0(\xi)\) with \(g_0(0)=c_{\mathrm{D}}>0\).
We first prove the limit case \((p,q)=(1,+\infty)\). Motivated by the quadratic oscillatory structure appearing in the low-frequency zone, we introduce the scaling variable \(\eta=\sqrt{t}\,\xi\), which leads to \[\begin{align} \mathcal{K}_{1,\ell}(t,x) = t^{\frac{1}{2}-\frac{\ell}{2}} \widetilde{\mathcal{K}}_{1,\ell,t} \left( \tfrac{x}{\sqrt{t}} \right) \end{align}\] with \[\begin{align} \widetilde{\mathcal{K}}_{1,\ell,t}(y):= \int_{\mathbb{R}}\chi_{\mathop{\mathrm{int}}} \left(\tfrac{\eta}{\sqrt{t}}\right) \mathrm{e}^{iy\eta}\, |\eta|^{\ell}\, \frac{ \sin\big( |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) \big) }{ |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) } \,\mathrm{d}\eta. \end{align}\] We now derive estimates for \(\widetilde{\mathcal{K}}_{1,\ell,t}(y)\) uniformly in \(y\). We separately estimate the bounded region \(|\eta|\leqslant2\) and the oscillatory region \(|\eta|\geqslant1\). For the region \(|\eta|\leqslant2\), the boundedness of \(\frac{\sin y}{y}\) and \(\mathrm{e}^{iy\eta}\) yields \[\begin{align} \left| \int_{|\eta|\leqslant2} \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt{t}} \right) \mathrm{e}^{iy\eta}\, |\eta|^{\ell}\, \frac{ \sin\big( |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) \big) }{ |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) } \,\mathrm{d}\eta \right| \lesssim \int_{|\eta|\leqslant2} |\eta|^{\ell} \,\mathrm{d}\eta \lesssim1. \end{align}\] For the oscillatory region \(|\eta|\geqslant1\), we rewrite the integral via Euler’s formula as \[\begin{align} & \int_{|\eta|\geqslant1} \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt{t}} \right) \mathrm{e}^{iy\eta}\, |\eta|^{\ell}\, \frac{ \sin\big( |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) \big) }{ |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) } \,\mathrm{d}\eta = \frac{1}{2i} \int_{|\eta|\geqslant1} \left( \mathrm{e}^{i\Phi_+(y,\eta)} - \mathrm{e}^{i\Phi_-(y,\eta)} \right) b_{\ell,t}(\eta) \,\mathrm{d}\eta, \end{align}\] where we considered \[\begin{align} \Phi_{\pm}(y,\eta) := y\eta \pm |\eta|^2g_0 \left( \tfrac{\eta}{\sqrt{t}} \right)\;\;and\;\; b_{\ell,t}(\eta) := \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt{t}} \right) \frac{ |\eta|^{\ell-2} }{ g_0(\frac{\eta}{\sqrt{t}}) }. \end{align}\] A direct computation gives \[\begin{align} \left| \partial_{\eta}^2 \Phi_{\pm}(y,\eta) \right| &= \left| 2g_0 \left( \tfrac{\eta}{\sqrt{t}} \right) + \frac{4\eta}{\sqrt{t}} g_0' \left( \tfrac{\eta}{\sqrt{t}} \right) + \frac{|\eta|^2}{t} g_0'' \left( \tfrac{\eta}{\sqrt{t}} \right) \right| \\ &\geqslant 2 \left| g_0 \left( \tfrac{\eta}{\sqrt{t}} \right) \right| - \frac{|\eta|}{\sqrt{t}} \left| 4g_0' \left( \tfrac{\eta}{\sqrt{t}} \right) + \frac{|\eta|}{\sqrt{t}} g_0'' \left( \tfrac{\eta}{\sqrt{t}} \right) \right| \\ &\geqslant C_0, \end{align}\] where we used the continuity of \(g_0\), the identity \(g_0(0)=c_{\mathrm{D}}>0\), which guarantees a non-degenerate quadratic oscillation in the low-frequency regime, and the small support property of \(\chi_{\mathop{\mathrm{int}}}\). Since \(|\eta|\geqslant1\), we immediately have \[\begin{align} \|b_{\ell,t}\|_{L^\infty(|\eta|\geqslant1)} \lesssim1 \end{align}\] for \(\ell\in\{0,1,2\}\). Moreover, the first-order derivative is controlled by \[\begin{align} |b'_{\ell,t}(\eta)| &= \left| \chi_{\mathop{\mathrm{int}}}' \left( \tfrac{\eta}{\sqrt{t}} \right) \frac{ |\eta|^{\ell-2} }{ \sqrt{t}\, g_0(\frac{\eta}{\sqrt{t}}) } + \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt{t}} \right) \frac{ (\ell-2)|\eta|^{\ell-3} }{ g_0(\frac{\eta}{\sqrt{t}}) } - \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt{t}} \right) \frac{ |\eta|^{\ell-2}\, g_0'(\frac{\eta}{\sqrt{t}}) }{ \sqrt{t}\, [g_0(\frac{\eta}{\sqrt{t}})]^2 } \right| \\ &\lesssim |\eta|^{\ell-3} + \frac{1}{\sqrt{t}} |\eta|^{\ell-2}. \end{align}\] Therefore, it leads to \[\begin{align} \|b'_{\ell,t}\|_{L^1(|\eta|\geqslant1)} &\lesssim \int_{1\leqslant|\eta|\leqslant\varepsilon_0\sqrt{t}} \left( |\eta|^{\ell-3} + \frac{1}{\sqrt{t}} |\eta|^{\ell-2} \right) \mathrm{d}\eta \lesssim1 \end{align}\] for \(\ell\in\{0,1,2\}\) and \(t\geqslant1\). By Lemma 1, we conclude that \[\begin{align} \left| \int_{|\eta|\geqslant1} \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt{t}} \right) \mathrm{e}^{iy\eta}\, |\eta|^{\ell}\, \frac{ \sin\big( |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) \big) }{ |\eta|^2g_0(\frac{\eta}{\sqrt{t}}) } \,\mathrm{d}\eta \right| & \lesssim \sum_{\pm} \left| \int_{|\eta|\geqslant1} \mathrm{e}^{i\Phi_{\pm}(y,\eta)}\, b_{\ell,t}(\eta) \,\mathrm{d}\eta \right| \\ & \lesssim \|b_{\ell,t}\|_{L^\infty(|\eta|\geqslant1)} + \|b'_{\ell,t}\|_{L^1(|\eta|\geqslant1)} \\ & \lesssim1. \end{align}\] Hence, \[\begin{align} \|\widetilde{\mathcal{K}}_{1,\ell,t}\|_{L^\infty} \lesssim1 \end{align}\] for all \(\ell\in\{0,1,2\}\) and \(t\geqslant1\), namely, \[\begin{align} \|\mathcal{K}_{1,\ell}(t,\cdot)\|_{L^\infty} \lesssim t^{\frac{1}{2}-\frac{\ell}{2}}. \end{align}\] The case \(0<t\leqslant1\) follows directly from \[\begin{align} \|\mathcal{K}_{1,\ell}(t,\cdot)\|_{L^\infty} \lesssim t \int_{\mathbb{R}} \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^\ell \,\mathrm{d}\xi \lesssim1. \end{align}\] Consequently, a summary of them addresses \[\begin{align} \|\mathcal{K}_{1,\ell}(t,\cdot)\|_{L^\infty} \lesssim (1+t)^{\frac{1}{2}-\frac{\ell}{2}} \end{align}\] for all \(\ell\in\{0,1,2\}\) and \(t>0\). By Young’s inequality, \[\begin{align} \label{Est-01} \|\mathcal{K}_{1,\ell}(t,\cdot) \ast_{(x)}f(\cdot)\|_{L^\infty} \lesssim (1+t)^{\frac{1}{2}-\frac{\ell}{2}} \|f\|_{L^1}. \end{align}\tag{7}\]
We next consider the case \((p,q)=(2,2)\). Separating the regions \(|\xi|\leqslant t^{-\frac{1}{2}}\) and \(t^{-\frac{1}{2}}\leqslant|\xi|\leqslant\varepsilon_0\), we are able to derive \[\begin{align} \|\widehat{\mathcal{K}}_{1,\ell}(t,\xi)\|_{L^\infty} &\lesssim \left\| |\xi|^\ell\, \frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}} \right\|_{L^\infty} \lesssim (1+t)^{1-\frac{\ell}{2}}, \end{align}\] where we used \(|\sin y|\leqslant|y|\) in the first region and \(|\sin y|\leqslant1\) in the second one. By Plancherel’s identity, \[\begin{align} \label{Est-06} \|\mathcal{K}_{1,\ell}(t,\cdot) \ast_{(x)}f(\cdot)\|_{L^2} &\lesssim \|\widehat{\mathcal{K}}_{1,\ell}(t,\xi)\|_{L^\infty} \|f\|_{L^2} \notag\\ &\lesssim (1+t)^{1-\frac{\ell}{2}} \|f\|_{L^2}. \end{align}\tag{8}\]
The estimates 7 and 8 correspond respectively to the dispersive endpoint \((p,q)=(1,+\infty)\) and the energy endpoint \((p,q)=(2,2)\). Finally, the Riesz-Thorin interpolation theorem between 7 and 8 completes the proof. ◻
Remark 11. The growth factor \((1+t)^{1-\frac{\ell}{2}}\) originates from the amplitude singularity \(|\xi|^{\ell-2}\), whereas the dispersive scaling \((1+t)^{-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}\) is generated by the quadratic oscillation.
