1 Introduction↩︎

In this manuscript, we consider the semilinear nonlocal heat exchanger system in the whole space \(\mathbb{R}^n\) with any dimension \(n\geqslant 1\), namely, \[\begin{align} \label{Eq-Semi-Nonlocal-Heat-System} \begin{cases} u_t+(-\Delta)^{\sigma}u+\mu u-\nu v=u^p,&x\in\mathbb{R}^n,\;t>0,\\ v_t+(-\Delta)^{\sigma}v-\mu u+\nu v=v^q,&x\in\mathbb{R}^n,\;t>0,\\ u(0,x)=\varepsilon u_0(x),\;v(0,x)=\varepsilon v_0(x),&x\in\mathbb{R}^n, \end{cases} \end{align}\tag{1}\] with the power of fractional Laplacian \(\sigma>0\), the constants \(\mu,\nu>0\), and \(p,q>1\), where the parameter \(\varepsilon>0\) denotes the size of initial data. Particularly, the fractional Laplacian \((-\Delta)^{\sigma}:\;H^{2\sigma}\to L^2\) is defined via \((-\Delta)^{\sigma}f:=\mathcal{F}^{-1}(|\xi|^{2\sigma}f)\). Our main purpose is to derive large time asymptotic profiles of global in-time solutions in the super-critical case \[\begin{align} \min\{p,q\}>p_{\mathrm{Fuj}}\left(\frac{n}{\sigma}\right):=1+\frac{2\sigma}{n}\;\;with any\;\;\sigma>0. \end{align}\] As our byproduct, the sharp lower bound estimates of lifespan (with respect to \(\varepsilon\)) will be given in the sub-critical and critical cases, i.e. \(\min\{p,q\}\leqslant p_{\mathrm{Fuj}}(\frac{n}{\sigma})\), in which the lifespan \(T_{\varepsilon}\) of solutions is understood as the quantity by \[\begin{align} T_{\varepsilon}&:=\sup\{T\in(0,+\infty):\;there exist unique local in-time solutions (u,v) to the Cauchy\\ &\qquad\qquad\qquad\qquad\qquad\;problem \eqref{Eq-Semi-Nonlocal-Heat-System} on [0,T) with a fixed parameter \varepsilon>0\}. \end{align}\]

Taking \(\sigma=1\) in the semilinear Cauchy problem 1 , it will immediately turn into the Fujita-type semilinear local heat exchanger system (cf. [1]), which is strongly motivated by the biological issue, precisely, the field-road reaction-diffusion system (cf. [2], [3]). In this recent paper [1], the author derived a global in-time existence result (associated with \(L^{\infty}\) estimates of solutions) if \(\min\{p,q\}>p_{\mathrm{Fuj}}(n)\), and a finite time blow-up result if \(\min\{p,q\}<p_{\mathrm{Fuj}}(n)\) to the semilinear Cauchy problem 1 with \(\sigma=1\). Note that \(p_{\mathrm{Fuj}}(n):=1+\frac{2}{n}\) is the well-known critical exponent (cf. [4], [5] and references therein) for the semilinear heat equation \(U_t-\Delta U=U^p\) in \(\mathbb{R}^n\). However, to the best of knowledge of authors, some detailed information of solutions, including large time behavior of global in-time solutions and sharp estimates of lifespan, are generally unknown questions, even in the special (local) case \(\sigma=1\). We will partly answer these questions for the Fujita-type semilinear nonlocal heat exchanger system 1 in this manuscript.

The corresponding linearized model to 1 with \(\varepsilon=1\), i.e. \[\begin{align} \label{Eq-Linear-Heat-Exchanger-System} \begin{cases} \bar{u}_t+(-\Delta)^{\sigma}\bar{u}+\mu \bar{u}-\nu \bar{v}=0,&x\in\mathbb{R}^n,\;t>0,\\ \bar{v}_t+(-\Delta)^{\sigma}\bar{v}-\mu \bar{u}+\nu \bar{v}=0,&x\in\mathbb{R}^n,\;t>0,\\ \bar{u}(0,x)= \bar{u}_0(x),\;\bar{v}(0,x)= \bar{v}_0(x),&x\in\mathbb{R}^n, \end{cases} \end{align}\tag{2}\] has the heat exchanger from the coupling diffusive components \((-\nu \bar{v},-\mu \bar{u})^{\mathrm{T}}\). As mentioned in [1], the nonlocal coupled system 2 may be interpreted as a population dynamics model for a single species dispersing on two parallel environments and switching from one to another. Note that the fractional Laplacian \((-\Delta)^{\sigma}\) substitutes for the classical one \(-\Delta\), resulting a different (sometimes wider and faster) spread [6]. We later will analyze sharp large time behavior of solutions by the Fourier analysis (see, for example, [7]).

