June 01, 2026
Abstract: We are interested in the asymptotic behavior of solutions toward strong rarefaction waves for a parabolic-hyperbolic system arising from chemotaxis. Suppose that the Riemann problem to the corresponding inviscid system admits rarefaction waves. We show that if the initial data is a small perturbation of an approximate rarefaction wave, then the Cauchy problem has a unique global solution that converges to the rarefaction wave asymptotically in time. The waves can be either a single rarefaction wave or a superposition of two rarefaction waves. Furthermore, the stability results hold regardless of the wave strengths. The proofs are based on the energy method, where the key observations are the monotonicity of the approximate rarefaction waves with respect to both space and time, and the appropriate replacement of spatial derivative of the hyperbolic component with the parabolic component in proper forms.
Key words: Chemotaxis; parabolic-hyperbolic system; strong rarefaction wave; composite wave; asymptotic stability.
AMS(2020) Subject Classification: 35B35, 35B40, 35Q92, 92C17
In this paper, we are concerned with the large time behaviors of solutions to the Cauchy problem of the following parabolic-hyperbolic system: \[\label{1461}
\begin{cases}
n_{t} -(nq)_{x}=n_{xx}, &\;x\in\mathbb{R},t>0,\\
q_{t}-n_{x}=0, &\;x\in\mathbb{R},t>0,
\end{cases}\tag{1}\]
with the initial data \[\label{i32d}
(n,q)(x,0)=(n_0,q_0)(x)\rightarrow (n_\pm ,q_\pm ) \;\text{ as } x\rightarrow \pm\infty.\tag{2}\] The system 1 is derived from the following PDE-ODE hybrid chemotaxis model with logarithmic sensitivity
\[\label{chemotaxis32model}
\left\{
\begin{align}
&n_t=D n_{xx}-\chi(n(\ln c)_x)_x,\\
&c_t=- nc+\beta c,
\end{align}
\right.\tag{3}\] which was first proposed by Levine et al. H.A.?, Levine?, 00? to model the interactions between vascular endothelial cells and the signaling molecule vascular endothelial growth factor (VEGF)
during the initiation of tumor angiogenesis. In this system, the unknowns \(n(x,t)>0\) and \(c(x, t)>0\) denote the density of vascular endothelial cells and the concentration of VEGF,
respectively. In contrast to random diffusion, chemotaxis represents a biased movement of organisms in response to chemical stimuli. The parameter \(D>0\) denotes the diffusion coefficient of vascular endothelial cells.
The parameter \(\chi>0\) stands for the chemotactic coefficient measuring the strength of chemotaxis, and the parameter \(\beta\geq0\) denotes the growth rate of chemical VEGF. The
logarithmic sensitivity function \(\ln c\) of the model 3 reflects that the chemotactic response of cells to the chemical signal adheres to the Weber-Fechner law, a principle with
prominent implications in biological modelings (cf. W.?, Alt?, 87?, F.W.?, Dahlquist?, 72?, E.F.?, Keller?, 71?). Since the sensitivity function \(\ln c\) is singular at \(c=0\),
it is challenging to study the model 3 directly. To resolve this singularity, Levine and Sleeman H.A.?, Levine?, 97? introduced an effective Hopf-Cole transformation \[\begin{align}
q:=-(\ln c)_x=-\frac{c_x}{c},
\end{align}\] which transform the chemotaxis model 3 to the following parabolic-hyperbolic system \[\left\{
\begin{align}
&n_t-\chi(nq)_x=D n_{xx},\\
&q_t-n_x=0.
\end{align}
\right.\] Then applying the scaling transformations \[\begin{align}
\tilde{x}=\frac{x}{\sqrt{\chi}},\quad \tilde{q}=\sqrt{\chi}q,\quad
\tilde{D}=\frac{D}{\chi},
\end{align}\] and taking \(\tilde{D}=1\), one can obtain 1 by dropping the tildes.
The first analytical work on the well-posedness of the system 1 2 was established by Guo et al. [1], where it was shown that the system admits a unique global strong solution for large initial data. Subsequently, by taking a positive constant state as the background solution, Zhang et al. Y.?, Zhang?, 13? constructed global classical small solutions and obtained an algebraic convergence rate. By establishing an entropy type estimate relative to a constant state, Li et al. D.Li?, 15? further proved the global well-posedness of large classical solutions. In the multi-dimensional case, owing to the complexity of higher dimensionality, all existing results on global well-posedness have been established for small solutions around a constant state. See [2], [3], D.Li?, 15? for results concerning global well-posedness in various working spaces.
All the above works are concerned with the stability of constant steady states. Once \(n_+\neq n_-\), it turns out that the large time behaviors of solutions to 1 are intrinsically related to the Riemann problem for inviscid system: \[\label{conservation32law} \left\{ \begin{align} &n_{t} -(nq)_{x}=0,\\ &q_{t}-n_{x}=0, \end{align} \right.\tag{4}\] with initial data \[\label{n044q0} (n,q)(x,0)=(n_0^r,q_0^r)(x)= \left\{ \begin{align} &(n_-,q_-),\;\;\;\;x<0,\\ &(n_+,q_+),\;\;\;\;x>0. \end{align} \right.\tag{5}\] To analyze its mathematical structure, we rewrite 4 as \[\label{equivalent32system} \left(\begin{align} &n\\&q \end{align} \right)_t +\left(\begin{aligned} &-q\;\;-n\\&-1\;\;\;\;\;0 \end{aligned}\right) \left(\begin{align} &n\\&q \end{align} \right)_x=0.\tag{6}\] A direct calculation shows that the matrix \[J=\left(\begin{align} &-q\;\;-n\\&-1\;\;\;\;\;0 \end{align}\right)\] has two distinct eigenvalues \[\label{1468} \lambda_1(n,q)=\frac{-q-\sqrt{q^2+4n}}{2} \;<\;\frac{-q+\sqrt{q^2+4n}}{2}=\lambda_2(n,q),\tag{7}\] if \(q^2+4n>0\), and the corresponding right eigenvectors are given by \[r_k(n,q)=(-\lambda_k(n,q), 1)^T,\; k=1,2.\] It is easy to verify that \[\label{genuinely32nonlinear} \begin{align} &\nabla \lambda_1(n,q)\cdot r_1(n,q)= -\frac{q}{\sqrt{q^{2}+4n}}-1\neq 0,\\ &\nabla \lambda_2(n,q)\cdot r_2(n,q)= \frac{q}{\sqrt{q^{2}+4n}}-1\neq 0, \end{align}\tag{8}\] where \(\nabla \lambda_k(n,q)\triangleq(\frac{\partial\lambda_k}{\partial n}, \frac{\partial\lambda_k}{\partial q})^T ,k=1,2\). Thus the system 4 is strictly hyperbolic and the characteristic fields are genuinely nonlinear if \(q^2+4n>0\).
According to the theory of hyperbolic conservation laws J.?, A.?, 83?, the solution to the Riemann problem 4 5 consists of shock waves, rarefaction waves, and their linear superpositions. Wang and Hillen [4] were the first to construct shock waves for the inviscid system 4 and explicitly derived the traveling waves for the parabolic-hyperbolic system 1 . Subsequently, using the energy method, Li and Wang T.Li?, 09? proved the asymptotic stability of traveling waves for 1 away from the vacuum state under zero-mass perturbations. Li et al. J.Li?, 13? generalized the work of T.Li?, 09? to composite two traveling waves. On the basis of weighted energy estimates, Jin et al. Jin?, 13? further derived the asymptotic stability of traveling waves with vacuum end state under zero mass perturbations. The first multidimensional study was conducted by Chae et al. [5], who obtained the stability of planar traveling waves on an infinite strip domain under zero-mass perturbations. All these works were restricted to zero-mass perturbations as their methods essentially rely on the anti-derivative approach. Very recently, a breakthrough was achieved by Choi et al. [6], [6], 20? where the so-called \(L^2\) contraction property was successfully derived for general perturbations of traveling waves via the relative entropy method. Based on such relative entropy method, along with some intrinsic cancellation mechanisms of the chemotaxis model, the authors of the current paper [7] proved the nonlinear stability of planar traveling waves in the three-dimensional case under general perturbations.
Compared with the fruitful works available for traveling waves of 1 , the understanding of rarefaction waves is quite limited. To our knowledge, Rascle [8] was the first to construct rarefaction waves for the inviscid system 4 , where a strictly hyperbolic region and a linearly degenerate region were found. Very recently, Li and Mathur T.Li?, 22? as well as He and Wang F.He?, 24? presented a comprehensive study of the Riemann problem for 4 and constructed rarefaction waves, shock waves, contact discontinuities, and their superpositions. However, it remains an open question whether these basic waves are stable under appropriate perturbations. Furthermore, for the parabolic-hyperbolic system 1 , it is unknown whether its large-time behaviors could include a single rarefaction wave and its superposition with rarefaction waves or shock waves from other fields.
The purpose of this paper is to prove the asymptotic stability of rarefaction waves and the superposition of two rarefaction waves. We note that as early as the 1980s, Xin Z.P?, Xin?, 88?, Z.P?, Xin?, 89? presented a general stability theory of weak rarefaction waves and the superposition of two weak rarefaction waves for \(2\times2\) viscous conservation laws. In contrast to the works in Z.P?, Xin?, 88?, Z.P?, Xin?, 89?, we are interested in the strong rarefaction waves.
We now state our main results. Define the rarefaction curve \(R_k\) in a suitable neighborhood of \((n_-,q_-)\) as \[\label{Rk} \begin{align} &R_k(n_-,q_-)=\{(n,q)\in \mathbb{R}^2;\; h_k(n,q)=h_k(n_-,q_-);\; \lambda_k(n,q)\geq\lambda_k(n_-,q_-)\}, \end{align}\tag{9}\] where \(h_k\) is a \(k\)-Riemann invariant. Following F.He?, 24? or T.Li?, 22?, we derive our Riemann invariants as follows: \[\label{RI} \begin{align} &h_1(n,q)=(\sqrt{q^2+4n}+q)(\sqrt{q^2+4n}-2q)^2,\\ &h_2(n,q)=(\sqrt{q^2+4n}-q)(\sqrt{q^2+4n}+2q)^2. \end{align}\tag{10}\] It is easy to verify that the Riemann invariants satisfy \[\label{RId} \begin{align} & \nabla h_k\cdot r_k=0,\; k=1,2. \end{align}\tag{11}\]
Our first result is about the asymptotic stability of single rarefaction waves.