By applying the same argument as in Lemma 3, one obtains the following auxiliary result. Compared with Lemma 3, the proof is simpler since the multiplier \(\cos(\lambda_{\mathrm{I},2}t)\) does not contain the singular factor \(\lambda_{\mathrm{I},2}^{-1}\) at \(|\xi|=0\).
Lemma 4. Suppose that \(f\in L^p\) and \(\ell\in\{0,1,2\}\). Then, the Fourier multiplier satisfies the following \(L^p-L^q\) estimate: \[\begin{align} \left\| \mathcal{F}^{-1}_{\xi\to x} \left( |\xi|^{\ell}\cos(\lambda_{\mathrm{I},2}t) \right) \ast_{(x)}f(\cdot) \right\|_{L^q_{\chi}} \lesssim (1+t)^{ -\frac{\ell}{2} -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|f\|_{L^p} \end{align}\] for \(1\leqslant p\leqslant2\leqslant q\leqslant+\infty\).
In contrast to the low-frequency regime, the high-frequency multipliers do not contain the strong singular structure appearing in the low-frequency region, since \(\lambda_{\mathrm I,2}\approx |\xi|\) for \(|\xi|\gg1\). Consequently, the polynomial growth disappears in the high-frequency estimates. We first derive \(L^p-L^q\) estimates for the dominant oscillatory multipliers associated with the sine terms.
Lemma 5.
Suppose that \(f\in H^{s,p}\) with \(s> \ell-1 + \frac{3}{2}\left(\frac{1}{p}-\frac{1}{q}\right)\) for \(\ell\in\{0,1,2\}\). Then, the Fourier multiplier satisfies the following \(L^p-L^q\) estimate: \[\begin{align} \left\| \mathcal{F}^{-1}_{\xi\to x} \left( |\xi|^\ell\, \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right) \ast_{(x)} f(\cdot) \right\|_{L^q_{1-\chi}} \lesssim (1+t)^{ -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|f\|_{H^{s,p}} \end{align}\] for \(1\leqslant p\leqslant2\leqslant q\leqslant+\infty\). The same estimate remains valid with \(\lambda_{\mathrm I,2}\) replaced by \(\lambda_{\mathrm I,1}\).
Proof. Let us rewrite the Fourier multiplier as \[\begin{align} \mathcal{K}_{2,\ell}(t,x) &:= \big(1-\chi_{\mathop{\mathrm{int}}}(D)\big) \mathcal{F}^{-1}_{\xi\to x} \left( |\xi|^{\ell}\, \frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}} \right) \\ &\,= \frac{1}{2i} \sum_{j\geqslant j_0} \int_{\mathbb{R}} \chi_j(\xi) \,\mathrm{e}^{ix\xi} \left( \mathrm{e}^{i\lambda_{\mathrm{I},2}t} - \mathrm{e}^{-i\lambda_{\mathrm{I},2}t} \right) |\xi|^\ell\, \lambda_{\mathrm{I},2}^{-1} \,\mathrm d\xi, \end{align}\] where we use the dyadic decomposition \[1-\chi_{\mathop{\mathrm{int}}}(\xi) = \sum_{j\geqslant j_0}\chi_j(\xi) \;\;with\;\; \chi_j(\xi):=\chi_0(2^{-j}\xi),\] and \(|\xi|\approx 2^j\) on \(\operatorname{supp}\chi_j\). For each dyadic block, we consider \[\mathcal{K}_{2,\ell,j}^{\pm}(t,x) := \int_{\mathbb{R}} \mathrm{e}^{i(x\xi\pm\lambda_{\mathrm{I},2}t)}\, a_j(\xi) \,\mathrm d\xi,\] where \[a_j(\xi) := \chi_j(\xi) |\xi|^\ell\, \lambda_{\mathrm{I},2}^{-1}.\] For high-frequencies, the asymptotic expansion of \(\lambda_{\mathrm{I},2}\) yields \(|\lambda_{\mathrm{I},2}''(\xi)| \approx |\xi|^{-3} \approx 2^{-3j}\) on \(\operatorname{supp}\chi_j\). Therefore, applying Lemma 1 with \(\mu\approx t\,2^{-3j}\), we obtain \[\begin{align} \|\mathcal{K}_{2,\ell,j}^{\pm}(t,\cdot)\|_{L^\infty} &\lesssim t^{-\frac{1}{2}}\, 2^{\frac{3}{2}j} \left( \|a_j\|_{L^\infty} + \|a_j'\|_{L^1} \right). \end{align}\] Since \(\lambda_{\mathrm{I},2}\approx |\xi|\) and \(|\xi|\approx 2^j\) on \(\operatorname{supp}\chi_j\), we have \[\|a_j\|_{L^\infty} + \|a_j'\|_{L^1} \lesssim 2^{(\ell-1)j}.\] That is to say, \[\begin{align} \|\mathcal{K}_{2,\ell,j}^{\pm}(t,\cdot)\|_{L^\infty} \lesssim t^{-\frac{1}{2}}\, 2^{(\ell+\frac{1}{2})j}. \end{align}\] The dispersive decay \(t^{-\frac{1}{2}}\) originates from the oscillatory structure of the high-frequency phase, while the dyadic factor \(2^{(\ell+\frac{1}{2})j}\) reflects the regularity requirement generated by the multiplier amplitude. By Young’s inequality and summing over \(j\), we get \[\begin{align} \|\mathcal{K}_{2,\ell}(t,\cdot)\ast_{(x)}f(\cdot)\|_{L^\infty} &\lesssim t^{-\frac{1}{2}} \sum_{j\geqslant j_0} 2^{(\ell+\frac{1}{2})j} \|\chi_j(D)f\|_{L^1} \\ &\lesssim t^{-\frac{1}{2}} \|f\|_{B^{\ell+\frac{1}{2}}_{1,1}}. \end{align}\] Using the Sobolev–Besov embedding \(H^{s,1} \hookrightarrow B^{\ell+\frac{1}{2}}_{1,1}\) for \(s>\ell+\frac{1}{2}\), we arrive at \[\begin{align} \label{Est-Large-01} \|\mathcal{K}_{2,\ell}(t,\cdot)\ast_{(x)}f(\cdot)\|_{L^\infty} \lesssim t^{-\frac{1}{2}} \|f\|_{H^{s,1}} \end{align}\tag{9}\] for \(s>\ell+\frac{1}{2}\).
On the other hand, by Plancherel’s identity, \[\begin{align} \|\mathcal{K}_{2,\ell}(t,\cdot)\ast_{(x)}f(\cdot)\|_{L^2} &\lesssim \left\|\big( 1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) |\xi|^\ell\, \lambda_{\mathrm{I},2}^{-1} \widehat f(\xi) \right\|_{L^2} \notag\\ &\lesssim \|f\|_{H^{\ell-1,2}},\label{Est-Large-02} \end{align}\tag{10}\] because \(\lambda_{\mathrm{I},2}\approx|\xi|\) in the high-frequency region.
The estimates 9 and 10 correspond respectively to the dispersive endpoint \((p,q)=(1,+\infty)\) and the energy endpoint \((p,q)=(2,2)\). Interpolating between 9 and 10 by the Riesz–Thorin theorem yields \[\begin{align} \|\mathcal{K}_{2,\ell}(t,\cdot)\ast_{(x)}f(\cdot)\|_{L^q} \lesssim t^{-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)} \|f\|_{H^{s,p}}, \end{align}\] provided that \(s> \ell-1 + \frac{3}{2}\left(\frac{1}{p}-\frac{1}{q}\right)\). Replacing \(t^{-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}\) by \((1+t)^{-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}\) also covers the case \(0<t\leqslant1\). This completes the proof. ◻
By the same argument as above, one obtains the following auxiliary result for the cosine multipliers.
Lemma 6. Suppose that \(f\in H^{s,p}\) with \(s> \ell + \frac{3}{2} \left( \frac{1}{p}-\frac{1}{q} \right)\) for \(\ell\in\{0,1,2\}\). Then, the Fourier multiplier satisfies the following \(L^p-L^q\) estimate: \[\begin{align} \left\| \mathcal{F}^{-1}_{\xi\to x} \left( |\xi|^{\ell}\cos(\lambda_{\mathrm{I},2}t) \right) \ast_{(x)}f(\cdot) \right\|_{L^q_{1-\chi}} \lesssim (1+t)^{ -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|f\|_{H^{s,p}} \end{align}\] for \(1\leqslant p\leqslant2\leqslant q\leqslant+\infty\). The same estimate remains valid with \(\lambda_{\mathrm{I},2}\) replaced by \(\lambda_{\mathrm{I},1}\).