Let us briefly recall the decoupled model, i.e. \(\mu=\nu=0\) in the Cauchy problem 1 , namely, \[\begin{align} \label{Eq-Single-Fractional} \begin{cases} U_t+(-\Delta)^{\sigma}U=U^p,&x\in\mathbb{R}^n,\;t>0,\\ U(0,x)=\varepsilon U_0(x),&x\in\mathbb{R}^n, \end{cases} \end{align}\tag{3}\] with \(\sigma>0\) in general. The classical works [8]–[11] and references therein derived the critical exponent \(p=p_{\mathrm{Fuj}}(\frac{n}{\sigma})\), namely, a global in-time small data solution uniquely exists if \(p>p_{\mathrm{Fuj}}(\frac{n}{\sigma})\) whereas local in-time solutions blow up in finite time with \(L^1\) data if \(1<p\leqslant p_{\mathrm{Fuj}}(\frac{n}{\sigma})\). Furthermore, the lifespan \(T_{\varepsilon}^U\) of \(U=U(t,x)\) in 3 as \(0<\varepsilon\ll 1\) is precisely estimated by \[\begin{align} T_{\varepsilon}^U \begin{cases} \approx \displaystyle{\varepsilon^{-\frac{p-1}{1-\frac{n}{2\sigma}(p-1)}}}&if\;\;p<p_{\mathrm{Fuj}}(\frac{n}{\sigma}),\\ \approx \exp\left(C\varepsilon^{-\frac{2\sigma}{n}}\right)&if\;\;p=p_{\mathrm{Fuj}}(\frac{n}{\sigma}),\\ =+\infty&if\;\;p>p_{\mathrm{Fuj}}(\frac{n}{\sigma}). \end{cases} \end{align}\] The above sharp relation \(f\approx g\) holds if and only if \(g\lesssim f\lesssim g\), where the unexpressed multiplicative constants are independent of \(\varepsilon\). Later, some papers (e.g. [12], [13]) generalized to weakly coupled systems whose coupling effect arises in the nonlinear part \((V^p,U^q)^{\mathrm{T}}\), namely, reaction-coupled systems. Nevertheless, our situation is more complicated due to the coupling diffusive structure \((-\nu v,-\mu u)^{\mathrm{T}}\) in the linear part of the diffusion-coupled system 1 . Especially, it seems not clear so far how do the coupling coefficients \(\mu,\nu\) influence on asymptotic behavior of solutions.

Due to some technical difficulties from the nonlocal operator \((-\Delta)^{\sigma}\) with any \(\sigma>0\), our proof of global in-time existence in the super-critical case is different from the one in [1]. Note that our situation differs from \(\sigma=1\) due to the lack of comparison principle in general (e.g. \(\sigma>1\)). Precisely, we are going to apply the Banach fixed point argument to the suitable time-weighted solution space \(\mathcal{C}([0,+\infty),L^1\cap L^{\infty})\) instead of nonlinear differential inequalities. This will bring some benefits, for example, the study of large time asymptotic behavior of global in-time solutions and lower bound estimates of lifespan in sub-critical and critical case.

2 Main results↩︎

Let us denote the fractional diffusion (heat) kernel \[\begin{align} G=G(t,x):=\mathcal{F}^{-1}_{\xi\to x}\left(\mathrm{e}^{-|\xi|^{2\sigma}t}\right), \end{align}\] which also can be understood by \(\mathcal{G}(t,|D|)\) associated with its Fourier transform \(\widehat{\mathcal{G}}(t,|\xi|)=\mathrm{e}^{-|\xi|^{2\sigma}t}\). The differential operator \(|D|\) has its symbol \(|\xi|\). We next introduce the solutions’ operators \[\begin{align} K_0^u(t,|D|)&:=\frac{\nu+\mu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\mathcal{G}(t,|D|),\;\;K_1^u(t,|D|):=\frac{\nu-\nu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\mathcal{G}(t,|D|),\\ K_0^v(t,|D|)&:=\frac{\mu-\mu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\mathcal{G}(t,|D|),\;\;K_1^v(t,|D|):=\frac{\mu+\nu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\mathcal{G}(t,|D|). \end{align}\]

We next state large time profiles for the global in-time solutions \((u,v)\) defined in Theorem 1. To be specific, by subtracting their corresponding profiles associated with the integrals of summable functions, i.e. \(P_f:=\int_{\mathbb{R}^n}f(x)\,\mathrm{d}x\) and \(\mathcal{P}_f:=\int_{0}^{+\infty}\int_{\mathbb{R}^n}f(t,x)\,\mathrm{d}x\,\mathrm{d}t\), some faster decay estimates arise, comparing with ?? .

As we conjectured in Remark 5 and from the blow-up phenomenon derived in [1] when \(\sigma=1\), in the sub-critical case \(\min\{p,q\}<p_{\mathrm{Fuj}}(\frac{n}{\sigma})\) and the critical case \(\min\{p,q\}=p_{\mathrm{Fuj}}(\frac{n}{\sigma})\), every non-trivial local in-time solution to the semilinear Cauchy problem 1 will blow up in finite time. For this reason, we in the next result describe more detailed information of lifespan \(T_{\varepsilon}\). Notice that \(T_{\varepsilon}=+\infty\) if \(\min\{p,q\}>p_{\mathrm{Fuj}}(\frac{n}{\sigma})\) and \(\varepsilon\in(0,\varepsilon_0]\) according to Theorem 1.