Theorem 1. Let \(k=1, 2\). For each fixed \((n_-, q_-)\) with \(n_->0\), there exists a positive constant \(\delta_0\), such that if \((n_+,q_+)\in R_k(n_-,q_-)\) \((k\)-rarefaction curve\()\), \(n_+>0\) and \[\label{n0-n0r} \|n_{0}-N_{k0}, q_{0}-Q_{k0}\|_{H^1} \leq\delta_0,\qquad{(1)}\] then the initial value problem 1 2 has a unique global solution \((n,q)(x,t)\) satisfying \[\label{solutio} (n-n^r_k, q-q^r_k)\in C^0([0, +\infty); L^2),\;(n, q)_x\in C^0([0, +\infty); L^2),\;q_{xx}\in L^2((0, +\infty); L^2),\qquad{(2)}\] and \[\label{approximate} \lim\limits_{t\rightarrow \infty} \sup\limits_{x\in \mathbb{R}} \left|(n,q)(x,t)-(n^r_k,q^r_k)(x,t)\right|=0,\qquad{(3)}\] where \((N_{k0},Q_{k0})(x)\) is the initial value of smooth approximate rarefaction wave \((N_k,Q_k)(x,t)\) constructed in 21 .
We proceed to investigate the stability of the composite wave of two rarefaction waves. According to the theory of hyperbolic conservation laws J.?, A.?, 83? (see the works of F.He?, 24?, T.Li?, 22? for details), for each fixed state \((n_-,q_-)\) with \(n_->0\), there exists a region \(RR(n_-,q_-)\) such that for any state \((n_+,q_+)\in RR(n_-,q_-)\), the Riemann problem 4 5 has a unique solution denoted by \((n^r,q^r)(x,t)\) which can be constructed as follows.
One can find a unique state \((\bar{n},\bar{q})\) on the \(1\)-rarefaction wave curve \(R_1(n_-,q_-)\), i.e., \((\bar{n},\bar{q})\in R_1(n_-,q_-)\), such that \((n_+,q_+)\) is on the \(2\)-rarefaction wave curve \(R_2(\bar{n},\bar{q})\). Let \((n_1^r, q_1^r)(x,t)\) denote the \(1\)-rarefaction wave connecting \((n_-,q_-)\) to \((\bar{n},\bar{q})\) and \((n_2^r, q_2^r)(x,t)\) denote the \(2\)-rarefaction wave connecting \((\bar{n},\bar{q})\) to \((n_+,q_+)\). Then the composite rarefaction wave \((n^r, q^r)\) \((x,t)\) is a linear superposition of \((n_1^r, q_1^r)(x,t)\) and \((n_2^r, q_2^r)(x,t)\): \[\label{n143n2} (n^r, q^r)(x,t):=(n_1^r, q_1^r)(x,t)+(n_2^r, q_2^r)(x,t)-(\bar{n},\bar{q}).\tag{12}\]
Our second result is to show that when the initial data \((n_0,q_0)(x)\) and \((n_0^r,q_0^r)(x)\) are suitably close, the solution of the system 1 2 will tend to the composite rarefaction wave \((n^r,q^r)(x,t)\) as \(t\rightarrow +\infty\).
Theorem 2. For each fixed \((n_-, q_-)\) with \(n_->0\), assume that \((n_+,q_+)\in RR(n_-,q_-)\) and \(n_+>0\). Then the composite rarefaction wave \((n^r,q^r)(x,t)\) constructed in 12 is nonlinearly stable in the sense that there exists a constant \(\delta_1>0\) such that if \[\|n_0-N_0, q_0-Q_0\|_{H^1}\leq\delta_1,\] then the initial value problem 1 2 has a unique global solution \((n,q)(x,t)\) satisfying \[\label{1469} (n-n^r, q-q^r)\in C^0([0, +\infty); L^2),\; (n, q)_x\in C^0([0, +\infty); L^2),\; q_{xx}\in L^2((0, +\infty); L^2),\qquad{(4)}\] and \[\label{stable} \lim\limits_{t\rightarrow \infty} \sup\limits_{x\in \mathbb{R}} \left|(n,q)(x,t)-(n^r,q^r)(x,t)\right|=0,\qquad{(5)}\] where \((N_0,Q_0)(x)\) is the initial value of smooth approximate rarefaction wave \((N,Q)(x,t)\) constructed in 33 .
****Remark** 1**. In biology our results imply that the bacterial distribution will become increasingly sparse in the form of the rarefaction wave if initially it is close to \((N_0,Q_0)\).
****Remark** 2**. Our stability results hold true regardless of the strengths of the rarefaction waves, i.e. the amplitude \(|n_--n_+|+|q_--q_+|\) can be arbitrarily large.
Yang and Zhao T.?, Yang?, 05? proposed a general stability theory of strong rarefaction wave for \(2\times2\) conservation laws with positive definite viscosity coefficient matrix. Since the system 1 only exhibits partial viscosity, its viscosity coefficient matrix is not positive definite but degenerate. Hence, the theory established in T.?, Yang?, 05? cannot be applied. We prove our two theorems by observing two key ingredients: One is the monotonicity of the approximations of waves in both \(x\) and \(t\), which provides a ‘’good’’sign in the proof of basic energy estimate (see Lemma 6). The other is the management of \((q-Q_k)_x\), i.e. \(\psi_x\) in the perturbation equations. Since \(\psi\) satisfies a first-order equation (see 47 ), it is challenging to estimate \(\psi_x\) directly. Instead, thanks to the specific coupling structure of the perturbation equations, we replace \(\psi_x\) in terms of \(\phi\) in the \(L^2\) estimate (see 53 ) and \(H^1\) estimate (see 75 ). Then, by exploring the strong dissipation of the parabolic equation of \(\phi\), we successfully close the a priori estimate.
Before concluding this section, we mention some other works comparable to the current work. Matsumura and Nishihara Matsumura?, 86? were the first to show the stability of rarefaction waves to the isentropic Navier-Stokes equations, a typical system of viscous conservation laws. In Matsumura?, 86?, they required the waves to be weak and the initial perturbations to be small; subsequently, in Matsumura?, 92?, they removed both the smallness of the wave amplitude and the strength of the perturbations by assuming that the adiabatic constant \(\gamma\) satisfies \(1\leq\gamma\leq2\). The methodology proposed in Matsumura?, 86?, Matsumura?, 92? has been extensively refined by numerous researchers in the context of more complex systems, including the one dimensional full Navier-Stokes equations [9], Nishihara?, 04?, R.?, Duan?, 09? and the Navier-Stokes-Poisson equations [10]. In particular, the work in R.?, Duan?, 09? also includes the global stability of strong rarefaction waves of isentropic Navier-Stokes equations with a very general pressure. By observing some essential cancellations in the perturbation system, Li and Wang [11] and Li et al. [12] obtained the stability of planar rarefaction waves to the multidimenional Navier-Stokes equations.
The rest of this paper is organized as follows. In Section 2, we construct smooth approximations of the single rarefaction waves and the composite rarefaction waves of the inviscid system 4 by following the framework established by Matsumura and Nishihara Matsumura?, 92?. Moreover, we present some basic properties, such as monotonicity, of the approximations. In Section 3, we prove the asymptotic stability of single rarefaction waves. Section 4 is devoted to the proof of stability of composite rarefaction waves.
This section is devoted to constructing a smooth approximation \((N,Q)(x,t)\) of the rarefaction wave \((n^r,q^r)(x,t)\) and presenting some preliminary estimates for \((N,Q)(x,t)\).
We first construct a smooth approximation of a single rarefaction wave. Suppose \((n_+,q_+)\in R_k(n_-,q_-)\), \(k=1,2\), and consider the Riemann problem for the inviscid Burgers equation: \[\label{Burgers} \left\{\begin{align} &\frac{\partial{w^r_k}}{\partial_t} +w^r_k\frac{\partial{w^r_k}}{\partial_x}=0,\\ &w^r_k(x,0)=w^r_{k0}(x), \end{align} \right.\tag{13}\] where the initial value \(w^r_{k0}(x)\) satisfies \[\label{wr0} w^r_{k0}(x)= \left\{\begin{align} &\lambda_k(n_-,q_-),\;\;\;\;\;x<0,\\ &\lambda_k(n_+,q_+),\;\;\;\;\;x>0. \end{align} \right.\tag{14}\] Here \(\lambda_k\) \((k=1,2)\) is given by 7 . It is well-known (cf. J.?, A.?, 83?) that 13 14 has a continuous weak solution \(w^r_k(x,t)\) in the form of \[\label{wr} w^r_k(x,t)= \left\{\begin{align} &\lambda_k(n_-,q_-),\;\;\;\;\;\frac{x}{t}\leq\lambda_k(n_-,q_-),\\ &\frac{x}{t},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\lambda_k(n_-,q_-)\leq\frac{x}{t}\leq \lambda_k(n_+,q_+),\\ &\lambda_k(n_+,q_+),\;\;\;\;\;\frac{x}{t}\geq\lambda_k(n_+,q_+). \end{align} \right.\tag{15}\] Set \[\label{nr44qr} \begin{align} \lambda_k((n^r_k,q^r_k)(x,t))\triangleq w^r_k(x,t), \;h_k((n^r_k,q^r_k)(x,t))\triangleq h_k(n_-,q_-), \end{align}\tag{16}\] where \(h_k\) \((k=1,2)\) is the \(k\)-th Riemann invariant given by 10 . Since \((n_+,q_+)\in R_k(n_-,q_-)\), it holds that \(\lambda_k((n^r_k,q^r_k)(x,t)) \geq \lambda_k(n_-,q_-)\), which implies \[(n^r_k,q^r_k)(x,t) \in R_k(n_-,q_-).\] Hence, the unique solution \((n^r_k,q^r_k)(x,t)\) of the Riemann problem 4 5 is given by 16 .
We approximate \(w_k^r(x,t)\) by smooth functions \(w_k(x,t)\) that are the solutions of the following initial value problem \[\label{smooth} \left\{\begin{align} &\frac{\partial{w_k}}{\partial_t} +w_k\frac{\partial{w_k}}{\partial_x}=0,\\ &w_k(x,0)=w_{k0}(x) \end{align} \right.\tag{17}\] with \[\label{w0} \begin{align} w_{k0}(x):=\frac{\lambda_k(n_+,q_+)+\lambda_k(n_-,q_-)}{2} +\frac{\lambda_k(n_+,q_+)-\lambda_k(n_-,q_-)}{2} \kappa_\theta\int^{\varepsilon x}_0(1+y^2)^{-\theta}dy, \end{align}\tag{18}\] where \(\varepsilon>0\) is a small parameter and \(\kappa_\theta\) is the constant such that \(\kappa_\theta\int^\infty_0(1+y^2)^{-\theta}dy=1\) for each \(\theta>\frac{3}{2}\). As in Matsumura?, 92?, one can see that \(w_k(x,t)\) has the following estimates.