According to the representations of solutions in the Fourier space, we first estimate the low-frequency part. The dominant contributions are controlled by Lemma 3, whereas the remaining terms are controlled by Lemma 4. Consequently, for any \(1\leqslant p\leqslant2\leqslant q\leqslant+\infty\), we obtain \[\begin{align} \|\varphi(t,\cdot)\|_{L^q_{\chi}} &\lesssim (1+t)^{ 1-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\varphi_0,\varphi_1)\|_{L^p\times L^p} + (1+t)^{ \frac{1}{2}-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\psi_0,\psi_1)\|_{L^p\times L^p}, \\ \|\psi(t,\cdot)\|_{L^q_{\chi}} &\lesssim (1+t)^{ \frac{1}{2}-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\varphi_0,\varphi_1)\|_{L^p\times L^p} + (1+t)^{ -\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right) } \|(\psi_0,\psi_1)\|_{L^p\times L^p}. \end{align}\]
We next estimate the part away from the low-frequency zone \(\mathcal{Z}_{\mathop{\mathrm{int}}}(\varepsilon_0)\). Applying Lemma 5 and Lemma 6, we derive the corresponding high-frequency estimates with suitable regularities of the initial data. In contrast to the low-frequency region, the polynomial growth disappears in the high-frequency analysis with the decay rate \(-\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)\), since the oscillatory multipliers no longer contain the singular low-frequency structure.
Combining the low-frequency estimates and high-frequency estimates, we directly complete the proof of Theorem 1.
Remark 12. The required regularities depend on whether the wave speeds are equal, because the denominator \(\lambda_{\mathrm{I},1}^2-\lambda_{\mathrm{I},2}^2\) has different high-frequency behavior described in 6 . More precisely, the equal speed configuration produces an additional cancellation of the leading high-frequency terms, which improves the regularity requirements.
We begin with several asymptotic estimates for the dominant Fourier multipliers appearing in the representations of solutions. The following lemma shows that the quadratic oscillation generated by \(\frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2}\) provides the leading low-frequency asymptotic profile.
Lemma 7. Suppose that \(\ell\in\{0,1\}\). Then, the following asymptotic estimate: \[\begin{align} &\left\|\chi_{\mathop{\mathrm{int}}}(\xi)|\xi|^{\ell} \left(\frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}}-\frac{\sin(c_{\mathrm{D}}|\xi|^2t)}{c_{\mathrm{D}}|\xi|^2}\right) \right\|_{L^2}^2 = o(t^{\frac{3}{2}-\ell}) \end{align}\] holds as \(t\to+\infty\).
Proof. By the low-frequency expansion of \(\lambda_{\mathrm I,2}\), we have \[\begin{align} \chi_{\mathop{\mathrm{int}}}(\xi) \left| \frac{1}{\lambda_{\mathrm I,2}} - \frac{1}{c_{\mathrm{D}}|\xi|^2} \right| = \chi_{\mathop{\mathrm{int}}}(\xi) \left| \frac{O(|\xi|^6)}{c_{\mathrm{D}}|\xi|^2\lambda_{\mathrm I,2}} \right| \lesssim \chi_{\mathop{\mathrm{int}}}(\xi)|\xi|^2, \end{align}\] which suggests \[\begin{align} \label{Triangle-01} & \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^{\ell} \left( \frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}} - \frac{\sin(\lambda_{\mathrm{I},2}t)}{c_{\mathrm{D}}|\xi|^2} \right) \right\|_{L^2}^2 \lesssim \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^{\ell+2}\, \right\|_{L^2}^2 \lesssim1. \end{align}\tag{11}\]
Next, we approximate \(\sin(\lambda_{\mathrm{I},2}t)\) by \(\sin(c_{\mathrm{D}}|\xi|^2t)\) in the \(L^2\) framework, rather than relying on pointwise estimates in the Fourier space. Indeed, the error estimate in the Fourier space \[\begin{align} \chi_{\mathop{\mathrm{int}}}(\xi) \left| \sin(\lambda_{\mathrm{I},2}t) - \sin(c_{\mathrm{D}}|\xi|^2t) \right| \lesssim \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^4t \end{align}\] produces the additional growth factor \(t\), which cannot be compensated in the absence of the dissipative structure. This is substantially different from the dissipative Timoshenko system 2 studied in [12], in which the damping factor \(\mathrm{e}^{-\frac{\gamma}{2\rho}|\xi|^2t}\) provides the additional decay rate (but fails when \(\gamma=0\)) as follows: \[\begin{align} \chi_{\mathop{\mathrm{int}}}(\xi)|\xi|^4t \,\mathrm{e}^{-\frac{\gamma}{2\rho}|\xi|^2t}\lesssim t^{-1}\chi_{\mathop{\mathrm{int}}}(\xi) \,\mathrm{e}^{-c|\xi|^2t}. \end{align}\] In the vanishing dissipation case, we introduce the scaled variable \(\eta=\sqrt{t}\,\xi\) to obtain \[\begin{align} & \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^{\ell} \left( \frac{\sin(\lambda_{\mathrm{I},2}t)}{c_{\mathrm{D}}|\xi|^2} - \frac{\sin(c_{\mathrm{D}}|\xi|^2t)}{c_{\mathrm{D}}|\xi|^2} \right) \right\|_{L^2}^2 \\ &= t^{\frac{3}{2}-\ell} \int_{\mathbb{R}} \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt t} \right) |\eta|^{2(\ell-2)} \left[ \sin \left( |\eta|^2 g_0 \left( \tfrac{\eta}{\sqrt t} \right) \right) - \sin(c_{\mathrm D}|\eta|^2) \right]^2 \mathrm d\eta, \end{align}\] where \(\lambda_{\mathrm I,2} = |\xi|^2g_0(\xi)\) with \(g_0(0)=c_{\mathrm D}>0\). As \(t\to+\infty\), we have \[g_0 \left( \tfrac{\eta}{\sqrt t} \right) \to g_0(0)=c_{\mathrm D}\] for every fixed \(\eta\). Hence, the next convergence: \[\begin{align} & \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt t} \right) |\eta|^{2(\ell-2)} \left[ \sin \left( |\eta|^2 g_0 \left( \tfrac{\eta}{\sqrt t} \right) \right) - \sin(c_{\mathrm D}|\eta|^2) \right]^2 \to0 \end{align}\] holds pointwise as \(t\to+\infty\). By the elementary bounds \(|\sin y| \leqslant |y|\) and \(|\sin y| \leqslant1\), we derive \[\begin{align} & \chi_{\mathop{\mathrm{int}}} \left( \tfrac{\eta}{\sqrt t} \right) \left| |\eta|^{2(\ell-2)} \left[ \sin \left( |\eta|^2 g_0 \left( \tfrac{\eta}{\sqrt t} \right) \right) - \sin(c_{\mathrm D}|\eta|^2) \right]^2 \right| \lesssim \min \left\{ |\eta|^{2\ell}, |\eta|^{2(\ell-2)} \right\}. \end{align}\] Since \[\begin{align} \int_{\mathbb{R}} \min \left\{ |\eta|^{2\ell}, |\eta|^{2(\ell-2)} \right\} \mathrm d\eta <+\infty \end{align}\] for \(\ell\in\{0,1\}\), the dominated convergence theorem yields \[\begin{align} \label{Triangle-02} & \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^{\ell} \left( \frac{\sin(\lambda_{\mathrm{I},2}t)}{c_{\mathrm{D}}|\xi|^2} - \frac{\sin(c_{\mathrm{D}}|\xi|^2t)}{c_{\mathrm{D}}|\xi|^2} \right) \right\|_{L^2}^2 = o(t^{\frac{3}{2}-\ell}) \end{align}\tag{12}\] as \(t\to+\infty\). Finally, since \(1=o(t^{\frac{3}{2}-\ell})\) for \(\ell\in\{0,1\}\), combining 11 and 12 completes the proof immediately. ◻
The next optimal estimate reveals the precise low-frequency scaling responsible for the growth rates \(t^{\frac{3}{4}}\) and \(t^{\frac{1}{4}}\).
Lemma 8.
Suppose that \(\ell\in\{0,1\}\). Then, the Fourier multiplier satisfies the following sharp scaling asymptotics: \[\begin{align} \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^{\ell}\, \frac{ \sin(c_{\mathrm D}|\xi|^2t) }{ c_{\mathrm D}|\xi|^2 } \right\|_{L^2}^2 \approx t^{\frac{3}{2}-\ell} \end{align}\] as \(t\to+\infty\).
Remark 13. The scaling \(\eta=\sqrt{t}\,\xi\) shows that the singular amplitude \(|\xi|^{-2}\) combined with the quadratic oscillation \(\sin(c_{\mathrm D}|\xi|^2t)\) produces the growth factor \(t^{\frac{3}{4}}\). Similarly, the additional factor \(|\xi|\) reduces the growth rate to \(t^{\frac{1}{4}}\).