3 Linear nonlocal heat exchanger system↩︎

3.1 Preliminaries in the Fourier space↩︎

Applying the partial Fourier transform with respect to spatial variable to the Cauchy problem 2 , one may arrive at \[\begin{align} \label{Eq-Fourier-Imagine} \begin{cases} \widehat{\bar{u}}_t+|\xi|^{2\sigma}\widehat{\bar{u}}+\mu\widehat{\bar{u}}-\nu\widehat{\bar{v}}=0,&\xi\in\mathbb{R}^n,\;t>0,\\ \widehat{\bar{v}}_t+|\xi|^{2\sigma}\widehat{\bar{v}}-\mu\widehat{\bar{u}}+\nu\widehat{\bar{v}}=0,&\xi\in\mathbb{R}^n,\;t>0,\\ \widehat{\bar{u}}(0,\xi)=\widehat{\bar{u}}_0(\xi),\;\widehat{\bar{v}}(0,\xi)=\widehat{\bar{v}}_0(\xi),&\xi\in\mathbb{R}^n. \end{cases} \end{align}\tag{4}\] Let us now apply the so-called reduction method developed by Wenhui Chen in (thermo-)elasticity, firstly in the thermoelastic plate systems [15]–[17], later in the thermoelastic systems [18]–[20], in the Timoshenko systems [21], [22], and in the compressible Navier-Stokes system [23]. By the reduction methodology (e.g. acting \(\partial_t+|\xi|^{2\sigma}+\nu\) to 4 \(_1\) and combining with 4 \(_2\) directly), one notices that the unknown \(\widehat{\bar{w}}\in\{\widehat{\bar{u}},\widehat{\bar{v}}\}\) fulfills the same second-order in-time differential equation with different initial data. To be specific, \[\begin{align} \begin{cases} \widehat{\bar{w}}_{tt}+(2|\xi|^{2\sigma}+\mu+\nu)\widehat{\bar{w}}_t+|\xi|^{2\sigma}(|\xi|^{2\sigma}+\mu+\nu)\widehat{\bar{w}}=0,&\xi\in\mathbb{R}^n,\;t>0,\\ \widehat{\bar{w}}(0,\xi)=\widehat{\bar{w}}_0(\xi),\;\widehat{\bar{w}}_t(0,\xi)=\widehat{\bar{w}}_1(\xi),&\xi\in\mathbb{R}^n, \end{cases} \end{align}\] where the initial data are defined by \[\begin{align} \widehat{\bar{w}}_0(\xi):=\begin{cases} \widehat{\bar{u}}_0(\xi)&if\;\;\widehat{\bar{w}}=\widehat{\bar{u}},\\ \widehat{\bar{v}}_0(\xi)&if\;\;\widehat{\bar{w}}=\widehat{\bar{v}}, \end{cases}\;\;\;\; \widehat{\bar{w}}_1(\xi):=\begin{cases} -(|\xi|^{2\sigma}+\mu)\widehat{\bar{u}}_0(\xi)+\nu\widehat{\bar{v}}_0(\xi)&if\;\;\widehat{\bar{w}}=\widehat{\bar{u}},\\ -(|\xi|^{2\sigma}+\nu)\widehat{\bar{v}}_0(\xi)+\mu\widehat{\bar{u}}_0(\xi)&if\;\;\widehat{\bar{w}}=\widehat{\bar{v}}. \end{cases} \end{align}\] By carrying out explicit computations, the solutions are expressed via \[\begin{align} \widehat{\bar{u}}(t,\xi)&=\mathrm{e}^{-|\xi|^{2\sigma}t}\left(\frac{\nu+\mu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\widehat{\bar{u}}_0(\xi)+\frac{\nu-\nu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\widehat{\bar{v}}_0(\xi)\right),\\ \widehat{\bar{v}}(t,\xi)&=\mathrm{e}^{-|\xi|^{2\sigma}t}\left(\frac{\mu-\mu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\widehat{\bar{u}}_0(\xi)+\frac{\mu+\nu\,\mathrm{e}^{-(\mu+\nu)t}}{\mu+\nu}\,\widehat{\bar{v}}_0(\xi)\right), \end{align}\] where these kernels in the operator sense are defined at the beginning of Section 2.

To end this part, let us recall the sharp \(L^m\) estimates for the Fourier multiplier by using modified Bessel functions in [24] and [25].

3.2 Sharp asymptotic behavior of solutions for large time↩︎

For the sake of simplicity, concerning \(\gamma=\nu\) if \(\bar{w}=\bar{u}\); \(\gamma=\mu\) if \(\bar{w}=\bar{v}\), we take \[\begin{align} \bar{w}^{\mathrm{prof}}(t,x):=\frac{\gamma}{\mu+\nu}\,\mathcal{G}(t,|D|)\big(\bar{u}_0(x)+\bar{v}_0(x)\big), \end{align}\] which are the solutions to the anomalous diffusion equation \(\bar{U}_t+(-\Delta)^{\sigma}\bar{U}=0\) with suitable initial data \(\frac{\gamma}{\mu+\nu}(\bar{u}_0+\bar{v}_0)\). When \(\sigma=1\), they are the same as those in [1].

In the next result, we show that by subtracting these profiles, we gain exponentially faster decay estimates, which are non-vanishing provided that \(\mu\bar{u}_0\neq\nu\bar{v}_0\). Remark that the case \(\mu\bar{u}_0\equiv \nu\bar{v}_0\) will provide an equilibrium such that \(\mu \bar{u}\equiv \nu \bar{v}\) for any \(t>0\). This is the difference comparing with the single diffusion equation for \(\bar{U}\) in the above.

We are going to consider optimal large time estimates by rigorously verifying their lower bounds in the \(L^2\) framework with \(L^1\) data.

However, the profiles \(\bar{u}^{\mathrm{prof}}\), \(\bar{v}^{\mathrm{prof}}\) cannot be devoted to the semilinear problem 1 . For this reason, we are going to introduce another kind of large time profiles in the sense of integral of initial data. By the same way as 6 and using Lemma 1, one may easily get the next large time behavior for \(1\leqslant m\leqslant +\infty\): \[\begin{align} \|\mathcal{G}(t,|D|)\bar{g}_0(\cdot)-G(t,\cdot)P_{\bar{g}_0}\|_{L^m}=o(t^{-\frac{n}{2\sigma}(1-\frac{1}{m})}). \end{align}\] Then, the next result for their large time asymptotic profiles in the \(L^m\) framework can be deduced.