Lemma 1 (cf. Matsumura?, 92?-Lemma 2.1). Fix \(k=1,2\). If \(\lambda_k(n_-,q_-)<\lambda_k(n_+,q_+)\), then the problem 17 has a unique global smooth solution \(w_k(x,t)\) satisfying the following:
\(\lambda_k(n_-,q_-)< w_k(x,t)<\lambda_k(n_+,q_+)\), \(w_{kx}(x,t)>0\), \(\forall\) \((x,t)\in \mathbb{R}\times\mathbb{R}_+\).
For any \(p\in[0,+\infty]\), there exists a constant \(C_{p,\theta}>0\) such that \[\label{wkx} \begin{align} &\left\|w_{k}(\cdot, t)\right\|_{L^p} \leq C_{p,\theta} \min\{\varepsilon^{1-1/p}, t^{-1+1/p}\}, \forall\;t> 0,\\ &\left\|w_{kxx}(\cdot, t)\right\|_{L^p} \leq C_{p,\theta} \min\{\varepsilon^{2-1/p}, \varepsilon^{(1-\frac{1}{2\theta})(1-\frac{1}{p})} t^{-1-\frac{p-1}{2p\theta}}\}, \forall \;t>0. \end{align}\qquad{(6)}\]
If \(\lambda_k(n_-,q_-)>0\), then for \(x\leq0\), \(t\in \mathbb{R}_+\), it holds that \[\label{wk-w-} \begin{align} \left|w_k(x,t)-\lambda_k(n_-,q_-)\right| &\leq C_\theta (1+(\varepsilon x)^2)^{-\theta/3} [1+(\varepsilon \lambda_k(n_-,q_-)t)^2]^{-\theta/3},\\ \left|w_{kx}(x, t)\right| &\leq C_\theta \varepsilon(1+(\varepsilon x)^2)^{-\theta/2} [1+(\varepsilon \lambda_k(n_-,q_-)t)^2]^{-\theta/2}. \end{align}\qquad{(7)}\]
If \(\lambda_k(n_+,q_+)<0\), then for \(x\leq0\), \(t\in \mathbb{R}_+\), it holds that \[\label{wk-w43} \begin{align} \left|w_k(x,t)-\lambda_k(n_+,q_+)\right| &\leq C_\theta (1+(\varepsilon x)^2)^{-\theta/3} [1+(\varepsilon \lambda_k(n_+,q_+)t)^2]^{-\theta/3},\\ \left|w_{kx}(x, t)\right| &\leq C_\theta \varepsilon(1+(\varepsilon x)^2)^{-\theta/2} [1+(\varepsilon \lambda_k(n_+,q_+)t)^2]^{-\theta/2}. \end{align}\qquad{(8)}\]
\(\lim\limits_{t\rightarrow \infty} \sup\limits_{x\in \mathbb{R}} \left|w_k(x,t)-w_k^r(x,t)\right|=0.\)
By the characteristic method, one can find that the solution of 17 is expressed by the form \[\label{w} \begin{align} w_k(x,t)=w_{k0}(x_0(x,t)), \end{align}\tag{19}\] where \(x_0(x,t)\) is given by the relation \[\label{x} \begin{align} x=x_0(x,t)+w_{k0}(x_0(x,t))t. \end{align}\tag{20}\] By 19 and 20 , one can easily show Lemma 1. We refer Matsumura?, 92? for the detail of the proof.
Now we define \((N_k,Q_k)(x,t)\) \((k=1,2)\) by \[\label{U} \begin{align} (N_k,Q_k)(x,t)\in R_k(n_-,q_-),\quad \lambda_k((N_k,Q_k)(x,t))=w_k(x,t), \;k=1,2. \end{align}\tag{21}\] The next lemma shows that \((N_k,Q_k)(x,t)\) is the desired smooth approximation of \((n^r_k,q^r_k)(x,t)\).
Lemma 2. \((N_k,Q_k)(x,t)\) \((k=1,2)\) is an approximation of \((n^r_k,q^r_k)(x,t)\) \((k=1,2)\) in the following sense:
\((N_k,Q_k)(x,t)\) satisfies the system 6 .
\(\lim\limits_{t\rightarrow \infty} \sup\limits_{x\in \mathbb{R}} \left|(N_k,Q_k)(x,t)-(n^r_k,q^r_k)(x,t)\right|=0.\)
Proof. By 8 and 21 , we have \[\left(\begin{align} &N_k\\&Q_k \end{align} \right)_t=\frac{w_{kt}}{\nabla \lambda_k\cdot r_k} r_k,\quad \left(\begin{align} &N_k\\&Q_k \end{align} \right)_x=\frac{w_{kx}}{\nabla \lambda_k\cdot r_k} r_k,\;\;\;\;k=1,2.\] Thus, \[\label{24613} \begin{align} \left(\begin{aligned} &N_k\\&Q_k \end{aligned} \right)_t +\left(\begin{align} &-Q_k\;\;-N_k\\&-1\;\;\;\;\;\;\;0 \end{align}\right) \left(\begin{align} &N_k\\&Q_k \end{align} \right)_x &=\frac{1}{\nabla \lambda_k\cdot r_k} (w_{kt}+\lambda_kw_{kx}) r_k\\ &=\frac{1}{\nabla \lambda_k\cdot r_k} (w_{kt}+w_k(x,t)w_{kx}) r_k=0, \end{align}\tag{22}\] which shows (1).
The conclusion (2) follows from 9 , 16 , 21 , the fact that \(\nabla \lambda_k\) and \(\nabla h_k\) are linearly independent and Lemma 1-(5). ◻
Lemma 3. The smooth functions \((N_k,Q_k)(x,t)\), \(k=1,2\), constructed in 21 have the following properties:
\(\frac{\partial N_k}{\partial x}<0\),\(\frac{\partial Q_k}{\partial x}<0\),\(\frac{\partial N_k}{\partial t}<0\),\(\frac{\partial Q_k}{\partial t}<0\),\(\forall \;x\in \mathbb{R},\;t>0.\)
For any \(p\in[0,+\infty]\), there exists \(\;C_p>0\) such that \[\label{Ux} \begin{align} &\left\|\frac{\partial N_k}{\partial x}\right\|_{L^p}, \; \left\|\frac{\partial Q_k}{\partial x}\right\|_{L^p} \leq C_p \min\{\varepsilon^{1-1/p},\; t^{-1+1/p}\},\forall \;t> 0, \\ &\left\|\frac{\partial N_k}{\partial x}\right\|_{L^\infty},\; \left\|\frac{\partial Q_k}{\partial x}\right\|_{L^\infty} \leq C_p \varepsilon, \forall \;t> 0. \end{align}\qquad{(9)}\]
For any \(p\in[0,+\infty]\), there exists \(\;C_p>0\) such that \[\label{Ul} \begin{align} \left\|\frac{\partial^2 N_k}{\partial x^2}\right\|_{L^p},\; \left\|\frac{\partial^2 Q_k}{\partial x^2}\right\|_{L^p} \leq C_p \min\{\varepsilon^{2-1/p},\;t^{-2+1/p}\}, \forall \;t> 0, \end{align}\qquad{(10)}\] and \[\label{Nxx} \begin{align} \int_{0}^{t} \left\|\frac{\partial^2 N_k}{\partial x^2}\right\|_{L^p}d\tau,\;\int_{0}^{t} \left\|\frac{\partial^2 Q_k}{\partial x^2}\right\|_{L^p}d\tau \leq C_p \varepsilon^{1-1/p}, \forall \;t> 0. \end{align}\qquad{(11)}\]
\(|N_{kt}|\leq C|N_{kx}|\), \(|Q_{kt}|\leq C|Q_{kx}|\), \(|N_{kt}|\leq C|Q_{kx}|\) on \(\mathbb{R}\times[0,\infty)\), where \(C>0\) is a constant independent of \((x,t)\).
Proof. We only investigate the scenario for \(k=1\) case, since the analysis for \(k=2\) case is similar. From 9 and 21 , we get \[\begin{align} \nabla \lambda_1\cdot (N_{1x}\;Q_{1x})^T=w_{1x},\; \nabla h_1\cdot (N_{1x}\;Q_{1x})^T=0. \end{align}\] Hence, \[\label{NxQx} \begin{align} N_{1x}=-\frac{h_{1q}}{h_{1n}}Q_{1x},\; (\lambda_{1q}-\lambda_{1n} \frac{h_{1q}}{h_{1n}})Q_{1x}=w_{1x}. \end{align}\tag{23}\] A direct calculation yields \[\begin{align} \frac{h_{1q}}{h_{1n}}=- \frac{\sqrt{q^2+4n}+q}{2}<0,\;\lambda_{1q}-\lambda_{1n} \frac{h_{1q}}{h_{1n}}= -1-\frac{q}{\sqrt{q^2+4n}}<0, \end{align}\] which along with 23 and Lemma 1-(1) implies that \[\frac{\partial Q_1}{\partial x}<0, \;\frac{\partial N_1}{\partial x}<0.\] Then it follows from Lemma 2-(1) that \(\frac{\partial Q_1}{\partial t}= \frac{\partial N_1}{\partial x}<0\). To determine the sign of \(\frac{\partial N_1}{\partial t}\), we take the partial derivative of 9 and 21 with respect to \(t\), which leads to \[\begin{align} \nabla \lambda_1\cdot (N_{1t}\;Q_{1t})^T=w_{1t},\; \nabla h_1\cdot (N_{1t}\;Q_{1t})^T=0. \end{align}\] This directly yields that \[\begin{align}\label{NtQt} N_{1t}=-\frac{h_{1q}}{h_{1n}} Q_{1t}, \end{align}\tag{24}\] which along with \(\frac{h_{1q}}{h_{1n}}<0\) and \(\frac{\partial Q_1}{\partial t}<0\) implies \(\frac{\partial N_1}{\partial t}<0\). This completes the proof of (1).
It follows from 23 that \[\begin{align} \frac{\partial N_1}{\partial x}= -\frac{\sqrt{Q_1^2+4N_1}}{2}w_{1x},\; \frac{\partial Q_1}{\partial x}= -\frac{\sqrt{Q_1^2+4N_1}}{\sqrt{Q_1^2+4N_1}+Q_1}w_{1x}. \end{align}\] Then by Lemma 1-(2), we get the estimate ?? of (2).