Proof. Concerning the upper bound, the approach in [12] is not applicable here due to the absence of the crucial exponentially decaying factor \(\mathrm{e}^{-\frac{\gamma}{2\rho}|\xi|^2t}\), which produces an uncontrollable growth factor in time. To overcome this difficulty, we introduce two time-dependent separating lines \(|\xi|=t^{-\alpha_{\ell}}\) and \(|\xi|=t^{-\beta_{\ell}}\), where \[\alpha_{\ell}:=\frac{1}{2}\;\;and\;\; \beta_{\ell}:=\frac{3-2\ell}{4(2-\ell)}\;\;satisfying\;\;\alpha_{\ell}>\beta_{\ell}>0.\] The first region captures the quadratic oscillatory scaling, whereas the third region is dominated by the singular weight, and the intermediate region balances these two mechanisms. Consequently, we implement the last philosophy to arrive at \[\begin{align} &\left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^{\ell}\, \frac{ \sin(c_{\mathrm D}|\xi|^2t) }{ c_{\mathrm D}|\xi|^2 } \right\|_{L^2}^2\\ &\lesssim \left( \int_{|\xi|\leqslant t^{-\alpha_{\ell}}} + \int_{t^{-\alpha_{\ell}}\leqslant|\xi|\leqslant t^{-\beta_{\ell}}} + \int_{t^{-\beta_{\ell}}\leqslant|\xi|\leqslant\varepsilon_0} \right) \frac{ |\sin(c_{\mathrm D}|\xi|^2t)|^2 }{ |\xi|^{4-2\ell} } \,\mathrm d\xi \\ &\lesssim t^2 \int_{|\xi|\leqslant t^{-\alpha_{\ell}}} \left| \frac{ \sin(c_{\mathrm D}|\xi|^2t) }{ c_{\mathrm D}|\xi|^2t } \right|^2 |\xi|^{2\ell} \,\mathrm d\xi + \int_{t^{-\alpha_{\ell}}\leqslant|\xi|\leqslant t^{-\beta_{\ell}}} |\xi|^{-4+2\ell} \,\mathrm d\xi + t^{2\beta_{\ell}(2-\ell)} \int_{t^{-\beta_{\ell}}\leqslant|\xi|\leqslant\varepsilon_0} \mathrm d\xi \\ &\lesssim t^2 \int_0^{t^{-\alpha_{\ell}}} |\xi|^{2\ell} \,\mathrm d|\xi| + \int_{t^{-\alpha_{\ell}}}^{t^{-\beta_{\ell}}} |\xi|^{-4+2\ell} \,\mathrm d|\xi| + t^{2\beta_{\ell}(2-\ell)} \int_{t^{-\beta_{\ell}}}^{\varepsilon_0} \mathrm d|\xi| \\ &\lesssim t^{2-\alpha_{\ell}(1+2\ell)} + t^{\alpha_{\ell}(3-2\ell)} + t^{2\beta_{\ell}(2-\ell)}. \end{align}\] The choices of \(\alpha_{\ell}\) and \(\beta_{\ell}\) are made so that all three contributions have the same asymptotic order \(t^{\frac{3}{2}-\ell}\), in other words, \[t^{2-\alpha_{\ell}(1+2\ell)}+t^{\alpha_{\ell}(3-2\ell)}+t^{2\beta_{\ell}(2-\ell)} \approx t^{\frac{3}{2}-\ell},\] which proves the desired upper bound.
Turning to the lower bound, we shrink the integration region to the dominant low-frequency zone \(\{|\xi|\leqslant\delta t^{-\frac{1}{2}}\}\), where \(\delta>0\) is sufficiently small. Indeed, since \(\frac{\sin y}{y}\to1\) as \(y\to0\), choosing \(\delta\) sufficiently small, we guarantee \[\left| \frac{ \sin(c_{\mathrm D}|\xi|^2t) }{ c_{\mathrm D}|\xi|^2t } \right| \geqslant \frac{1}{\sqrt2}.\] It follows that \[\begin{align} \left\|\chi_{\mathop{\mathrm{int}}}(\xi)|\xi|^{\ell}\,\frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2}\right\|_{L^2}^2 &\gtrsim t^2 \int_{|\xi|\leqslant\delta t^{-\frac{1}{2}}}\left|\frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2t}\right|^2|\xi|^{2\ell}\,\mathrm d\xi\\ &\gtrsim t^2\int_0^{\delta t^{-\frac{1}{2}}}|\xi|^{2\ell}\,\mathrm d|\xi|\\ &\gtrsim t^{\frac{3}{2}-\ell}. \end{align}\] This completes the proof. ◻
Following the asymptotic approach in [16], we compare the exact Fourier multipliers \[\begin{align} \label{Notation-K-ell} \mathcal{K}_0^{\ell}(t,x) := \chi_{\mathop{\mathrm{int}}}(D) \mathcal{F}^{-1}_{\xi\to x} \left( (i\xi)^{\ell}\, \frac{\sin(c_{\mathrm{D}}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} \right) \end{align}\tag{13}\] acting on the initial data (in the sense of convolution) with the corresponding asymptotic kernels multiplied by the moments of the initial data.
Lemma 9. Suppose that \(f\in L^1\) and \(\ell\in\{0,1\}\). Then, the following approximation: \[\begin{align} \left\| \mathcal{K}_0^{\ell}(t,\cdot)\ast_{(x)}f(\cdot) - \mathcal{K}_0^{\ell}(t,\cdot)P_f \right\|_{L^2}^2 = o(t^{\frac{3}{2}-\ell}) \end{align}\] holds as \(t\to+\infty\).
Proof. Applying the mean value theorem to the spatial variable, we have \[|\mathcal{K}_0^{\ell}(t,x-y) - \mathcal{K}_0^{\ell}(t,x)| \lesssim |y| \,|\partial_x\mathcal{K}_0^{\ell}(t,x-\eta_0y)|\] for some \(\eta_0\in(0,1)\). We separate it into two parts \[\begin{align} &\left\| \mathcal{K}_0^{\ell}(t,\cdot)\ast_{(x)}f(\cdot) - \mathcal{K}_0^{\ell}(t,\cdot)P_f \right\|_{L^2}^2\\ &\leqslant \left\| \int_{|y|\leqslant t^{\frac{1}{8}}} \big( \mathcal{K}_0^{\ell}(t,\cdot-y) - \mathcal{K}_0^{\ell}(t,\cdot) \big) f(y)\,\mathrm dy \right\|_{L^2}^2 + \left\| \int_{|y|\geqslant t^{\frac{1}{8}}} \big( |\mathcal{K}_0^{\ell}(t,\cdot-y)| + |\mathcal{K}_0^{\ell}(t,\cdot)| \big) |f(y)|\,\mathrm dy \right\|_{L^2}^2. \end{align}\] The time-dependent threshold \(|y|=t^{\frac{1}{8}}\) is chosen so that the near-field contribution remains asymptotically negligible compared with the leading order \(t^{\frac{3}{2}-\ell}\). Using also Plancherel’s identity, we arrive at \[\begin{align} &\left\| \mathcal{K}_0^{\ell}(t,\cdot)\ast_{(x)}f(\cdot) - \mathcal{K}_0^{\ell}(t,\cdot)P_f \right\|_{L^2}^2\\ &\lesssim t^{\frac{1}{4}} \left\| \chi_{\mathop{\mathrm{int}}}(\xi) (i\xi)^{\ell+1}\, \frac{\sin(c_{\mathrm{D}}|\xi|^2t)}{c_{\mathrm{D}}|\xi|^2} \right\|_{L^2}^2 \|f\|_{L^1}^2 + \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi|^{\ell}\, \frac{\sin(c_{\mathrm{D}}|\xi|^2t)}{c_{\mathrm{D}}|\xi|^2} \right\|_{L^2}^2 \|f\|_{L^1(|x|\geqslant t^{\frac{1}{8}})}^2 \\ &\lesssim t^{\frac{3-2\ell}{4}} \|f\|_{L^1}^2 + o(t^{\frac{3}{2}-\ell}), \end{align}\] as \(t\to+\infty\), thanks to Lemma 8. Since our assumption \(f\in L^1\), we have \[\lim_{t\to+\infty} \int_{|x|\geqslant t^{\frac{1}{8}}} |f(x)|\,\mathrm dx = 0.\] Moreover, \(\frac{3-2\ell}{4} < \frac{3}{2}-\ell\) for all \(\ell\in\{0,1\}\), so that the first term is lower-order. This completes the proof directly. ◻
Remark 14. Actually, Lemma 9 shows that the leading large time behavior is completely determined by the zeroth moment \(P_f\). This is the fundamental mechanism behind the asymptotic profiles derived later for \(\varphi\) and \(\psi\).
The next result shows that when the zeroth moment vanishes, the leading asymptotic behavior is determined by the first moment.
Lemma 10. Suppose that \(f\in L^{1,1}\) such that \(P_f=0\). Then, the following approximation: \[\begin{align} \left\| \mathcal{K}_0^{0}(t,\cdot)\ast_{(x)}f(\cdot) - \partial_x\mathcal{K}_0^0(t,\cdot)M_f \right\|_{L^2}^2 = o(t^{\frac{1}{2}}) \end{align}\] holds as \(t\to+\infty\).