4 Semilinear nonlocal heat exchanger system↩︎

4.1 Philosophy of our proofs↩︎

Before proving the local/global in-time existence results and the sharp lower bound estimates of lifespan \(T_{\varepsilon}\), as preparations, we at first explain our philosophy.

For any \(T>0\), we introduce the evolution space of solution \(\mathcal{U}=\mathcal{U}(t,x)\) in the vector sense that \(\mathcal{U}:=(u,v)^{\mathrm{T}}\) by \[\begin{align} X_T:=\big(\mathcal{C}([0,T],L^1\cap L^{\infty})\times \mathcal{C}([0,T],L^1\cap L^{\infty})\big)^{\mathrm{T}} \end{align}\] equipping the time-weighted norm \[\begin{align} \|\mathcal{U}\|_{X_T}:=\sup\limits_{t\in[0,T]}\sum\limits_{w\in\{u,v\}}\left(\|w(t,\cdot)\|_{L^1}+(1+t)^{\frac{n}{2\sigma}}\|w(t,\cdot)\|_{L^{\infty}}\right). \end{align}\] From Duhamel’s principle, concerning the semilinear Cauchy problem 1 in the vector version, let us construct the following nonlinear integral operator: \[\begin{align} \mathcal{N}:\;\mathcal{U}\in X_T\to \mathcal{N}[\mathcal{U}]:=\varepsilon \mathcal{U}_{\mathop{\mathrm{lin}}}+\mathcal{U}_{\mathop{\mathrm{nlin}}} \end{align}\] for any \(t\in[0,T]\) and \(x\in\mathbb{R}^n\), with \(\mathcal{U}_{\mathop{\mathrm{lin}}}=(\bar{u},\bar{v})^{\mathrm{T}}\) and \(\mathcal{U}_{\mathop{\mathrm{nlin}}}=(u^{\mathop{\mathrm{nlin}}},v^{\mathop{\mathrm{nlin}}})^{\mathrm{T}}\) defined by \[\begin{align} u^{\mathop{\mathrm{nlin}}}(t,x)&:=\int_0^t\big(K_0^u(t-\tau,|D|)[u(\tau,x)]^p+K_1^u(t-\tau,|D|)[v(\tau,x)]^q\big)\,\mathrm{d}\tau,\\ v^{\mathop{\mathrm{nlin}}}(t,x)&:=\int_0^t\big(K_0^v(t-\tau,|D|)[u(\tau,\cdot)]^p+K_1^v(t-\tau,|D|)[v(\tau,x)]^q\big)\,\mathrm{d}\tau. \end{align}\]

The assumption \((u_0,v_0)\in (L^{\infty}\cap L^1)^2\) indicates \(\mathcal{U}_{\mathop{\mathrm{lin}}}\in X_{T}\) for any \(T>0\) and the uniform estimate (i.e. Proposition 1 with \(m=1\) or \(m=+\infty\) and \(r=1\)) \[\begin{align} \|\mathcal{U}_{\mathop{\mathrm{lin}}}\|_{X_T}\leqslant C_0\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2} \end{align}\] for a suitable constant \(C_0>0\).

Our goal is to prove the existence of unique fixed point \(\mathcal{U}\) of nonlinear integral operator \(\mathcal{N}\) in the space \(X_T\), which is equivalent to the unique solutions \((u,v)\) to the semilinear problem 1 in \(X_T\). So, we are going to apply the Banach contraction principle for any \(\mathcal{U}\) and \(\mathcal{V}\) in the set \[\begin{align} \mathcal{B}_{\kappa}(X_T):=\big\{\mathcal{U}\in X_T:\;\|\mathcal{U}\|_{X_T}\leqslant \kappa:=2\varepsilon C_0\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}>0\big\}. \end{align}\] In other words, we will prove that the following fundamental inequalities: \[\begin{align} \|\mathcal{N}[\mathcal{U}]\|_{X_T}&\leqslant \varepsilon C_0\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}+C_1(T)\|\mathcal{U}\|_{X_T}^{\min\{p,q\}},\tag{8}\\ \|\mathcal{N}[\mathcal{U}]-\mathcal{N}[\mathcal{V}]\|_{X_T}&\leqslant C_1(T)\|\mathcal{U}-\mathcal{V}\|_{X_T}\big(\|\mathcal{U}\|_{X_T}^{\min\{p,q\}-1}+\|\mathcal{V}\|_{X_T}^{\min\{p,q\}-1}\big),\tag{9} \end{align}\] hold with a suitable constant \(C_1(T)>0\).

In the forthcoming subsection, we will demonstrate the crucial estimate \[\begin{align} \label{Est-Important-03} C_1(T)\lesssim \sum\limits_{r\in\{p,q\}}\left(\int_0^T(1+\tau)^{-\frac{n}{2\sigma}(r-1)}\,\mathrm{d}\tau+(1+T)^{1-\frac{n}{2\sigma}(r-1)}\right). \end{align}\tag{10}\] We next separate our discussion according to the value of \(\min\{p,q\}\), motivated by the recent work [26].