A straightforward calculation yields \[\begin{align} \frac{\partial^2 N_1}{\partial x^2}&=- \left(\frac{\sqrt{Q_1^2+4N_1}}{2}\right)_x w_{1x} -\frac{\sqrt{Q_1^2+4N_1}}{2}w_{1xx}\\ &=f_1(N_1,Q_1)w_{1x}^2 -\frac{\sqrt{Q_1^2+4N_1}}{2}w_{1xx},\\ \frac{\partial^2 Q_1}{\partial x^2}&= -\left(\frac{\sqrt{Q_1^2+4N_1}}{\sqrt{Q_1^2+4N_1}+Q_1}\right)_x w_{1_x}-\frac{\sqrt{Q_1^2+4N_1}}{\sqrt{Q_1^2+4N_1}+Q_1}w_{1_{xx}} \\&=f_2(N_1,Q_1)w_{1x}^2 -\frac{\sqrt{Q_1^2+4N_1}}{\sqrt{Q_1^2+4N_1}+Q_1}w_{1xx}, \end{align}\] where \(f_1(N_1,Q_1)\) and \(f_2(N_1,Q_1)\) are defined by \[\begin{align} f_1(N_1,Q_1)\triangleq\frac{Q_1+1}{2(\sqrt{Q_1^2+4N_1}+Q_1)},\; f_2(N_1,Q_1)\triangleq\frac{Q_1-4N_1}{(\sqrt{Q_1^2+4N_1}+Q_1)^3}. \end{align}\] It then follows that \[\begin{align}\label{w1x} \left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p}, \;\left\|\frac{\partial^2 Q_1}{\partial x^2}\right\|_{L^p} &\leq C (\left\|w_{1x}\right\|_{L^{2p}}^2 +\left\|w_{1xx}\right\|_{L^p})\\ &\leq C_{p,\theta}\min\{\varepsilon^{2-1/p}, \;t^{-2+1/p},\; \varepsilon^{(1-\frac{1}{2\theta})(1-\frac{1}{p})} t^{-1-\frac{p-1}{2p\theta}}\}. \end{align}\tag{25}\] If \(t\leq\varepsilon^{-1}\triangleq t_0\), one has \(\varepsilon^{2-1/p}\leq t^{-2+1/p}\) and \(\varepsilon^{2-1/p}\leq \varepsilon^{(1-\frac{1}{2\theta})(1-\frac{1}{p})} t^{-1-\frac{p-1}{2p\theta}}\). It then follows from 25 that \[\label{t60t0} \begin{align} \left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p},\;\left\|\frac{\partial^2 Q_1}{\partial x^2}\right\|_{L^p}\leq C_p\varepsilon^{2-1/p} \;\text{ for } t\leq t_0, \end{align}\tag{26}\] and \[\label{t60t0Nxx} \begin{align} &\int_{0}^{t} \left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p}d\tau \leq\int_{0}^{t_0}\left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p}d\tau \leq C_p\varepsilon^{2-1/p}t_0 \leq C_p\varepsilon^{1-1/p} \;\text{ for } t\leq t_0,\\ &\int_{0}^{t} \left\|\frac{\partial^2 Q_1}{\partial x^2}\right\|_{L^p}d\tau \leq\int_{0}^{t_0} \left\|\frac{\partial^2 Q_1}{\partial x^2}\right\|_{L^p}d\tau \leq C_p\varepsilon^{2-1/p}t_0 \leq C_p\varepsilon^{1-1/p} \;\text{ for } t\leq t_0. \end{align}\tag{27}\] If \(t>t_0\), i.e. \(t^{-1}<\varepsilon\), we have \(t^{-2+1/p}< \varepsilon^{2-1/p}\) and \(t^{-2+1/p}< \varepsilon^{(1-\frac{1}{2\theta})(1-\frac{1}{p})} t^{-1-\frac{p-1}{2p\theta}}\), which implies \[\label{t62t0} \begin{align} \left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p}\leq C_pt^{-2+1/p},\;\left\|\frac{\partial^2 Q_1}{\partial x^2}\right\|_{L^p} &\leq C_pt^{-2+1/p} \;\text{ for } t>t_0, \end{align}\tag{28}\] and \[\label{t62t0Nxx} \begin{align} \int_{0}^{t} \left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p}d\tau &=\int_{0}^{t_0}\left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p}d\tau +\int_{t_0}^{t}\left\|\frac{\partial^2 N_1}{\partial x^2}\right\|_{L^p}d\tau\\ &\leq C_p\varepsilon^{2-1/p}t_0 +C_p \int_{t_0}^{t}\tau^{-2+1/p}d\tau\\ &\leq C_p\varepsilon^{2-1/p}t_0 +C_p \varepsilon^{3/4-1/p}t_0^{-1/4}\\ &\leq C_p\varepsilon^{1-1/p} \;\text{ for } t>t_0. \end{align}\tag{29}\] Similarly, we have \[\label{t62t0Qxx} \begin{align} \int_{0}^{t} \left\|\frac{\partial^2 Q_1}{\partial x^2}\right\|_{L^p}d\tau\leq C_p\varepsilon^{1-1/p} \;\text{ for } t>t_0.\end{align}\tag{30}\] Hence, ?? follows from 26 and 28 , and ?? follows from 27 , 29 and 30 .
We finally prove (4). Substituting \(\frac{h_{1q}}{h_{1n}}=- \frac{\sqrt{q^2+4n}+q}{2}\) into 24 , and noting that \(Q_{1t}-N_{1x}=0\), we get \[\label{N1tN1x} \begin{align} N_{1t}=\frac{\sqrt{Q_1^2+4N_1}+Q_1}{2}N_{1x}. \end{align}\tag{31}\] Moreover, by 23 and 31 , we have \[\label{Q1t} \begin{align} &Q_{1t}=N_{1x} =\frac{\sqrt{Q_1^2+4N_1}+Q_1}{2}Q_{1x},\\ &N_{1t}=\left(\frac{h_{1q}}{h_{1n}}\right)^2 Q_{1x}=\frac{(\sqrt{Q_1^2+4N_1}+Q_1)^2}{4}Q_{1x}. \end{align}\tag{32}\] Therefore, the last property of \((N_k,Q_k)(x,t)\) holds. ◻
We proceed to construct smooth approximations for the composite wave. For the composite rarefaction wave \((n^r, q^r)(x,t)\) constructed in 12 , its smooth approximation \((N,Q)(x,t)\) can be defined by \[\label{N44Q} \begin{align} (N,Q)(x,t)=(N_1,Q_1)(x,t)+(N_2,Q_2)(x,t) -(\bar{n},\bar{q}). \end{align}\tag{33}\] Here \((N_i,Q_i)(x,t)\) \((i=1,2)\) are defined by the following relations \[\label{Ni44Qi} \begin{align} &(N_1,Q_1)(x,t)\in R_1(n_-,q_-),\quad & \lambda_1((N_1,Q_1)(x,t))=W_1(x,t);\\ &(N_2,Q_2)(x,t)\in R_2(\bar{n},\bar{q}),\quad & \lambda_2((N_2,Q_2)(x,t))=W_2(x,t), \end{align}\tag{34}\] and \(W_i(x,t)\) \((i=1,2)\) are the solutions of the following initial value problems for the inviscid Burgers equation, \[\label{smooth39} \left\{\begin{align} &W_{it}+W_iW_{ix}=0,\\ &W_i(x,0)=W_{i0}(x), \end{align} \right.\tag{35}\] with \[\label{wi0} \begin{align} &W_{1 0}(x)=\frac{\lambda_1(\bar{n},\bar{q}) +\lambda_1(n_-,q_-)}{2} +\frac{\lambda_1(\bar{n},\bar{q})-\lambda_1(n_-,q_-)}{2} \kappa_\theta\int^{\varepsilon x}_0(1+y^2)^{-\theta}dy,\\ &W_{2 0}(x)=\frac{\lambda_2(n_+,q_+) +\lambda_2(\bar{n},\bar{q})}{2} +\frac{\lambda_2(n_+,q_+)-\lambda_2(\bar{n},\bar{q})}{2} \kappa_\theta\int^{\varepsilon x}_0(1+y^2)^{-\theta}dy, \end{align}\tag{36}\] where \(\varepsilon>0\) is a small parameter and \(\kappa_\theta\) is the constant such that \(\kappa_\theta\int^\infty_0(1+y^2)^{-\theta}dy=1\) for \(\theta>\frac{3}{2}\).
As in the single-mode case, by using the implicit function theorem and the characteristic method, one can easily show that 34 gives smooth functions \[\label{Ni} \begin{align} (N_i,Q_i)(x,t)=(N_i,Q_i)(x_{i 0}(x,t)),\;\;\;\;i=1,2, \end{align}\tag{37}\] where \[\label{x94i} \begin{align} x=x_{i 0}(x,t)+W_{i 0}(x_{i 0}(x,t))t,\;\;\;\;i=1,2. \end{align}\tag{38}\] By Lemma 2, one can see that \((N_i,Q_i)(x,t)\) satisfy the system 6 and \[\label{limNi} \lim\limits_{t\rightarrow \infty} \sup\limits_{x\in \mathbb{R}} \left|(N_i,Q_i)(x,t)-(n_i^r,q_i^r)(x,t)\right|=0 ,\;\;\;\;i=1,2.\tag{39}\] Thus, the smooth approximation \((N,Q)\) satisfies \[\label{N39Q39} \left\{ \begin{align} &N_{t} -(NQ)_{x}=-g(N,Q)_x,\\ &Q_{t}-N_{x}=0, \end{align} \right.\tag{40}\] and \[\label{limN} \lim\limits_{t\rightarrow \infty} \sup\limits_{x\in \mathbb{R}} \left|(N,Q)(x,t)-(n^r,q^r)(x,t)\right|=0,\tag{41}\] where \(g(N,Q)=NQ-N_1Q_1-N_2Q_2\).
Analogous to Lemma 3, we have the following estimates for \((N,Q)\).