Proof. Recalling the notation in 13 and applying the mean value theorem, we have \[|\mathcal{K}_0^{0}(t,x-y) - \mathcal{K}_0^{0}(t,x) -(-y) \partial_x\mathcal{K}_0^0(t,x)| \lesssim |y|^2 |\partial_x^2\mathcal{K}_0^{0}(t,x-\eta_1y)|\] for some \(\eta_1\in(0,1)\). Therefore, using \(\mathcal{K}_0^{0}(t,x)P_f=0\) from \(P_f=0\), we estimate \[\begin{align} & \left\| \mathcal{K}_0^{0}(t,\cdot)\ast_{(x)}f(\cdot) - \partial_x\mathcal{K}_0^0(t,\cdot)M_f \right\|_{L^2}^2 \\ &\leqslant \left\| \int_{|y|\leqslant t^{\frac{1}{16}}} \big( \mathcal{K}_0^{0}(t,\cdot-y) - \mathcal{K}_0^{0}(t,\cdot) -(-y)\partial_x\mathcal{K}_0^0(t,\cdot) \big) f(y)\,\mathrm dy \right\|_{L^2}^2 \\ &\quad + \left\| \int_{|y|\geqslant t^{\frac{1}{16}}} \big( |\mathcal{K}_0^{0}(t,\cdot-y) - \mathcal{K}_0^{0}(t,\cdot)| + |y\partial_x\mathcal{K}_0^{0}(t,\cdot)| \big) |f(y)|\,\mathrm dy \right\|_{L^2}^2. \end{align}\] The time-dependent threshold \(|y|=t^{\frac{1}{16}}\) is chosen so that the near-field contribution remains asymptotically negligible compared with the leading order \(t^{\frac{1}{2}}\). Using also Plancherel’s identity, we may deduce \[\begin{align} &\left\| \mathcal{K}_0^{0}(t,\cdot)\ast_{(x)}f(\cdot) - \partial_x\mathcal{K}_0^0(t,\cdot)M_f \right\|_{L^2}^2\\ &\lesssim t^{\frac{1}{4}} \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \sin(c_{\mathrm{D}}|\xi|^2t) \right\|_{L^2}^2 \|f\|_{L^1}^2 + \left\| \chi_{\mathop{\mathrm{int}}}(\xi) i\xi \frac{\sin(c_{\mathrm{D}}|\xi|^2t)}{c_{\mathrm{D}}|\xi|^2} \right\|_{L^2}^2 \|xf\|_{L^1(|x|\geqslant t^{\frac{1}{16}})}^2 \\ &\lesssim t^{\frac{1}{4}} \|f\|_{L^1}^2 + o(t^{\frac{1}{2}}), \end{align}\] as \(t\to+\infty\), thanks to Lemma 8. Here the singular factor \(|\xi|^{-2}\) disappears after differentiation, which reduces the growth order. Since our assumption \(f\in L^{1,1}\), we have \[\lim_{t\to+\infty} \int_{|x|\geqslant t^{\frac{1}{16}}} |xf(x)|\,\mathrm dx = 0.\] This completes the proof. ◻
The combination of Lemma 9 with \(\ell=1\) and Lemma 10 gives the following final auxiliary lemma by applying the triangle inequality straightforwardly.
Lemma 11. Suppose that \(f_0\in L^{1,1}\) and \(f_1\in L^1\) such that \(P_{f_0}=0\). Then, the following approximation: \[\begin{align} \left\| \mathcal{K}_0^0(t,\cdot)\ast_{(x)}f_0(\cdot) + \mathcal{K}_0^1(t,\cdot)\ast_{(x)}f_1(\cdot) - \mathcal{K}_0^1(t,\cdot) \big(M_{f_0}+P_{f_1}\big) \right\|_{L^2}^2 = o(t^{\frac{1}{2}}) \end{align}\] holds as \(t\to+\infty\).
The leading large time behavior of solutions is generated by the low-frequency sine multipliers associated with the initial velocity \(\varphi_1\). In particular, the low-frequency cosine multipliers produce only lower-order contributions.
We first extract the dominant low-frequency contributions from the representations of solutions. Obviously, they fulfill the following pointwise estimates in the Fourier space: \[\begin{align} \chi_{\mathop{\mathrm{int}}}(\xi) \left( |\widehat{K}_{\varphi,0}(t,|\xi|)| + |\widehat{K}_{\psi,0}(t,|\xi|)| \right) &\lesssim \chi_{\mathop{\mathrm{int}}}(\xi), \\ \chi_{\mathop{\mathrm{int}}}(\xi) |\widehat{K}_{\psi,1}(t,|\xi|)| &\lesssim \chi_{\mathop{\mathrm{int}}}(\xi) |\xi| \left| \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right|, \\ \chi_{\mathop{\mathrm{int}}}(\xi) |\widehat{K}_{\varphi,1}(t,|\xi|)| &\lesssim \chi_{\mathop{\mathrm{int}}}(\xi) \left| \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right|. \end{align}\] For the low-frequency analysis, we occasionally employ the \(H^{-1}\) framework instead of the \(L^2\) framework in order to match the high-frequency regularity structure (to be explained later). Indeed, \[\begin{align} \|\chi_{\mathop{\mathrm{int}}}(\xi)\widehat{f}(\xi)\|_{L^2}^2 &\lesssim \|\chi_{\mathop{\mathrm{int}}}(\xi)\langle\xi\rangle\|_{L^{\infty}}^2 \|\langle\xi\rangle^{-1}\widehat{f}(\xi)\|_{L^2}^2 \\ &\lesssim \|f\|_{H^{-1}}^2. \end{align}\] Applying Lemma 2 and the Hausdorff–Young inequality, we obtain \[\begin{align} \|\varphi(t,\cdot)\|_{L^2_{\chi}}^2 &\lesssim \|\varphi_0\|_{L^2}^2 + \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \left| \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right| \big( |P_{\varphi_1}| + |\xi|\,\|\varphi_1\|_{L^{1,1}} \big) \right\|_{L^2}^2 \\ &\quad + \|\psi_0\|_{H^{-1}}^2 + \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi| \left| \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right| \right\|_{L^2}^2 \|\psi_1\|_{L^1}^2. \end{align}\] By Lemma 7 and Lemma 8, we further deduce that \[\begin{align} \|\varphi(t,\cdot)\|_{L^2_{\chi}}^2 &\lesssim \|\varphi_0\|_{L^2}^2 + \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \left| \frac{\sin(c_{\mathrm{D}} |\xi|^2t)}{c_{\mathrm{D}} |\xi|^2} \right| \right\|_{L^2}^2 |P_{\varphi_1}|^2 + o(t^{\frac{3}{2}}) |P_{\varphi_1}|^2 + \|\psi_0\|_{H^{-1}}^2 \\ &\quad + \left\| \chi_{\mathop{\mathrm{int}}}(\xi) |\xi| \left| \frac{\sin(c_{\mathrm{D}} |\xi|^2t)}{c_{\mathrm{D}} |\xi|^2} \right| \right\|_{L^2}^2 \left( \|\varphi_1\|_{L^{1,1}}^2 + \|\psi_1\|_{L^1}^2 \right) + o(t^{\frac{1}{2}}) \left( \|\varphi_1\|_{L^{1,1}}^2 + \|\psi_1\|_{L^1}^2 \right) \\ &\lesssim \|\varphi_0\|_{L^2}^2 + t^{\frac{1}{2}} \left( t\,|P_{\varphi_1}|^2 + \|\varphi_1\|_{L^{1,1}}^2 \right) + \|\psi_0\|_{H^{-1}}^2 + t^{\frac{1}{2}} \|\psi_1\|_{L^1}^2 \end{align}\] as \(t\to+\infty\). Furthermore, a direct subtraction implies \[\begin{align} & \left\| \varphi(t,\cdot) - \mathcal{F}^{-1}_{\xi\to x} \left( \frac{\lambda_{\mathrm I,1}^2}{\lambda_{\mathrm I,1}^2-\lambda_{\mathrm I,2}^2} \, \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right) \ast_{(x)} \varphi_1(\cdot) \right\|_{L^2_{\chi}}^2 \lesssim \|(\varphi_0,\varphi_1)\|_{L^2\times L^2}^2 + t^{\frac{1}{2}} \|(\psi_0,\psi_1)\|_{H^{-1}\times L^2}^2. \end{align}\] Here we used that \[\frac{\lambda_{\mathrm I,1}^2}{\lambda_{\mathrm I,1}^2-\lambda_{\mathrm I,2}^2} = 1+O(|\xi|^2)\] for low-frequencies. Motivated by Lemma 7, and applying Lemma 9 with \(\ell=0\), we obtain \[\begin{align} & \left\| \mathcal{F}^{-1}_{\xi\to x} \left( \frac{\lambda_{\mathrm I,1}^2}{\lambda_{\mathrm I,1}^2-\lambda_{\mathrm I,2}^2} \, \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right) \ast_{(x)} \varphi_1(\cdot) - \mathcal{F}^{-1}_{\xi\to x} \left( \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} \right) P_{\varphi_1} \right\|_{L^2_{\chi}}^2 = o(t^{\frac{3}{2}}), \end{align}\] which yields the asymptotic approximation \[\begin{align} \label{Est-02} \left\| \varphi(t,\cdot) - \mathcal{F}^{-1}_{\xi\to x} \left( \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} \right) P_{\varphi_1} \right\|_{L^2_{\chi}}^2 = o(t^{\frac{3}{2}}) \end{align}\tag{14}\] as \(t\to+\infty\).