Super-critcial Case: \(\min\{p,q\}>p_{\mathrm{Fuj}}(\frac{n}{\sigma})\).

It is trivial that \[\begin{align} \int_0^{T}(1+\tau)^{-\frac{n}{2\sigma}(r-1)}\,\mathrm{d}\tau\lesssim 1\;\;and\;\;(1+T)^{1-\frac{n}{2\sigma}(r-1)}\lesssim 1, \end{align}\] for \(r\in\{p,q\}\), uniformly in-time \(T\), namely, \(C_1(T)\leqslant C_2\) for any \(T>0\), where \(C_2>0\) is uniformly bounded with respect to \(T\). Carrying \(T=+\infty\), the estimates 8 and 9 deduce, respectively, \[\begin{align} \|\mathcal{N}[\mathcal{U}]\|_{X_{+\infty}}\leqslant\frac{3}{4}\kappa\;\; and\;\; \|\mathcal{N}[\mathcal{U}]-\mathcal{N}[\mathcal{V}]\|_{X_{+\infty}}\leqslant\frac{1}{2}\|\mathcal{U}-\mathcal{V}\|_{X_{+\infty}}, \end{align}\] if \(4C_2\kappa^{\min\{p,q\}-1}\leqslant 1\). That is to say, the choice of small size \(\varepsilon\) leads to \[\begin{align} 0<\varepsilon\leqslant \varepsilon_0:=(4C_2)^{-\frac{1}{\min\{p,q\}-1}}\left( 2C_0\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}\right)^{-1}. \end{align}\] It follows \((u,v)^{\mathrm{T}}\in\mathcal{B}_{\kappa}(X_{+\infty})\). As a byproduct, we also find \[\begin{align} \sup\limits_{t\in[0,+\infty)}\sum\limits_{w\in\{ u,v\}}\left(\|w(t,\cdot)\|_{L^1}+(1+t)^{\frac{n}{2\sigma}}\|w(t,\cdot)\|_{L^{\infty}}\right)\lesssim \varepsilon\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}. \end{align}\] Finally, the interpolation completes the desired estimates ?? .

Sub-critcial Case: \(\min\{p,q\}< p_{\mathrm{Fuj}}(\frac{n}{\sigma})\).

We may get \[\begin{align} C_1(T)\leqslant C_3 (1+T)^{1-\frac{n}{2\sigma}(\min\{p,q\}-1)}, \end{align}\] with a uniformly in-time suitable constant \(C_3>0\). Similarly to the super-critical case, the operator \(\mathcal{N}\) is a contraction on the ball \(\mathcal{B}_{\kappa}(X_T)\) for some \(T>0\) satisfying \[\begin{align} 4C_3(1+T)^{1-\frac{n}{2\sigma}(\min\{p,q\}-1)}\kappa^{\min\{p,q\}-1}\leqslant 1, \end{align}\] namely, \[\begin{align} \label{Est-02} T\lesssim \varepsilon^{-\frac{\min\{p,q\}-1}{1-\frac{n}{2\sigma}(\min\{p,q\}-1)}}. \end{align}\tag{11}\] Thus, for any \(T>0\) such that 11 holds, a direct application of Banach fixed point argument leads to the existence of uniquely local in-time solutions \((u,v)\in\mathcal{B}_{\kappa}(X_T)\). The lower bound estimate of lifespan \(T_{\varepsilon}\) in the sub-critical case has been derived.

Critcial Case: \(\min\{p,q\}= p_{\mathrm{Fuj}}(\frac{n}{\sigma})\).

For a uniformly in-time suitable constant \(C_4>0\), it follows \[\begin{align} C_1(T)\leqslant C_4 \ln (1+T). \end{align}\] From the needed condition (to ensure the local in-time existence of solutions) \[\begin{align} 4C_4\ln (1+T)\kappa^{\min\{p,q\}-1}\leqslant 1, \end{align}\] we analogously to the above sub-critical case arrive at \[\begin{align} T\lesssim \exp\left(C\varepsilon^{-\min\{p,q\}+1}\right)=\exp\left(C\varepsilon^{1-p_{\mathrm{Fuj}}(\frac{n}{\sigma})}\right), \end{align}\] which shows our desired lower bound estimate of lifespan \(T_{\varepsilon}\) in the critical case.

All in all, to complete Theorem 1 and Theorem 3, it remains to justify the crucial estimate 10 .