Lemma 4. The smooth function \((N,Q)(x,t)\) constructed in 33 has the following properties:
\(\frac{\partial N}{\partial x}<0\),\(\frac{\partial Q}{\partial x}<0\),\(\frac{\partial N}{\partial t}<0\),\(\frac{\partial Q}{\partial t}<0\),\(\forall \;x\in \mathbb{R},\;t>0.\)
\(\forall\) \(p\in[0,+\infty]\), \(\exists \;C_p>0\) such that \[\label{Ux39} \begin{align} &\left\|\frac{\partial N}{\partial x}\right\|_{L^p},\; \left\|\frac{\partial Q}{\partial x}\right\|_{L^p} \leq C_p \min\{\varepsilon^{1-1/p},\;t^{-1+1/p}\}, \;\forall \;t> 0,\\ &\left\|\frac{\partial N}{\partial x}\right\|_{L^\infty},\; \left\|\frac{\partial Q}{\partial x}\right\|_{L^\infty} \leq C_p \varepsilon, \;\forall \;t> 0. \end{align}\qquad{(12)}\]
\(\forall\) \(p\in[0,+\infty]\), \(\exists \;C_p>0\) such that \[\label{Ul39} \begin{align} \left\|\frac{\partial^2 N}{\partial x^2}\right\|_{L^p},\; \left\|\frac{\partial^2 Q}{\partial x^2}\right\|_{L^p} \leq C_p \min\{\varepsilon^{2-1/p},\;t^{-2+1/p}\}, \;\forall \;t> 0, \end{align}\qquad{(13)}\] and \[\label{N39xx} \begin{align} \int_{0}^{t} \left\|\frac{\partial^2 N}{\partial x^2}\right\|_{L^p}d\tau,\;\int_{0}^{t} \left\|\frac{\partial^2 Q}{\partial x^2}\right\|_{L^p}d\tau \leq C_p \varepsilon^{1-1/p}, \;\forall \;t> 0. \end{align}\qquad{(14)}\]
\(|N_t|\leq C|N_x|\), \(|Q_t|\leq C|Q_x|\), \(|N_t|\leq C|Q_x|\) on \(\mathbb{R}\times[0,\infty)\) where \(C>0\) is a constant independent of \((x,t)\).
\(\forall\) \(p\in[0,+\infty]\), \(\exists \;C_{p\theta}>0\) such that \[\label{gx} \begin{align} \left\|g(N,Q)_x\right\|_{L^p} \leq C_{p\theta}\varepsilon^{2-1/p} (1+(\varepsilon t)^2)^{-\theta/3}, \;\forall \;t> 0, \end{align}\qquad{(15)}\] and \[\label{tgx} \begin{align} \int_{0}^{\infty} \left\|g(N,Q)_x\right\|_{L^p} \leq C_p\varepsilon^{1-1/p}, \;\forall \;t> 0. \end{align}\qquad{(16)}\]
Proof. We only prove the property (5). A direct calculation yields \[\label{gx39} \begin{align} g(N,Q)_x=N_{1x}(Q_2-\bar{q}) +N_{2x}(Q_1-\bar{q}) +Q_{1x}(N_2-\bar{n}) +Q_{2x}(N_1-\bar{n}). \end{align}\tag{42}\] Owing to 34 , there exist smooth functions \(f_i\;(i=1,2)\) and \(z_i\;(i=1,2)\) such that \(N_i=f_i(W_i)\) and \(Q_i=z_i(W_i)\). Then we have \[\label{gx60} \begin{align} |g(N,Q)_x|&\leq C\left(| N_{1x}||W_2-\lambda_2(\bar{n},\bar{q})| +|N_{2x}||W_1-\lambda_1(\bar{n},\bar{q})|\right.\\ &\quad \left.+|Q_{1x}||W_2-\lambda_2(\bar{n},\bar{q})| +|Q_{2x}||W_1-\lambda_1(\bar{n},\bar{q})|\right)\\ &\leq C\left(| W_{1x}||W_2-\lambda_2(\bar{n},\bar{q})| +|W_{2x}||W_1-\lambda_1(\bar{n},\bar{q})|\right), \end{align}\tag{43}\] where we have used 23 in the last inequality. Hence, it follows from Lemma 1-(3) that for \(x\leq 0\), \[\label{w2x} \begin{align} \left|W_2(x,t)-\lambda_2(\bar{n},\bar{q})\right| &\leq C_\theta (1+(\varepsilon x)^2)^{-\theta/3} [1+(\varepsilon \lambda_2(\bar{n},\bar{q})t)^2]^{-\theta/3},\\ \left|W_{2x}(x, t)\right| &\leq C_\theta \varepsilon(1+(\varepsilon x)^2)^{-\theta/2} [1+(\varepsilon \lambda_2(\bar{n},\bar{q})t)^2]^{-\theta/2}. \end{align}\tag{44}\] In the same way, owing to \(\lambda_1(\bar{n},\bar{q})<0\), and by Lemma 1-(4), we have for \(x\geq0\), \[\label{W1x} \begin{align} \left|W_1(x,t)-W_{1+}\right| &\leq C_\theta (1+(\varepsilon x)^2)^{-\theta/3} [1+(\varepsilon \lambda_1(\bar{n},\bar{q})t)^2]^{-\theta/3},\\ \left|W_{1x}(x, t)\right| &\leq C_\theta \varepsilon(1+(\varepsilon x)^2)^{-\theta/2} [1+(\varepsilon \lambda_1(\bar{n},\bar{q})t)^2]^{-\theta/2}. \end{align}\tag{45}\] Substituting 44 and 45 into 43 leads to \[\begin{align} \left\|g(N,Q)_x\right\|_{L^p} \leq &C_{p\theta}\varepsilon^{2-1/p} {[1+(\varepsilon \lambda_2(\bar{n},\bar{q})t)^2]^{-\theta/3} +[1+(\varepsilon \lambda_1(\bar{n},\bar{q})t)^2]^{-\theta/3}}\\ \leq &C_{p\theta}\varepsilon^{2-1/p} (1+(\varepsilon t)^2)^{-\theta/3}, \end{align}\] and \[\begin{align} \int_{0}^{\infty} \left\|g(N,Q)_x\right\|_{L^p} \leq C_{p\theta}\int_{0}^{\infty} \varepsilon^{2-1/p} (1+(\varepsilon t)^2)^{-\theta/3}d\tau \leq C_p\varepsilon^{1-1/p}. \end{align}\] We complete the proof. ◻
In this section, we investigate the stability of single rarefaction waves for the system 1 2 and prove Theorem 1. In what follows, we denote \(\|\cdot\|_k:=\|\cdot\|_{H^k(\mathbb{R})}\) and \(\|\cdot\|:=\|\cdot\|_{L^2(\mathbb{R})}\).
We decompose \((n,q)\) as \[\label{decompose} (n,q)=(N_k+\phi,Q_k+\psi).\tag{46}\] Then by 1 and 22 , one can see that \((\phi,\psi)\) satisfies \[\label{rewrite} \left\{ \begin{align} &\phi_{t} -(\phi\psi+\phi Q_k+N_k\psi)_{x}=\phi_{xx}+N_{kxx},\\ &\psi_{t}-\phi_{x}=0, \end{align} \right.\tag{47}\] with initial value \[\label{phi044psi0} \begin{align} (\phi_0,\psi_0)(x):=(\phi,\psi)(x,0)=(n_{k0}-N_k(x,0),q_{k0}-Q_k(x,0)). \end{align}\tag{48}\] We look for solutions of the system 47 in the space \[\label{solution32space} \begin{align} X(0,T):=\{(\phi,\psi)|(\phi,\psi)\in C^0([0,T];H^1), \phi_x\in L^2((0,T);H^1), \psi_x\in L^2((0,T);L^2), \end{align}\tag{49}\] for \(T\in(0,+\infty]\). Set \[\label{assumption} E(t):=\sup\limits_{0\leq \tau\leq t}\|(\phi,\psi)(\cdot,\tau)\|_1.\tag{50}\] By the Sobolev embedding theorem, we have \[\label{3466} \sup\limits_{0\leq \tau\leq t}(\|\phi(\cdot,\tau)\|_{L^\infty}+\|\psi(\cdot,\tau)\|_{L^\infty})\leq CE(t).\tag{51}\] We have the following global well-posedness of solutions to the system 47 48 .
****Proposition** 1**. Let \(n_+>0\). Suppose that \((\phi_0,\psi_0)\in H^1(\mathbb{R})\). Then there exists a constant \(\delta_{1}>0\) such that if \(E(0)\leq \delta_{1}\), then the system 47 48 has a unique global solution \((\phi,\psi)\in X(0,\infty)\) satisfying \[\label{priori32estimates} \begin{align} \left\|(\phi,\psi)(\cdot,t) \right\|_1^2 +&\int_{0}^{t} \Big(\||Q_{kx}|^{1/2}\phi(\cdot,\tau) \|^{2} +\||N_{kt}|^{1/2}\psi(\cdot,\tau) \|^{2}\\&+\left\|\phi_x(\cdot,\tau) \right\|_1^{2}+\left\|\psi_x(\cdot,\tau) \right\|^{2}\Big)d\tau\leq C_0 (\|(\phi_0,\psi_0)\|_1^2+\varepsilon), \end{align}\qquad{(17)}\] for any \(t\in [0,\infty)\), where \(\varepsilon>0\) is the small constant given in 18 . Moreover, \((\phi,\psi)(x,t)\) has the following asymptotic behavior \[\label{asymptotic} \sup _{x \in \mathbb{R}}|(\phi,\psi)(x, t)| \rightarrow 0 \text{ as } t \rightarrow+\infty.\qquad{(18)}\]
The global existence of \((\phi,\psi)\) can be proved by the local existence result and the a priori estimate given below.
****Proposition** 2** (Local existence). Let \(n_+>0\). For any \(\Xi_0>0\), if the initial data \((\phi_0,\psi_0)\) satisfy \(\|(\phi_0,\psi_0)\|_1\leq \Xi_0\), then there exists a positive constant \(T_0\) depending on \(\Xi_0\), such that the system 47 48 has a unique solution \((\phi,\psi)\) in \(X(0, T_0)\) satisfying \[\label{local32estimates} \begin{align} \left\|(\phi,\psi)(\cdot,t) \right\|_1^2 +&\int_{0}^{t} \Big(\||Q_{kx}|^{1/2}\phi(\cdot,\tau) \|^{2} +\||N_{kt}|^{1/2}\psi(\cdot,\tau) \|^{2}\\&+\left\|\phi_x(\cdot,\tau) \right\|_1^{2}+\left\|\psi_x(\cdot,\tau) \right\|^{2}\Big)d\tau\leq 4 \|(\phi_0,\psi_0)\|_1^2, \;\;\forall \;t \in [0, T_0]. \end{align}\qquad{(19)}\]
The local existence in Proposition 2 can be shown by a standard iteration method and we omit the detail. To extend the local solution globally, it suffices to establish the following a priori estimate.
****Proposition** 3** (A priori estimate). Let \(n_+>0\). Suppose that the system 47 48 has a solution \((\phi,\psi)\in X(0,T)\) for some \(T>0\). Then there exist positive constants \(\chi_0\leq 1\) and \(C_0\) independent of \(T\) such that if \[E(T)\leq \chi_0,\] then the estimate ?? holds for any \(t\in[0,T]\).