Analogously, thanks to \[\begin{align} \chi_{\mathop{\mathrm{int}}}(\xi) \left( |\widehat{G}_{\varphi,0}(t,|\xi|)| + |\widehat{G}_{\psi,0}(t,|\xi|)| + |\widehat{G}_{\psi,1}(t,|\xi|)| \right) &\lesssim \chi_{\mathop{\mathrm{int}}}(\xi), \\ \chi_{\mathop{\mathrm{int}}}(\xi) |\widehat{G}_{\varphi,1}(t,|\xi|)| &\lesssim \chi_{\mathop{\mathrm{int}}}(\xi) |\xi| \left| \frac{\sin(\lambda_{\mathrm I,2}t)}{\lambda_{\mathrm I,2}} \right|, \end{align}\] we conclude \[\begin{align} \|\psi(t,\cdot)\|_{L^2_{\chi}}^2 \lesssim \|\varphi_0\|_{H^{-1}}^2 + \left( t^{\frac{1}{2}} |P_{\varphi_1}|^2 + \|\varphi_1\|_{L^{1,1}}^2 \right) + \|\psi_0\|_{L^2}^2 + \|\psi_1\|_{H^{-1}}^2 \end{align}\] as \(t\to+\infty\). Moreover, applying Lemma 9 with \(\ell=1\), we derive \[\begin{align} \label{Est-03} \left\| \psi(t,\cdot) - \mathcal{F}^{-1}_{\xi\to x} \left( i\xi \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} \right) P_{\varphi_1} \right\|_{L^2_{\chi}}^2 = o(t^{\frac{1}{2}}) \end{align}\tag{15}\] as \(t\to+\infty\).
We next employ the asymptotic expansions of characteristic roots for \(|\xi|\gg1\). Here, the high-frequency behavior depends on whether the wave speeds are equal. Moreover, thanks to the separation property \[\lambda_{\mathrm{I},1}\neq\lambda_{\mathrm{I},2} \;\;for \;\;\xi\in\mathcal{Z}_{\mathop{\mathrm{bdd}}}(\varepsilon_0,N_0),\] the coefficients appearing in the representations of \(\widehat{\varphi}\) and \(\widehat{\psi}\) remain uniformly bounded away from singularities.
If \(c_{\mathrm S}\neq c_{\mathrm R}\), then \[\begin{align} \big(1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) |\widehat{\varphi}| &\lesssim \big(1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) \left( |\widehat{\varphi}_0| + \langle\xi\rangle^{-1} |\widehat{\varphi}_1| + \langle\xi\rangle^{-1} |\widehat{\psi}_0| + \langle\xi\rangle^{-2} |\widehat{\psi}_1| \right), \\ \big(1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) |\widehat{\psi}| &\lesssim \big(1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) \left( \langle\xi\rangle^{-1} |\widehat{\varphi}_0| + \langle\xi\rangle^{-2} |\widehat{\varphi}_1| + |\widehat{\psi}_0| + \langle\xi\rangle^{-1} |\widehat{\psi}_1| \right). \end{align}\]
Consequently, Plancherel’s identity gives \[\begin{align} \|\varphi(t,\cdot)\|_{L^2_{1-\chi}}^2 &\lesssim \|(\varphi_0,\varphi_1)\|_{L^2\times H^{-1}}^2 + \|(\psi_0,\psi_1)\|_{H^{-1}\times H^{-2}}^2, \\ \|\psi(t,\cdot)\|_{L^2_{1-\chi}}^2 &\lesssim \|(\varphi_0,\varphi_1)\|_{H^{-1}\times H^{-2}}^2 + \|(\psi_0,\psi_1)\|_{L^2\times H^{-1}}^2. \end{align}\]
If \(c_{\mathrm S}=c_{\mathrm R}\), then \[\begin{align} \big(1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) |\widehat{\varphi}| + \big(1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) |\widehat{\psi}|\lesssim \big(1-\chi_{\mathop{\mathrm{int}}}(\xi)\big) \left( |\widehat{\varphi}_0| + \langle\xi\rangle^{-1} |\widehat{\varphi}_1| + |\widehat{\psi}_0| + \langle\xi\rangle^{-1} |\widehat{\psi}_1| \right). \end{align}\]
Consequently, Plancherel’s identity gives \[\begin{align} \|\varphi(t,\cdot)\|_{L^2_{1-\chi}}^2 + \|\psi(t,\cdot)\|_{L^2_{1-\chi}}^2 \lesssim \|(\varphi_0,\varphi_1)\|_{L^2\times H^{-1}}^2 + \|(\psi_0,\psi_1)\|_{L^2\times H^{-1}}^2. \end{align}\]
In both cases, no additional time growth appears in the high-frequency zone. Therefore, the intrinsic large time growth mechanism is generated entirely by the low-frequency singular structure. Moreover, the bounded property of sine functions suggests \[\begin{align} \left\| \mathcal{F}^{-1}_{\xi\to x} \left( (i\xi)^{\ell}\, \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} \right) P_{\varphi_1} \right\|_{L^2_{1-\chi}}^2 &\lesssim \int_{|\xi|\geqslant\varepsilon_0} |\xi|^{-4+2\ell} \,\mathrm d\xi \,|P_{\varphi_1}|^2\\ &\lesssim |P_{\varphi_1}|^2 \end{align}\] for all \(\ell\in\{0,1\}\). Hence, the asymptotic profiles obtained in 14 and 15 are generated exclusively by the low-frequency region.
Applying Plancherel’s identity and combining all previous estimates, we complete the proof of the desired upper bound estimates.
For \(u\in\{\varphi,\psi\}\), Plancherel’s identity implies \[\|u(t,\cdot)\|_{L^2}^2=\|\widehat{u}(t,\xi)\|_{L^2}^2\geqslant\|\chi_{\mathop{\mathrm{int}}}(\xi)\widehat{u}(t,\xi)\|_{L^2}^2,\] which allows us to restrict the analysis to the low-frequency zone.
Applying the triangle inequality, together with the estimate 14 in the Fourier space and Lemma 8, we obtain \[\begin{align} \|\varphi(t,\cdot)\|_{L^2}^2 &\gtrsim \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} P_{\varphi_1} \right\|_{L^2}^2 - \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \left( \widehat{\varphi}(t,\xi) - \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} P_{\varphi_1} \right) \right\|_{L^2}^2 \\ &\gtrsim t^{\frac{3}{2}} |P_{\varphi_1}|^2 - o(t^{\frac{3}{2}}) \end{align}\] as \(t\to+\infty\).
Similarly, combining the estimate 15 in the Fourier space with Lemma 8, we derive \[\begin{align} \|\psi(t,\cdot)\|_{L^2}^2 &\gtrsim \left\| \chi_{\mathop{\mathrm{int}}}(\xi) i\xi \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} P_{\varphi_1} \right\|_{L^2}^2 - \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \left( \widehat{\psi}(t,\xi) - i\xi \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} P_{\varphi_1} \right) \right\|_{L^2}^2 \\ &\gtrsim t^{\frac{1}{2}} |P_{\varphi_1}|^2 - o(t^{\frac{1}{2}}) \end{align}\] as \(t\to+\infty\).
Therefore, for sufficiently large time, the leading profile terms dominate the remainder contributions, which completes the proof of the lower bound estimates under the condition \(P_{\varphi_1}\neq0\).