4.2 Proofs of Theorem 1 and Theorem 3↩︎

Recalling the definition of \(X_T\), by the Riesz-Thorin interpolation between the \(L^1\) and \(L^{\infty}\) norms, we are able to conclude \[\begin{align} \|\,[u(\tau,\cdot)]^p\|_{L^m}\lesssim (1+\tau)^{-\frac{n}{2\sigma}(1-\frac{1}{mp})p}\|\mathcal{U}\|_{X_T}^p,\\ \|\,[v(\tau,\cdot)]^q\|_{L^m}\lesssim (1+\tau)^{-\frac{n}{2\sigma}(1-\frac{1}{mq})q}\|\mathcal{U}\|_{X_T}^q, \end{align}\] for any \(1\leqslant m\leqslant +\infty\). Concerning \(w\in\{u,v\}\), by using the derived (bounded) \(L^1-L^1\) estimate from Proposition 1, one obtains \[\begin{align} \|w^{\mathop{\mathrm{nlin}}}(t,\cdot)\|_{L^1}&\lesssim\int_0^t\big(\|\,[u(\tau,\cdot)]^p\|_{L^1}+\|\,[v(\tau,\cdot)]^q\|_{L^1}\big)\,\mathrm{d}\tau\\ &\lesssim\int_0^t(1+\tau)^{-\frac{n}{2\sigma}(p-1)}\,\mathrm{d}\tau\,\|\mathcal{U}\|_{X_T}^p+\int_0^t(1+\tau)^{-\frac{n}{2\sigma}(q-1)}\,\mathrm{d}\tau\,\|\mathcal{U}\|_{X_T}^q\\ &\lesssim\sum\limits_{r\in\{ p,q\}}\int_0^t(1+\tau)^{-\frac{n}{2\sigma}(r-1)}\,\mathrm{d}\tau\,\|\mathcal{U}\|_{X_T}^{\min\{p,q\}}, \end{align}\] thanks to the smallness condition in \(\mathcal{B}_{\kappa}(X_T)\). Next, with the aid of derived \((L^{\infty}\cap L^1)-L^{\infty}\) estimate in \([0,\frac{t}{2}]\) and \(L^{\infty}-L^{\infty}\) estimate in \([\frac{t}{2},t]\) from Proposition 1, one gets \[\begin{align} (1+t)^{\frac{n}{2\sigma}}\|w^{\mathop{\mathrm{nlin}}}(t,\cdot)\|_{L^{\infty}}&\lesssim(1+t)^{\frac{n}{2\sigma}}\int_0^{\frac{t}{2}}(1+t-\tau)^{-\frac{n}{2\sigma}}\big(\|\,[u(\tau,\cdot)]^p\|_{L^{\infty}\cap L^1}+\|\,[v(\tau,\cdot)]^q\|_{L^{\infty}\cap L^1}\big)\,\mathrm{d}\tau\\ &\quad+(1+t)^{\frac{n}{2\sigma}}\int_{\frac{t}{2}}^t\big(\|\,[u(\tau,\cdot)]^p\|_{L^{\infty}}+\|\,[v(\tau,\cdot)]^q\|_{L^{\infty}}\big)\,\mathrm{d}\tau\\ &\lesssim \sum\limits_{r\in\{p,q\}}\left(\int_0^{\frac{t}{2}}(1+\tau)^{-\frac{n}{2\sigma}(r-1)}\,\mathrm{d}\tau+(1+t)^{\frac{n}{2\sigma}}\int_{\frac{t}{2}}^t(1+\tau)^{-\frac{nr}{2\sigma}}\,\mathrm{d}\tau\right)\|\mathcal{U}\|_{X_T}^{\min\{p,q\}}\\ &\lesssim \sum\limits_{r\in\{ p,q\}}\left(\int_0^t(1+\tau)^{-\frac{n}{2\sigma}(r-1)}\,\mathrm{d}\tau+(1+t)^{1-\frac{n}{2\sigma}(r-1)}\right)\|\mathcal{U}\|_{X_T}^{\min\{p,q\}}, \end{align}\] where we employed the asymptotic relations \(1+t-\tau\approx 1+t\) for \(\tau\in[0,\frac{t}{2}]\) and \(1+\tau\approx1+t\) for \(\tau\in[\frac{t}{2},t]\). In other words, the summary of them shows \[\begin{align} \|\mathcal{U}^{\mathop{\mathrm{nlin}}}\|_{X_T}\lesssim \sum\limits_{r\in\{ p,q\}}\left(\int_0^T(1+\tau)^{-\frac{n}{2\sigma}(r-1)}\,\mathrm{d}\tau+(1+T)^{1-\frac{n}{2\sigma}(r-1)}\right)\|\mathcal{U}\|_{X_T}^{\min\{p,q\}}. \end{align}\] By the analogous way, it leads to \[\begin{align} \|\mathcal{N}[\mathcal{U}]-\mathcal{N}[\mathcal{V}]\|_{X_T}&\lesssim \sum\limits_{r\in\{ p,q\}}\left(\int_0^T(1+\tau)^{-\frac{n}{2\sigma}(r-1)}\,\mathrm{d}\tau+(1+T)^{1-\frac{n}{2\sigma}(r-1)}\right)\\ &\quad\times\|\mathcal{U}-\mathcal{V}\|_{X_T}\big(\|\mathcal{U}\|_{X_T}^{\min\{p,q\}-1}+\|\mathcal{V}\|_{X_T}^{\min\{p,q\}-1}\big). \end{align}\] All in all, our desired estimate 10 is obtained.

4.3 Proof of Theorem 2↩︎

Let us see that the global in-time solutions demonstrated in Theorem 1 have the integral forms ?? for \(w\in\{u,v\}\). Recalling the refined estimates for the linear problem in Proposition 3, in order to justify our desired error estimates, we have to consider each kernel in the nonlinear parts motivated by [27]. For the sake of readability, we are going to demonstrate the first part, namely, the refined estimate for \(\int_0^tK_0^u(t-\tau,|D|)[u(\tau,x)]^p\,\mathrm{d}\tau\). The other three parts can be proved similarly. In the following discussions, we consider large time \(t\gg1\) without more repetition.