Before proving Proposition 3, we first present an inequality of Gronwall’s type.
Lemma 5. Suppose \(F(t)\) is a continuously differentiable function on \([0,T]\), \(\beta(t)\geq0\) is integrable on \(\mathbb{R}^+\). If \(F(t)\) satisfies \[\label{24645} \begin{align} \left(F'(t)\right)^2 \leq\beta^2(t) F(t) \text{ and } F(0)>0,\;t\in [0,T], \end{align}\qquad{(20)}\] then the following estimate holds: \[\label{Gronwall39} \begin{align} F(t) \leq 2 F(0)+\frac{1}{2}\left(\int_{0}^{t}\beta(\tau)d\tau\right)^2 \text{ for all }t\in [0,T]. \end{align}\qquad{(21)}\]
Proof. Solving ?? yields \[F^{1/2}(t)\leq F^{1/2}(0)+\frac{1}{2}\int_{0}^{t}\beta(\tau)d\tau.\] Taking a square of this inequality, one obtains the desired estimate ?? . ◻
We now derive the \(L^2\) energy estimate.
Lemma 6. Let the assumptions of Proposition 3 hold. There exist two constants \(C>0\) and \(\chi_0>0\) such that if \(E(T)\leq\chi_0\), then \[\label{L2} \begin{align} &\left\|\phi(\cdot,t) \right\|^2 +\left\|\psi(\cdot,t) \right\|^2+\int_{0}^{t} \Big(\||Q_{kx}|^{1/2}\phi(\cdot,\tau) \|^{2} +\||N_{kt}|^{1/2}\psi(\cdot,\tau) \|^{2}+\left\|\phi_x(\cdot,\tau) \right\|^{2}\Big)\\ &\leq C(\|\phi_0\|^2 +\|\psi_0\|^2 +\varepsilon) \;\text{ for any } t\in[0,T]. \end{align}\qquad{(22)}\]
Proof. Multiplying the first equation of 47 by \(\phi\) and the second one by \(\psi N_k\), summing them up and integrating the result over \(\mathbb{R}\times[0,t]\), we have by integration by parts, \[\label{phi243psi2} \begin{align} &\left.\int_{\mathbb{R}}\left(\frac{\phi^2}{2} +\frac{\psi^2}{2} N_k \right)dx\right|_0^t -\frac{1}{2}\int_0^t\int_{\mathbb{R}}Q_{kx}\phi^2dxd\tau -\frac{1}{2}\int_0^t\int_{\mathbb{R}}N_{kt}\psi^2dxd\tau +\int_0^t\int_{\mathbb{R}}\phi_x^2dxd\tau\\ &=\int_0^t\int_{\mathbb{R}}\phi N_{kxx}dxd\tau +\int_0^t\int_{\mathbb{R}}\frac{\phi^2}{2} \psi_xdxd\tau. \end{align}\tag{52}\] To estimate the last term on the right hand side (RHS) of 52 , we rewrite the first equation of 47 as \[\label{psix} \begin{align} \psi_x=(N_k+\phi)^{-1}[\phi_t- (Q_k+\psi)\phi_{x}-N_{kx}\psi-\phi Q_{kx} -\phi_{xx}-N_{kxx}], \end{align}\tag{53}\] which implies \[\label{4465} \begin{align} \int_0^t\int_{\mathbb{R}}\frac{\phi^2}{2} \psi_x =\int_0^t\int_{\mathbb{R}}(N_k+\phi)^{-1} \frac{\phi^2}{2}[\phi_t- Q_k\phi_{x}-\psi\phi_{x}-(N_{kx}\psi+\phi Q_{kx}) -\phi_{xx}-N_{kxx}]. \end{align}\tag{54}\]
We next estimate each term on the RHS of 54 . Integrating by parts gives \[\label{34612} \begin{align} \int_0^t\int_{\mathbb{R}} \frac{\phi^2}{2}(N_k+\phi)^{-1}\phi_t =\int_{\mathbb{R}} \left.(N_k+\phi)^{-1}\frac{\phi^3}{6}\right| _{\tau=0}^{\tau=t}dx -\int_0^t\int_{\mathbb{R}} \partial_t\left((N_k+\phi)^{-1}\right) \frac{\phi^3}{6}. \end{align}\tag{55}\] By 51 and the fact that \(\frac{\partial N_k}{\partial x}<0\), we get \[N_k(x,t)+\phi(x,t)\geq\frac{n_+}{2} \;\text{ if } E(t)\ll1,\] which implies the first term on the RHS of 54 satisfies \[\label{I1461} \begin{align} \left|\int_{\mathbb{R}} \left.(N_k+\phi)^{-1}\frac{\phi^3}{6}\right| _{\tau=0}^{\tau=t}dx\right| \leq CE(t)\|\phi(t)\|^2+CE(0)\|\phi_0\|^2. \end{align}\tag{56}\] A direct calculation yields \[\label{I1462} \begin{align} \left|\int_0^t\int_{\mathbb{R}} \partial_t\left((N_k+\phi)^{-1}\right) \frac{\phi^3}{6}\right| \leq C \int_0^t\int_{\mathbb{R}}|N_{kt}||\phi|^3 +\left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}\phi_t\frac{\phi^3}{6}\right|. \end{align}\tag{57}\] By Lemma 3-(4) and 51 , we have \[\label{I1462461} \begin{align} \int_0^t\int_{\mathbb{R}}|N_{kt}||\phi|^3dxd\tau \leq CE(t)\int_0^t\int_{\mathbb{R}} |Q_{kx}|\phi^2dxd\tau. \end{align}\tag{58}\] Owing to the first equation of 47 , we get \[\label{I1462462} \begin{align} \left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}\phi_t\frac{\phi^3}{6}\right| =&\left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}N_{kxx} \frac{\phi^3}{6}\right|\\ &+\left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}(\phi\psi+\phi Q_k+N_k\psi+\phi_x)_{x} \frac{\phi^3}{6}\right|. \end{align}\tag{59}\] It follows from 51 that \[\label{I1462462463} \begin{align} \left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}N_{kxx} \frac{\phi^3}{6}\right| \leq CE(t)\int_0^t\int_{\mathbb{R}}|\phi N_{kxx}|dxd\tau. \end{align}\tag{60}\] Integrating by parts gives rise to \[\begin{align} \label{I146246246139} &\left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}(\phi\psi+\phi Q_k+N_k\psi+\phi_x)_{x} \frac{\phi^3}{6}\right| \nonumber\\ &=\left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-3}(N_{kx}+\phi_x) (\phi\psi+\phi Q_k+N_k\psi+\phi_x) \frac{\phi^3}{3}\right.\nonumber\\ &\quad \left.+\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}(\phi\psi+\phi Q_k+N_k\psi+\phi_x) \frac{\phi^2}{2}\phi_x\right|\\ &\leq C\int_0^t\int_{\mathbb{R}} \left|N_{kx}(\phi\psi+\phi Q_k+N_k\psi+\phi) \phi^3\right| +C\int_0^t\int_{\mathbb{R}} \left|\phi_x\phi^4\psi+\phi_x\phi^4Q_k+ \phi_xN_k\psi\phi^3\right|\nonumber\\ &\quad+C\int_0^t\int_{\mathbb{R}} \left|\phi_x\phi^3\psi+\phi_x\phi^3Q_k+ \phi_xN_k\psi\phi^2+\phi_x^2\phi^2+\phi_x^2\phi^3 \right|.\nonumber \end{align}\tag{61}\] By Lemma 3-(4) and 51 again, we get \[\label{I1462462461461} \begin{align} \int_0^t\int_{\mathbb{R}} \left|N_{kx}(\phi\psi+\phi Q_k+N_k\psi+\phi_x) \phi^3\right| \leq CE(t) \int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2 +CE(t)\int_0^t\|\phi_x(\tau)\|^2d\tau. \end{align}\tag{62}\] By Hölder’s inequality, Sobolev inequality \(\|\phi\|_{L^\infty}\leq \|\phi\|^{1/2}\|\phi_x\|^{1/2}\) and 51 , the second term on the RHS of 61 can be estimated as \[\label{I1462462461462} \begin{align} \int_0^t\int_{\mathbb{R}} \left|\phi_x\phi^4\psi+\phi_x\phi^4Q_k+ \phi_xN_k\psi\phi^3\right| &\leq CE(t)\int_0^t\int_{\mathbb{R}} \left|\phi_x\phi^3\right|\\ &\leq CE(t)\int_0^t \|\phi\|^2_{L^\infty}\|\phi\|\|\phi_x\|\\ &\leq CE(t)\int_0^t \|\phi\|^2\|\phi_x\|^2\\ &\leq CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau, \end{align}\tag{63}\] Similarly, it holds that \[\begin{align} \label{I1462462461463} &\int_0^t\int_{\mathbb{R}} \left|\phi_x\phi^3\psi+\phi_x\phi^3Q_k+ \phi_xN_k\psi\phi^2+\phi_x^2\phi^2+\phi_x^2\phi^3 \right| \nonumber\\ &\leq CE(t)\int_0^t\int_{\mathbb{R}} \left|\phi_x\phi^3\right|+ C\int_0^t\int_{\mathbb{R}} \left|\phi_x\phi^3\right|+ C\int_0^t\int_{\mathbb{R}} \left|\phi_x\psi\phi^2\right| +CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau\nonumber\\ &\leq CE(t)\int_0^t \|\phi\|^2_{L^\infty}\|\phi\|\|\phi_x\| +C\int_0^t \|\phi\|^2_{L^\infty}\|\psi\|\|\phi_x\| +CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau\\ &\leq CE(t)\int_0^t \|\phi\|^2\|\phi_x\|^2 +C\int_0^t \|\phi\|\|\psi\|\|\phi_x\|^2 +CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau \nonumber\\ &\leq CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau. \nonumber \end{align}\tag{64}\] Substituting 62 64 into 61 , and combing the result with 60 , we have \[\label{I1462462461} \begin{align} \left|\int_0^t\int_{\mathbb{R}} (N_k+\phi)^{-2}\phi_t\frac{\phi^3}{6}\right|\leq CE(t) \int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2 +CE(t)\int_0^t\|\phi_x(\tau)\|^2d\tau. \end{align}\tag{65}\] Then substituting 58 and 65 into 57 , we arrive at \[\begin{align} \left|\int_0^t\int_{\mathbb{R}} \partial_t\left((N_k+\phi)^{-1}\right) \frac{\phi^3}{6}\right| \leq CE(t) \int_0^t\int_{\mathbb{R}}(|Q_{kx}|\phi^2 +\phi_x^2+|\phi N_{kxx}|), \end{align}\] which along with 56 and 55 gives \[\label{I1} \begin{align} \int_0^t\int_{\mathbb{R}} \frac{\phi^2}{2}(N_k+\phi)^{-1}\phi_t \leq &CE(t)\|\phi(t)\|^2+CE(0)\|\phi_0\|^2 +CE(t) \int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2dxd\tau\\ &+CE(t)\int_0^t\|\phi_x(\tau)\|^2 +CE(t)\int_0^t\int_{\mathbb{R}}|\phi N_{kxx}|dxd\tau. \end{align}\tag{66}\] After integration