Combining 4 with the low-frequency asymptotic expansions of the characteristic roots, we obtain immediately \[\begin{align} & \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \left( \widehat{\varphi}(t,\xi) - \frac{ \lambda_{\mathrm{I},1}^2\,\widehat{\varphi}_1(\xi) - \frac{K}{\rho}i\xi\widehat{\psi}_1(\xi) }{ \lambda_{\mathrm{I},1}^2 - \lambda_{\mathrm{I},2}^2 } \, \frac{ \sin(\lambda_{\mathrm{I},2}t) }{ \lambda_{\mathrm{I},2} } \right) \right\|_{L^2}^2 \lesssim \|(\varphi_0,\varphi_1)\|_{L^2\times H^{-1}}^2 + \|(\psi_0,\psi_1)\|_{H^{-1}\times H^{-2}}^2. \end{align}\] Consequently, the triangle inequality associated with the last estimate shows \[\begin{align} \|\varphi(t,\cdot)\|_{L^2_{\chi}}^2 \gtrsim J_1(t) - \left( \|(\varphi_0,\varphi_1)\|_{L^2\times H^{-1}}^2 + \|(\psi_0,\psi_1)\|_{H^{-1}\times H^{-2}}^2 \right), \end{align}\] in which we took the notion \[\begin{align} J_1(t):= \left\|\chi_{\mathop{\mathrm{int}}}(\xi)\frac{\lambda_{\mathrm{I},1}^2\,\widehat{\varphi}_1(\xi)-\frac{K}{\rho}i\xi\widehat{\psi}_1(\xi)}{\lambda_{\mathrm{I},1}^2-\lambda_{\mathrm{I},2}^2}\,\frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}}\right\|_{L^2}^2. \end{align}\]
It remains to estimate lower bounds of \(J_1(t)\) for large time. Using the low-frequency asymptotic expansions of the characteristic roots, we may rewrite \[\begin{align} J_1(t) = \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \frac{ \big(c_{\mathrm O}^2+O(|\xi|^2)\big) \widehat{\varphi}_1(\xi) - \frac{K}{\rho}i\xi\widehat{\psi}_1(\xi) }{ c_{\mathrm O}^2+O(|\xi|^2) } \, \frac{ \sin(\lambda_{\mathrm{I},2}t) }{ \lambda_{\mathrm{I},2} } \right\|_{L^2}^2. \end{align}\] Applying the triangle inequality again, we further extract the following dominant contribution: \[\begin{align} J_1(t) &\gtrsim \left\| \chi_{\mathop{\mathrm{int}}}(\xi)\left(\widehat{\varphi}_1(\xi)-\frac{K}{\rho\,c_{\mathrm O}^2}i\xi\widehat{\psi}_1(\xi)\right) \frac{\sin(c_{\mathrm D}|\xi|^2t)}{c_{\mathrm D}|\xi|^2} \right\|_{L^2}^2\\ &\quad - \left\|\chi_{\mathop{\mathrm{int}}}(\xi)|\xi|^2 \left(|\widehat{\varphi}_1(\xi)|+|\xi|\,|\widehat{\psi}_1(\xi)|\right) \frac{\sin(\lambda_{\mathrm{I},2}t)}{\lambda_{\mathrm{I},2}} \right\|_{L^2}^2 \\ &\quad - \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \left( |\widehat{\varphi}_1(\xi)| + |\xi|\, |\widehat{\psi}_1(\xi)| \right) \left( \frac{ \sin(\lambda_{\mathrm{I},2}t) }{ \lambda_{\mathrm{I},2} } - \frac{ \sin(c_{\mathrm D}|\xi|^2t) }{ c_{\mathrm D}|\xi|^2 } \right) \right\|_{L^2}^2, \end{align}\] where we used \[\begin{align} &\chi_{\mathop{\mathrm{int}}}(\xi)\left|\frac{\big(c_{\mathrm O}^2+O(|\xi|^2)\big)\widehat{\varphi}_1(\xi)-\frac{K}{\rho}i\xi\widehat{\psi}_1(\xi)}{c_{\mathrm O}^2+O(|\xi|^2)}-\left(\widehat{\varphi}_1(\xi)-\frac{K}{\rho\, c_{\mathrm O}^2}i\xi\widehat{\psi}_1(\xi)\right)\right|\\ &\lesssim\chi_{\mathop{\mathrm{int}}}(\xi)|\xi|^2\left(|\widehat{\varphi}_1(\xi)|+|\xi|\,|\widehat{\psi}_1(\xi)|\right). \end{align}\] Since \(\frac{K}{\rho\,c_{\mathrm O}^2}=\frac{I_{\rho}}{\rho}\), Lemma 2 with \(P_{\varphi_1}=0\) and Lemma 7 with \(\ell=1\) yield \[\begin{align} J_1(t) &\gtrsim \left\| \chi_{\mathop{\mathrm{int}}}(\xi) \left( \widehat{\varphi}_1(\xi) - \frac{I_{\rho}}{\rho} i\xi\widehat{\psi}_1(\xi) \right) \frac{ \sin(c_{\mathrm D}|\xi|^2t) }{ c_{\mathrm D}|\xi|^2 } \right\|_{L^2}^2 - \left( \|\varphi_1\|_{L^{1,1}}^2 + \|\psi_1\|_{L^1}^2 \right) - o(t^{\frac{1}{2}}). \end{align}\] Applying Lemma 11 with \(f_0=\varphi_1\) and \(f_1=\frac{I_{\rho}}{\rho}\psi_1\), we further obtain \[\begin{align} J_1(t) &\gtrsim \left\| \mathcal{F}^{-1}_{\xi\to x} \left( i\xi \frac{ \sin(c_{\mathrm D}|\xi|^2t) }{ c_{\mathrm D}|\xi|^2 } \right) \right\|_{L^2_{\chi}}^2 \left| M_{\varphi_1} - \frac{I_{\rho}}{\rho} P_{\psi_1} \right|^2 - o(t^{\frac{1}{2}}) \\ &\gtrsim t^{\frac{1}{2}} \left| M_{\varphi_1} - \frac{I_{\rho}}{\rho} P_{\psi_1} \right|^2 \end{align}\] as \(t\to+\infty\). Therefore, we complete the proof of the lower bound estimate under the condition \(M_{\varphi_1} - \frac{I_{\rho}}{\rho} P_{\psi_1} \neq0.\) Moreover, the above derivation also yields the corresponding large time asymptotic profile. Since its proof follows exactly the same argument as in 14 , we omit the details.
As a consequence of the hyperbolic structure of the classical Timoshenko system 1 , one may prove the finite propagation property by the standard localized energy method, analogously to the classical wave equation.
Proposition 1. Suppose that \[\begin{align} \mathrm{supp}\, (\varphi_0,\varphi_1,\psi_0,\psi_1) \subset [-L,L] \;\;for some\;\; L>0 \end{align}\] for the classical Timoshenko system 1 . Then, the transversal displacement \(\varphi\) satisfies \[\begin{align} \mathrm{supp}\, \varphi(t,\cdot) \subset [-L-c_*t,L+c_*t] \end{align}\] for every \(t\geqslant0\), where \(c_* := \max\{ c_{\mathrm S}, c_{\mathrm R} \}\).
According to Proposition 1, the support of \(\varphi(t,\cdot)\) is contained in an interval of length \(2(L+c_*t)\). Therefore, for large time \(t\gg1\), Hölder’s inequality with \(\frac{1}{p}+\frac{1}{q}=1\) and \(q\geqslant 1\) implies \[\begin{align} \|\varphi(t,\cdot)\|_{L^2}^2 = \int_{-L-c_*t}^{L+c_*t} |\varphi(t,x)|^2 \,\mathrm dx &\lesssim (L+c_*t)^{\frac{1}{p}} \|\varphi(t,\cdot)\|_{L^{2q}}^{2} \\ &\lesssim t^{\frac{1}{p}} \|\varphi(t,\cdot)\|_{L^{2q}}^2. \end{align}\] Consequently, the lower bound estimate in Theorem 3 indicates \[\|\varphi(t,\cdot)\|_{L^{2q}} \gtrsim t^{\frac{1}{2q}-\frac{1}{2}} \|\varphi(t,\cdot)\|_{L^2} \gtrsim t^{\frac{1}{4}+\frac{1}{2q}} |P_{\varphi_1}|,\] which completes the proof.
In Section [sec:Section-Introduction], we pointed out that the singular low-frequency structure of the classical Timoshenko system generates an intrinsic plate-type asymptotic profile and is responsible for the polynomial growth structure of solutions. Moreover, as discussed in Remark 8, the frictional damping mechanism in the dissipative Timoshenko system 2 modifies the corresponding profile through a diffusive effect, while preserving the underlying plate-type structure. A similar phenomenon for the free wave equation and the strongly damped wave equation has recently been observed in [20].
Motivated by these observations, we next investigate the relation between the classical Timoshenko system 1 and the dissipative Timoshenko system 2 in the small damping regime \[\begin{align} \label{Damping-Regime} 0<\gamma<2\sqrt{\rho EI}, \end{align}\tag{16}\] where the low-frequency oscillatory structure is preserved. Throughout this section, \((\varphi^{\gamma=0},\psi^{\gamma=0})\) and \((\varphi^{\gamma>0},\psi^{\gamma>0})\) denote the solutions to 1 and 2 , respectively, associated with the same initial data \((\varphi_0,\varphi_1)\) as well as \((\psi_0,\psi_1)\), for which no superscripts are used.
More precisely, we establish a large time vanishing dissipation limit for time-normalized solutions. After normalization by the natural growth rates, the solutions to the dissipative Timoshenko system 2 converge to those of the classical Timoshenko system 1 as \(\gamma\to0\) in the large time regime.