First of all, let us carry out a suitable decomposition (into five parts) \[\begin{align} \int_0^tK_0^u(t-\tau,|D|)[u(\tau,x)]^p\,\mathrm{d}\tau-\frac{\nu}{\mu+\nu}\,G(t,x)\,\mathcal{P}_{u^p}=\sum\limits_{j\in\{1,\dots,5\}}A_{j}(t,x), \end{align}\] where we took \[\begin{align} A_1(t,x)&:=\int_0^{\frac{t}{2}}\left(K_0^u(t-\tau,|D|)-\frac{\nu}{\mu+\nu}\,\mathcal{G}(t-\tau,|D|)\right)[u(\tau,x)]^p\,\mathrm{d}\tau,\\ A_2(t,x)&:=\frac{\nu}{\mu+\nu}\int_0^{\frac{t}{2}}\big(\mathcal{G}(t-\tau,|D|)-\mathcal{G}(t,|D|)\big)[u(\tau,x)]^p\,\mathrm{d}\tau,\\ A_3(t,x)&:=\frac{\nu}{\mu+\nu}\int_0^{\frac{t}{2}}\left(\mathcal{G}(t,|D|)[u(\tau,x)]^p-\int_{\mathbb{R}^n}[u(\tau,y)]^p\,\mathrm{d}y\,G(t,x)\right)\mathrm{d}\tau,\\ A_4(t,x)&:=\int_{\frac{t}{2}}^tK_0^u(t-\tau,|D|)[u(\tau,x)]^p\,\mathrm{d}\tau,\\ A_5(t,x)&:=-\frac{\nu}{\mu+\nu}\,G(t,x)\int_{\frac{t}{2}}^{+\infty}\int_{\mathbb{R}^n}[u(\tau,y)]^p\,\mathrm{d}y\,\mathrm{d}\tau. \end{align}\]

• According to Proposition 1 (with \(m=m_1\) as well as \(r=1\)) and the decay estimate ?? of global in-time solution \(u\), the first term can be estimated by \[\begin{align} t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\|A_1(t,\cdot)\|_{L^{m_1}}&\lesssim t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\int_0^{\frac{t}{2}}\mathrm{e}^{-(\mu+\nu)(t-\tau)}\,(t-\tau)^{-\frac{n}{2\sigma}(1-\frac{1}{m_1})}\|\,[u(\tau,\cdot)]^p\|_{L^1}\,\mathrm{d}\tau\\ &\lesssim \varepsilon^p\,\mathrm{e}^{-ct}\,\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p. \end{align}\]

• By using the mean value theorem with respect to \(t\), i.e. \[\begin{align} \mathcal{G}(t-\tau,|D|)-\mathcal{G}(t,|D|)=-\tau\mathcal{G}_t(t-\theta_1\tau,|D|)\;\;with\;\;\theta_1\in(0,1), \end{align}\] and Lemma 1 (with \(m=m_1\) as well as \(s=2\sigma\)), the second term can be estimated by \[\begin{align} \|A_2(t,\cdot)\|_{L^{m_1}}&\lesssim\int_0^{\frac{t}{2}}\tau(t-\theta_1\tau)^{-\frac{n}{2\sigma}(1-\frac{1}{m_1})-1}\|\,[u(\tau,\cdot)]^p\|_{L^1}\,\mathrm{d}\tau\\ &\lesssim \varepsilon^p\,t^{-\frac{n}{2\sigma}(1-\frac{1}{m_1})-1}\int_0^{\frac{t}{2}}(1+\tau)^{1-\frac{n}{2\sigma}(p-1)}\,\mathrm{d}\tau\,\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p, \end{align}\] which shows \[\begin{align} t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\|A_2(t,\cdot)\|_{L^{m_1}}\lesssim \varepsilon^p\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p\times \begin{cases} t^{1-\frac{n}{2\sigma}(p-1)}&if\;\;p<p_{\mathrm{Fuj}}(\frac{n}{2\sigma}),\\ t^{-1}\ln t&if\;\;p=p_{\mathrm{Fuj}}(\frac{n}{2\sigma}),\\ t^{-1}&if\;\;p>p_{\mathrm{Fuj}}(\frac{n}{2\sigma}). \end{cases} \end{align}\] Its right-hand sides converge to zero as \(t\to+\infty\) due to \(p>p_{\mathrm{Fuj}}(\frac{n}{\sigma})\).