by parts, by Lemma 3-(4), one finds that \[\label{I2} \begin{align} -\int_0^t\int_{\mathbb{R}}(N_k+\phi)^{-1} \frac{\phi^2}{2} Q_k\phi_{x}dxd\tau &=\int_0^t\int_{\mathbb{R}} \partial_x \left((N_k+\phi)^{-1}Q_k\right) \frac{\phi^3}{6}dxd\tau\\ &\leq C \int_0^t\int_{\mathbb{R}}|Q_{kx}| |\phi|^3dxd\tau +C\int_0^t\int_{\mathbb{R}} |\phi_x||\phi|^3dxd\tau\\ &\leq C \int_0^t\int_{\mathbb{R}}|Q_{kx}| |\phi|^3dxd\tau +C\int_0^t \|\phi\|^2_{L^\infty}\|\phi\|\|\phi_x\|d\tau\\ &\leq C \int_0^t\int_{\mathbb{R}}|Q_{kx}| |\phi|^3dxd\tau +C\int_0^t \|\phi\|^2\|\phi_x\|^2d\tau\\ &\leq CE(t) \int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2dxd\tau +CE(t)\int_0^t\|\phi_x(\tau)\|^2 d\tau, \end{align}\tag{67}\] where we have used Sobolev inequality in the third inequality and 51 in the last inequality. Noting \(N_k(x,t)+\phi(x,t)>\frac{n_+}{2}\) when \(E(t)\ll1\), we have \[\label{I3} \begin{align} -\int_0^t\int_{\mathbb{R}}(N_k+\phi)^{-1} \frac{\phi^2}{2} \psi\phi_{x}dxd\tau &\leq C \int_0^t\int_{\mathbb{R}} \left|\phi^2\psi\phi_x\right|dxd\tau\\ &\leq C\int_0^t \|\phi\|^2_{L^\infty}\|\psi\|\|\phi_x\|\\ &\leq C\int_0^t \|\phi\|\|\psi\|\|\phi_x\|^2\\ &\leq CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau, \end{align}\tag{68}\] where we have used 51 in the last inequality. By Lemma 3-(4) again, one has \[\label{I4} \begin{align} -\int_0^t\int_{\mathbb{R}}(N_k+\phi)^{-1} \frac{\phi^2}{2} (N_{kx}\psi+\phi Q_{kx})dxd\tau&\leq C\int_0^t\int_{\mathbb{R}} |Q_{kx}|(|\phi|+|\psi|) \frac{|\phi|^2}{2}dxd\tau\\&\leq CE(t) \int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2dxd\tau. \end{align}\tag{69}\] A straightforward calculation by integration by parts yields \[\label{I5} \begin{align} -\int_0^t\int_{\mathbb{R}}(N_k+\phi)^{-1} \frac{\phi^2}{2}\phi_{xx}dxd\tau &=\int_0^t\int_{\mathbb{R}} \partial_x \left((N_k+\phi)^{-1}\frac{\phi^2}{2}\right) \phi_xdxd\tau\\ &\leq C\int_0^t\int_{\mathbb{R}} \left(|\phi|^2|\phi_x||N_{kx}| + \phi^2|\phi_x|^2 + |\phi||\phi_x|^2\right)dxd\tau\\ &\leq CE(t) \int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2dxd\tau +CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau, \end{align}\tag{70}\] where we have used \(|N_{kt}|\leq C|Q_{kx}|\) from Lemma 3-(4). By the fact that \(N_k(x,t)+\phi(x,t)>\frac{n_+}{2}\) and 51 , the following estimate holds \[\label{I6} \begin{align} -\int_0^t\int_{\mathbb{R}}(N_k+\phi)^{-1} \frac{\phi^2}{2}N_{kxx}\leq C\int_0^t\int_{\mathbb{R}} |\phi|^2|N_{kxx}| \leq CE(t)\int_0^t\int_{\mathbb{R}}|\phi N_{kxx}|. \end{align}\tag{71}\] Substituting 66 71 into 53 leads to \[\label{phi232psix} \begin{align} \left|\int_0^t\int_{\mathbb{R}}\frac{\phi^2}{2} \psi_xdxd\tau\right| \leq &CE(t)\|\phi(t)\|^2+CE(0)\|\phi_0\|^2 +CE(t)\int_0^t\int_{\mathbb{R}}|\phi N_{kxx}|dxd\tau\\ &+CE(t)\int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2dxd\tau +CE(t) \int_0^t\|\phi_x(\tau)\|^2d\tau. \end{align}\tag{72}\] Combining 52 and 72 , we have \[\label{4461} \begin{align} &\|\phi(\cdot,t)\|^2 +\|\psi(\cdot,t)\|^2 +\int_0^t\int_{\mathbb{R}}|Q_{kx}|\phi^2+\int_0^t\int_{\mathbb{R}}|N_{kt}|\psi^2+\int_0^t\|\phi_x(\tau)\|^2\\ &\leq \|\phi_0\|^2 +\|\psi_0\|^2 +C\int_0^t\int_{\mathbb{R}}|\phi N_{kxx}|dxd\tau\\ &\leq \|\phi_0\|^2 +\|\psi_0\|^2 +C\int_0^t \|\phi(\cdot,\tau)\|\|N_{kxx}(\cdot,\tau)\|d\tau, \end{align}\tag{73}\] provided that \(E(t)\) is suitably small. Now set \[\label{34635} \begin{align} F(t)\triangleq \|\phi_0\|^2 +\|\psi_0\|^2 +C\int_0^t \|\phi(\cdot,\tau)\|\|N_{kxx}(\cdot,\tau)\|d\tau, \;\beta(t)\triangleq C\|N_{kxx}(\cdot,t)\|. \end{align}\tag{74}\] It follows from 73 that \[(F'(t))^2\leq\beta^2(t)F(t), \;\forall t\in[0,T].\] Therefore, by ?? and Lemma 5, we obtain the estimate ?? and complete the proof. ◻
We next derive the estimate for \(\|\psi_x\|\) .
Lemma 7. Let the assumptions of Proposition 3 hold. There exist two constants \(C>0\) and \(\chi_0>0\) such that if \(E(T)\leq\chi_0\), then \[\label{H1461} \begin{align} \|\psi_x(\cdot,t)\|^2 +\int_0^t\|\psi_x(\cdot,\tau)\|^2 \leq &C(\|\phi_0\|^2 +\|\psi_0\|_1^2 +\varepsilon) \;\text{ for any } t\in[0,T]. \end{align}\qquad{(23)}\]
Proof. Substituting \(\psi_t=\phi_x\) into the first equation of 47 yields \[\label{psitx} \begin{align} \psi_{xt}=\phi_t -(\phi\psi+\phi Q_k+N_k\psi)_x-N_{kxx}. \end{align}\tag{75}\] Multiplying 75 by \(\psi_x\) and integrating the result over \(\mathbb{R}\times[0,t]\), noting \[\begin{align} \phi_t\psi_x&=(\phi\psi_x)_t-\phi\psi_{xt} =(\phi\psi_x)_t-\phi\phi_{xx} =(\phi\psi_x)_t-(\phi\phi_x)_x+\phi_x^2, \end{align}\] we have \[\label{psix2} \begin{align} &\left.\left(\int_{\mathbb{R}}\frac{\psi_x^2}{2} - \phi\psi_x dx\right)\right|_0^t +\int_0^t\int_{\mathbb{R}} N_k\psi_x^2dxd\tau\\ =&\int_0^t\int_{\mathbb{R}}\phi_x^2dxd\tau -\int_0^t\int_{\mathbb{R}}\psi_{x}(\phi Q_k)_{x} dxd\tau -\int_0^t\int_{\mathbb{R}}\psi_{x}(\phi\psi)_{x} dxd\tau\\ &+\int_0^t\int_{\mathbb{R}}\frac{\psi^2}{2}N_{kxx} dxd\tau -\int_0^t\int_{\mathbb{R}}\psi_{x}N_{kxx} dxd\tau. \end{align}\tag{76}\] We next estimate each term on the RHS of 76 . By Young’s inequality, one has \[\label{J2} \begin{align} -\int_0^t\int_{\mathbb{R}}\psi_x(\phi Q_k)_x dxd\tau =&-\int_0^t\int_{\mathbb{R}}\psi_x\phi_xQ_kdxd\tau -\int_0^t\int_{\mathbb{R}}\psi_x\phi Q_{kx}dxd\tau\\ \leq &\frac{1}{2}\int_0^t\int_{\mathbb{R}} |Q_k||\psi_x|^2dxd\tau +\frac{1}{2}\int_0^t\int_{\mathbb{R}} |Q_k||\phi_x|^2dxd\tau\\ &+\frac{1}{4}\int_0^t\int_{\mathbb{R}} |Q_{kx}|\phi^2dxd\tau +C_p \varepsilon \int_0^t\|\psi_x(\tau)\|^2d\tau, \end{align}\tag{77}\] where we have used \(\left\|N_{kx}\right\|_{L^\infty} \leq C_p \varepsilon\) from ?? . Using Young’s inequality and 51 , we have \[\label{J339} \begin{align} -\int_0^t\int_{\mathbb{R}}\psi_{x}(\phi \psi)_{x} dxd\tau &=-\int_0^t\int_{\mathbb{R}}\psi_{x}\phi_{x}\psi dxd\tau-\int_0^t\int_{\mathbb{R}}\phi\psi_{x}^2 dxd\tau\\&\leq CE(t) \int_0^t \left(\|\psi_x\|^{2} +\|\phi_x\|^2\right)d\tau+CE(t) \int_0^t\|\psi_x\|^2d\tau. \end{align}\tag{78}\] Thanks to the Sobolev inequality and 51 , we get \[\label{J439} \begin{align} \int_0^t\int_{\mathbb{R}}\frac{\psi^2}{2}N_{kxx} dxd\tau &\leq C\int_0^t \|\psi\|\|\psi_x\|\|N_{kxx}\|_{L_1} d\tau\\ &\leq CE\int_0^t\|\psi_x(\tau)\|^2d\tau +CE\int_0^t\|N_{kxx}\|_{L_1}^2d\tau. \end{align}\tag{79}\] From Lemma 3-(3), one has \(\|N_{kxx}\|_{L_1}^2\leq C_p \min\{\varepsilon^2,\;t^{-2}\}\). If \(t<\varepsilon^{-1}\triangleq t_0\), it then follows that \[\label{t60t039} \begin{align} \int_0^t\|N_{kxx}\|_{L_1}^2d\tau \leq \int_0^{t_0}\|N_{kxx}\|_{L_1}^2d\tau \leq C_p \varepsilon^2t_0 \leq C_p\varepsilon. \end{align}\tag{80}\] If \(t>t_0\), i.e. \(t^{-1}<\varepsilon\), the inequality \(\|N_{kxx}\|_{L_1}^2\leq C_p \;t^{-2}\) holds, which implies \[\label{t62t03939} \begin{align} \int_0^t\|N_{kxx}\|_{L_1}^2d\tau &=\int_0^{t_0}\|N_{kxx}\|_{L_1}^2d\tau +\int_{t_0}^t\|N_{kxx}\|_{L_1}^2d\tau\\ &\leq C_p \varepsilon^2t_0 +C_p\int_{t_0}^t\tau^{-2}d\tau \leq C_p\varepsilon. \end{align}\tag{81}\] A combination of 80 and 81 yields \[\label{NkxxL1} \begin{align} \int_0^t\|N_{kxx}\|_{L_1}^2d\tau \leq C_p\varepsilon. \end{align}\tag{82}\] Substituting 82 into the RHS of 79 leads to \[\label{J4} \begin{align} \int_0^t\int_{\mathbb{R}}\frac{\psi^2}{2}N_{kxx} dxd\tau &\leq CE(t) \int_0^t\|\psi_x(\tau)\|^2d\tau +C E(t)\varepsilon. \end{align}\tag{83}\]
Therefore, combining the estimates 77 , 78 , 83 and Lemma 6, we arrive at \[\label{psix239} \begin{align} \|\psi_x(\cdot,t)\|^2 +\int_0^t\|\psi_x(\cdot,\tau)\|^2 \leq &C(\|\phi_0\|^2 +\|\psi_0\|^2+\|\psi_{0x}\|^2) +C E\varepsilon +\int_0^t\int_{\mathbb{R}} |\psi_{x}N_{kxx}|, \end{align}\tag{84}\] provided that \(E(t)\) is suitably small. As in 74 , by Lemma 5 and ?? , we obtain the estimate ?? . ◻
We are now ready to establish the estimate for \(\|\phi_x\|\).