By applying the same low-frequency analysis as in [12] to the dissipative Timoshenko system 2 , one obtains the following large time asymptotic behavior in the small damping regime 16 , which coincides with [12]. We introduce the notations \[\begin{align} c_{\gamma} := \frac{\sqrt{4\rho EI-\gamma^2}}{2\rho} \;\;and\;\; \delta_{\gamma} := \frac{\gamma}{2\rho}. \end{align}\]
Proposition 2. Suppose that \((\varphi_0,\varphi_1)\in Z_1^{0}\) and \((\psi_0,\psi_1)\in Z_{2}^{0}\) such that \(P_{\varphi_1}\neq0\) for the dissipative Timoshenko system 2 with \(0<\gamma<2\sqrt{\rho EI}\). Then, the transversal displacement \(\varphi^{\gamma>0}\) satisfies the following optimal growth estimate: \[\begin{align} t^{\frac{3}{4}}|P_{\varphi_1}| \lesssim \|\varphi^{\gamma>0}(t,\cdot)\|_{L^2} \lesssim t^{\frac{3}{4}} \|(\varphi_0,\varphi_1)\|_{Z_1^0} + t^{\frac{1}{4}} \|(\psi_0,\psi_1)\|_{Z_2^0} \end{align}\] for sufficiently large time, and the following asymptotic relation: \[\begin{align} \lim\limits_{t\to+\infty} t^{-\frac{3}{4}} \left\| \varphi^{\gamma>0}(t,\cdot) - \sqrt t\, \widetilde{\mathcal{G}}_{0} \left( \tfrac{\cdot}{\sqrt t};\gamma \right) P_{\varphi_1} \right\|_{L^2} = 0, \end{align}\] where the diffusion plate-type asymptotic profile is given by \[\begin{align} \widetilde{\mathcal{G}}_{0}(y;\gamma) := \mathcal{F}^{-1}_{\eta\to y} \left( \frac{ \sin(c_{\gamma}|\eta|^2) }{ c_{\gamma}|\eta|^2 } \, \mathrm e^{-\delta_{\gamma}|\eta|^2} \right). \end{align}\]
Proposition 3. Suppose that \((\varphi_0,\varphi_1)\in Z_2^{0}\) and \((\psi_0,\psi_1)\in Z_{3}\) such that \(P_{\varphi_1}\neq0\) for the dissipative Timoshenko system 2 with \(0<\gamma<2\sqrt{\rho EI}\). Then, the rotation angle \(\psi^{\gamma>0}\) satisfies the following optimal growth estimate: \[\begin{align} t^{\frac{1}{4}}|P_{\varphi_1}| \lesssim \|\psi^{\gamma>0}(t,\cdot)\|_{L^2} \lesssim t^{\frac{1}{4}} \|(\varphi_0,\varphi_1)\|_{Z_2^0} + \|(\psi_0,\psi_1)\|_{Z_3} \end{align}\] for sufficiently large time, and the following asymptotic relation: \[\begin{align} \lim\limits_{t\to+\infty} t^{-\frac{1}{4}} \left\| \psi^{\gamma>0}(t,\cdot) - \widetilde{\mathcal{G}}_{1} \left( \tfrac{\cdot}{\sqrt t};\gamma \right) P_{\varphi_1} \right\|_{L^2} = 0, \end{align}\] where the derivative of diffusion plate-type asymptotic profile is given by \[\begin{align} \widetilde{\mathcal{G}}_{1}(y;\gamma) := \partial_y \widetilde{\mathcal{G}}_{0}(y;\gamma) = \mathcal{F}^{-1}_{\eta\to y} \left( i\eta \frac{ \sin(c_{\gamma}|\eta|^2) }{ c_{\gamma}|\eta|^2 } \, \mathrm e^{-\delta_{\gamma}|\eta|^2} \right). \end{align}\]
Remark 15. The asymptotic profiles \(\widetilde{\mathcal{G}}_0(y;\gamma)\) and \(\widetilde{\mathcal{G}}_1(y;\gamma)\) consist of two different components:
the oscillatory plate-type phase \(\frac{ \sin(c_\gamma |\eta|^2)}{c_\gamma |\eta|^2}\),
the diffusive factor \(\mathrm e^{-\delta_\gamma |\eta|^2}\).
The oscillatory structure is inherited from the conservative Timoshenko system 1 , whereas the Gaussian factor is generated by the frictional damping mechanism. In particular, as \(\gamma\to0\), one has \[c_\gamma\to c_{\mathrm D}\;\;and\;\;\delta_\gamma\to0,\] so that the dissipative profiles converge formally to the conservative profiles \(\mathcal{G}_0(y)\) and \(\mathcal{G}_1(y)\) derived in Theorem 2 and Theorem 3.
Remark 16. The restriction \(0<\gamma<2\sqrt{\rho EI}\) corresponds to the regime where the low-frequency characteristic roots of the dissipative Timoshenko system 2 remain complex-valued. Consequently, the oscillatory plate-type structure persists under the frictional damping mechanism, which makes it possible to compare the dissipative dynamics with those of the classical Timoshenko system 1 in the vanishing dissipation limit.
The next result shows that the frictional damping modifies the leading asymptotic behavior only through a diffusive factor and a shifted oscillation frequency, while preserving the singular plate-type structure responsible for the large time growth of solutions. The following statement is formulated under a common set of assumptions guaranteeing that all asymptotic profiles appearing in Theorem 2, Theorem 3, Proposition 2, and Proposition 3 are well-defined simultaneously.
Theorem 4. Suppose that \((\varphi_0,\varphi_1)\in Z_1^{0}\) and \((\psi_0,\psi_1)\in Z_{1}^0\) such that \(P_{\varphi_1}\neq0\) for the Timoshenko systems 1 and 2 . Then, the transversal displacements and the rotation angles, respectively, satisfy the following asymptotic relations: \[\begin{align} \lim\limits_{\gamma\to 0}\lim\limits_{t\to+\infty}t^{-\frac{3}{4}}\left\|\varphi^{\gamma>0}(t,\cdot)-\varphi^{\gamma=0}(t,\cdot)\right\|_{L^2}=0,\\[0.5em] \lim\limits_{\gamma\to 0}\lim\limits_{t\to+\infty}t^{-\frac{1}{4}}\left\|\psi^{\gamma>0}(t,\cdot)-\psi^{\gamma=0}(t,\cdot)\right\|_{L^2} =0. \end{align}\]
Proof. Let us begin with the first asymptotic relation for \(\varphi^{\gamma>0}\) and \(\varphi^{\gamma=0}\). By using the triangle inequality associated with Proposition 2 and Theorem 2, one notices \[\begin{align} t^{-\frac{3}{4}}\left\|\varphi^{\gamma>0}(t,\cdot)-\varphi^{\gamma=0}(t,\cdot)\right\|_{L^2} &\leqslant t^{-\frac{3}{4}}\left\|\varphi^{\gamma>0}(t,\cdot)-\sqrt{t}\,\widetilde{\mathcal{G}}_{0}\left(\tfrac{\cdot}{\sqrt t};\gamma\right)P_{\varphi_1}\right\|_{L^2}\\ &\quad+t^{-\frac{1}{4}}\left\|\widetilde{\mathcal{G}}_{0}\left(\tfrac{\cdot}{\sqrt t};\gamma\right)-\mathcal{G}_{0}\left(\tfrac{\cdot}{\sqrt t}\right)\right\|_{L^2}|P_{\varphi_1}|\\ &\quad+t^{-\frac{3}{4}}\left\|\varphi^{\gamma=0}(t,\cdot)-\sqrt{t}\,\mathcal{G}_{0}\left(\tfrac{\cdot}{\sqrt t}\right)P_{\varphi_1}\right\|_{L^2}\\ &\leqslant \underbrace{\left\| \frac{\sin(c_{\gamma}|\eta|^2)}{c_{\gamma}|\eta|^2} \,\mathrm{e}^{-\delta_{\gamma}|\eta|^2}-\frac{\sin(c_{\mathrm D}|\eta|^2)}{c_{\mathrm D}|\eta|^2} \right\|_{L^2}}_{=:A_1(\gamma)}|P_{\varphi_1}|+o(1) \end{align}\] for large time \(t\gg1\), where we used the scaling property of the \(L^2\) norm with \(\eta=\sqrt{t}\,\xi\).
Since \(c_{\gamma}\to c_{\mathrm D}\) and \(\delta_{\gamma}\to0\) as \(\gamma\to0\), the integrand converges pointwise to zero. Moreover, \[\begin{align} [A_1(\gamma)]^2 &=\int_{\mathbb{R}} \left|\frac{\sin(c_{\gamma}|\eta|^2) }{ c_{\gamma}|\eta|^2 } \,\mathrm{e}^{-\delta_{\gamma}|\eta|^2} - \frac{ \sin(c_{\mathrm D}|\eta|^2) }{ c_{\mathrm D}|\eta|^2 } \right|^2 \mathrm d\eta \\ &\lesssim \int_{|\eta|\leqslant1} \mathrm d\eta + \int_{|\eta|\geqslant1} |\eta|^{-4} \,\mathrm d\eta <+\infty. \end{align}\] Therefore, the dominated convergence theorem yields \(\lim\limits_{\gamma\to0}A_1(\gamma)=0\). Consequently, \[\begin{align} \lim\limits_{\gamma\to0} \lim\limits_{t\to+\infty} t^{-\frac{3}{4}} \left\| \varphi^{\gamma>0}(t,\cdot) - \varphi^{\gamma=0}(t,\cdot) \right\|_{L^2} =0. \end{align}\]
Concerning the second asymptotic relation, we follow the same argument associated with Proposition 3 and Theorem 3 to derive our aim. This completes the proof. ◻
Remark 17. Theorem 4 shows that, after normalization by the intrinsic growth rates \(t^{\frac{3}{4}}\) and \(t^{\frac{1}{4}}\), the dissipative Timoshenko system 2 converges to the classical Timoshenko system 1 in the small damping regime 16 . Therefore, the frictional damping modifies only the profile shape through the diffusive factor \(\mathrm{e}^{-\delta_{\gamma}|\eta|^2}\) and the shifted oscillation frequency \(c_{\gamma}\), while the singular plate-type growth mechanism itself remains stable as \(\gamma\to0\).
Remark 18. The large time limit and the vanishing dissipation limit are taken successively in Theorem 4. Since the dissipative correction appears through the factor \(\mathrm e^{-\delta_\gamma |\eta|^2}\), the corresponding asymptotic regimes depend on the interaction between the parameters \(\gamma\) and \(t\). The analysis of simultaneous limits, for example in the transitional regime \(\gamma \,t\approx 1\), remains an interesting open problem.
Wenhui Chen is supported in part by the National Natural Science Foundation of China (grant No. 12301270), Guangdong Basic and Applied Basic Research Foundation (grant No. 2025A1515010240). The author thanks Ryo Ikehata (Hiroshima University) for some suggestions in the preparation of this manuscript.
Wenhui Chen (wenhui.chen.math@gmail.com)↩︎