• By the same way as 6 , i.e. an application of mean value theorem with respect to \(x\) and an additional separation via \(t^{\frac{1}{4\sigma}}\), one derives \[\begin{align} \|A_3(t,\cdot)\|_{L^{m_1}}&\lesssim\left\|\int_0^{\frac{t}{2}}\int_{|y|\leqslant t^{\frac{1}{4\sigma}}}[G(t,\cdot-y)-G(t,\cdot)]\,[u(\tau,y)]^p\,\mathrm{d}y\,\mathrm{d}\tau\right\|_{L^{m_1}}\\ &\quad+\left\|\int_0^{\frac{t}{2}}\int_{|y|\geqslant t^{\frac{1}{4\sigma}}}[\,|G(t,\cdot-y)|+|G(t,\cdot)|\,]\,[u(\tau,y)]^p\,\mathrm{d}y\,\mathrm{d}\tau\right\|_{L^{m_1}}\\ &\lesssim\int_0^{\frac{t}{2}}t^{\frac{1}{4\sigma}}\|\nabla G(t,\cdot)\|_{L^{m_1}}\|\,[u(\tau,\cdot)]^p\|_{L^1}\,\mathrm{d}\tau+\int_0^{\frac{t}{2}}\|G(t,\cdot)\|_{L^{m_1}}\|\,[u(\tau,\cdot)]^p\|_{L^1(|x|\geqslant t^{\frac{1}{4\sigma}})}\,\mathrm{d}\tau\\ &\lesssim \varepsilon^p\,t^{-\frac{1}{4\sigma}-\frac{n}{2\sigma}(1-\frac{1}{m_1})}\int_0^{+\infty}(1+\tau)^{-\frac{n}{2\sigma}(p-1)}\,\mathrm{d}\tau\,\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p\\ &\quad+t^{-\frac{n}{2\sigma}(1-\frac{1}{m_1})}\int_0^{+\infty}\|\,[u(\tau,\cdot)]^p\|_{L^1(|x|\geqslant t^{\frac{1}{4\sigma}})}\,\mathrm{d}\tau. \end{align}\] According to the fact that \[\begin{align} \|\,[u(\tau,\cdot)]^p\|_{L^1([0,+\infty)\times \mathbb{R}^n)}\lesssim\varepsilon^p\int_0^{+\infty}(1+\tau)^{-\frac{n}{2\sigma}(p-1)}\,\mathrm{d}\tau\,\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p<+\infty, \end{align}\] we claim \[\begin{align} \lim\limits_{t\to+\infty}\int_0^{+\infty}\|\,[u(\tau,\cdot)]^p\|_{L^1(|x|\geqslant t^{\frac{1}{4\sigma}})}\,\mathrm{d}\tau=0. \end{align}\] That is to say, the third term can be estimated by \[\begin{align} \lim\limits_{t\to+\infty}t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\|A_3(t,\cdot)\|_{L^{m_1}}=0. \end{align}\]

• Applying the derived (bounded) \(L^{m_1}-L^{m_1}\) estimate in Proposition 1, the fourth term can be estimated by \[\begin{align} t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\|A_4(t,\cdot)\|_{L^{m_1}}&\lesssim t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\int_{\frac{t}{2}}^t\|\,[u(\tau,\cdot)]^p\|_{L^{m_1}}\,\mathrm{d}\tau\lesssim \varepsilon^p\,t^{1-\frac{n}{2\sigma}(p-1)}\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p. \end{align}\]

• Applications of Lemma 1 and the decay estimate ?? show \[\begin{align} t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\|A_5(t,\cdot)\|_{L^{m_1}}&\lesssim\varepsilon^p\,t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\int_{\frac{t}{2}}^{+\infty}(1+\tau)^{-\frac{n}{2\sigma}(p-1)}\,\mathrm{d}\tau\,\|G(t,\cdot)\|_{L^{m_1}} \|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p\\ &\lesssim \varepsilon^p\,t^{1-\frac{n}{2\sigma}(p-1)}\|(u_0,v_0)\|_{(L^{\infty}\cap L^1)^2}^p. \end{align}\]

Thanks to \(p>p_{\mathrm{Fuj}}(\frac{n}{\sigma})\), namely, \(1-\frac{n}{2\sigma}(p-1)<0\), the last obtained estimates conclude \[\begin{align} \lim\limits_{t\to+\infty}t^{\frac{n}{2\sigma}(1-\frac{1}{m_1})}\left\|\int_0^tK_0^u(t-\tau,|D|)[u(\tau,\cdot)]^p\,\mathrm{d}\tau-\frac{\nu}{\mu+\nu}\,G(t,\cdot)\,\mathcal{P}_{u^p}\right\|_{L^{m_1}}=0. \end{align}\] Analogously, due to the assumption \(\min\{p,q\}>p_{\mathrm{Fuj}}(\frac{n}{\sigma})\), the other kernels in the integral representation ?? can be treated by \[\begin{align} \lim\limits_{t\to+\infty}t^{\frac{n}{2\sigma}(1-\frac{1}{m})}\left\|w^{\mathop{\mathrm{nlin}}}(t,\cdot)-\frac{\gamma}{\mu+\nu}\,G(t,\cdot)\,\mathcal{P}_{u^p+v^q}\right\|_{L^{m}}=0, \end{align}\] where \(m=m_1\) and \(\gamma=\nu\) if \(w=u\); \(m=m_2\) and \(\gamma=\mu\) if \(w=v\).

Combining the derived estimates for \(\bar{u}\) and \(\bar{v}\) in Proposition 3 by taking \(m=m_1\) or \(m=m_2\), via the integral representation ?? again, we complete the derivation of asymptotic profiles for global in-time solutions \((u,v)\) in the \(L^m\) framework.

As \(m_1=m_2=2\), using the triangle inequality, for example, \[\begin{align} \|u(t,\cdot)\|_{L^2}&\geqslant \|G(t,\cdot)\|_{L^2}\left|\varepsilon P_{u_0+v_0}+\mathcal{P}_{u^p+v^q}\right|-\left\|u(t,\cdot)-\frac{\nu}{\mu+\nu}\,G(t,\cdot)\left(\varepsilon P_{u_0+v_0}+\mathcal{P}_{u^p+v^q}\right)\right\|_{L^{2}}\\ &\gtrsim t^{-\frac{n}{4\sigma}}\left|\varepsilon P_{u_0+v_0}+\mathcal{P}_{u^p+v^q}\right|-o(t^{-\frac{n}{4\sigma}}) \end{align}\] as large time \(t\gg1\) via 7 , the optimal large time lower bound estimates for the global in-time solutions \((u,v)\) in the \(L^2\) framework can be demonstrated. Our proof is completed.

Acknowledgments↩︎

The author thanks Wenhui Chen (Guangzhou University) for pointing out the reduction methodology and for some suggestions for Remark 2.5.

References↩︎