Lemma 8. Let the assumptions of Proposition 3 hold. There exist two constants \(C>0\) and \(\chi_0>0\) such that if \(E(T)\leq\chi_0\), then for any \(t\in[0,T]\) it holds that \[\label{H1462} \begin{align} \|\phi_x(\cdot,t)\|^2 +\int_0^t\|\phi_{xx}(\cdot,\tau)\|^2 \leq &C(\|\phi_0\|_1^2 +\|\psi_0\|_1^2 +\varepsilon). \end{align}\qquad{(24)}\]
Proof. Multiplying the first equation of 47 by \(-\phi_{xx}\) and integrating the result over \(\mathbb{R}\times[0,t]\), we have \[\label{phixx} \begin{align} \left.\int_{\mathbb{R}}\frac{\phi_x^2}{2} dx\right|_0^t +\int_0^t\int_{\mathbb{R}} \phi_{xx}^2 =&-\int_0^t\int_{\mathbb{R}} \phi_{xx}\left[(\phi\psi+\phi Q_k+N_k\psi)_{x} +N_{kxx}\right]\\ \leq &\frac{1}{2}\int_0^t\int_{\mathbb{R}} \phi_{xx}^2 +C\int_0^t\int_{\mathbb{R}} (|Q_k||\phi_x|^2+|Q_{kx}|^2|\phi|^2+|N_{kx}|^2|\psi|^2 \\&\quad+|N_k||\psi_x|^2 +|\phi_x|^2|\psi|^2+|\phi|^2|\psi_x|^2 ) +\frac{1}{2}\int_0^t\int_{\mathbb{R}} |N_{kxx}|^2. \end{align}\tag{85}\] By the Sobolev inequality, Young’s inequality and 51 , we get \[\label{K3} \begin{align} \int_0^t\int_{\mathbb{R}} |N_{kx}|^2|\psi|^2dxd\tau \leq &C\int_0^t \|\psi\|\|\psi_x\| \|N_{kx}\|^2d\tau\\ \leq &C\int_0^t\|\phi\| \left(\|\psi_x\|^2+\|N_{kx}\|^4\right)d\tau\\ \leq &CE(t)\int_0^t\|\psi_x\|^2d\tau +CE(t)\int_0^t\min(\varepsilon^2, \tau^{-2})d\tau\\ \leq &CE(t) \int_0^t \|\psi_x\|^2d\tau+CE(t)\varepsilon, \end{align}\tag{86}\] where we have used ?? of Lemma 3 in the third inequality. By Hölder’s inequality, we further have \[\label{K5} \begin{align} \int_0^t\int_{\mathbb{R}} |\phi_x|^2|\psi|^2dxd\tau \leq C \|\psi\|_{L^\infty}^2 \int_0^t\int_{\mathbb{R}} |\phi_x|^2d\tau \leq CE^2\int_0^t \|\phi_x\|^2d\tau, \end{align}\tag{87}\] and \[\label{K6} \begin{align} \int_0^t\int_{\mathbb{R}} |\phi|^2|\psi_x|^2dxd\tau \leq C \|\phi\|_{L^\infty}^2 \int_0^t\int_{\mathbb{R}} |\psi_x|^2d\tau \leq CE^2\int_0^t \|\psi_x\|^2d\tau. \end{align}\tag{88}\] Owing to ?? , it holds \[\label{K7} \begin{align} \frac{1}{2}\int_0^t\int_{\mathbb{R}} |N_{kxx}|^2dxd\tau \leq C\int_0^t \min\{\varepsilon^3, \varepsilon^{1/2}\tau^{-5/2}\}d\tau \leq C\varepsilon^2. \end{align}\tag{89}\] Now substituting 86 89 into 85 , by Lemmas 6 and 7, we obtain the desired estimate ?? . ◻
We are now in a position to prove Proposition 1.
Proof of Proposition 1. We define \[\label{deta0} \begin{align} \Xi_0:=\frac{\chi_0}{4}, \;\delta_1:= \frac{1}{4}\sqrt{\frac{\chi_0^2}{C_0}-16\varepsilon}, \end{align}\tag{90}\] where \(\chi_0\leq1\) and \(C_0>1\) are constants given in Proposition 3, and \(\varepsilon\) is suitably small.
Step 1. Since \(\|\phi_0,\psi_0\|_1 \leq \delta_1\leq\frac{\chi_0}{4}\), by the local existence result established in Proposition 2, there is a positive constant \(T_0=T_0(\chi_0)\) such that the system 47 48 has a unique solution on \([0,T_0]\), which satisfies \[\label{34652} \|(\phi,\psi)\|_1\leq 2\|\phi_0,\psi_0\|_1 \leq \frac{\chi_0}{2}<\chi_0 \text{ for } t \in [0,T_0].\tag{91}\] Applying the a priori estimate established in Proposition 3 with \(T=T_0\), we get from 90 that \[\begin{align} \|(\phi,\psi)(T_0)\|_1\leq \sqrt{C_0} \sqrt{\|\phi_0,\psi_0\|_1^2+\varepsilon} \leq \sqrt{C_0}\sqrt{\delta_1^2+\varepsilon} \leq \frac{\chi_0}{4}=\Xi_0. \end{align}\]
Step 2. Taking \(t=T_0\) as the new initial time and applying Proposition 2, one can see that the system 47 48 has a unique solution on \([T_0,2T_0]\), which satisfies \[\begin{align} \|(\phi,\psi)\|_1\leq 2\|(\phi,\psi)(T_0)\|_1 \leq \frac{\chi_0}{2} < \chi_0 \;\text{for} \;t \in [T_0,2T_0]. \end{align}\] This along with 91 gives rise to \(\|(\phi,\psi)\|_1 \leq \chi_0\) for \(t \in [0,2T_0]\). Hence, applying Proposition 3 with \(T=2T_0\), we get \[\begin{align} \|(\phi,\psi)\|_1\leq \sqrt{C_0} \sqrt{\|\phi_0,\psi_0\|_1^2+\varepsilon} \text{ for }t \in [0, 2T_0]. \end{align}\] Now it follows from 90 that \[\begin{align} \|(\phi,\psi)(2T_0)\|_1\leq \sqrt{C_0} \sqrt{\|\phi_0,\psi_0\|_1^2+\varepsilon} \leq \sqrt{C_0}\sqrt{\delta_0^2+\varepsilon} \leq \frac{\chi_0}{4}=\Xi_0. \end{align}\] When \(\chi_0\ll 1\), noting \(n_+>0\), \(N_k+\phi\) has a lower bound \[N_k+\phi>\frac{n_+}{2} \;\text{ for } t\in[0,2T_0],\] which ensures the feasibility of successive extensions. Then by repeating this continuation process, we can extend the solution to the whole time interval \([0, \infty)\) successively. Moreover, the solution satisfies \[\label{global} \begin{align} &\sup_{0\leq \tau< \infty} \left\|(\phi,\psi)(\tau,\cdot) \right\|_1^2 +\int_{0}^{\infty} \left(\left\||Q_{kx}|^{1/2}\phi(\tau) \right\|^{2} +\left\||N_{k\tau}|^{1/2}\psi(\tau) \right\|^{2}+\left\|\phi_x(\tau) \right\|_1^{2}+\left\|\psi_x(\tau) \right\|^{2}\right)d\tau\\ &\leq C (\|(\phi_0,\psi_0)\|_1^2+\varepsilon). \end{align}\tag{92}\]
Step 3. It remains to show the \(L^\infty\) convergence ?? . We first deduce from 92 that \[\label{phix243psix2} \begin{align} \int_{0}^{+\infty}\int_{\mathbb{R}} \left(\phi_x^2+\psi_x^2\right)dx \leq C (\|(\phi_0,\psi_0)\|_1^2+\varepsilon) <+\infty. \end{align}\tag{93}\] We next show \(\int_{0}^{+\infty} \left|\frac{d}{dt}\int_{\mathbb{R}} \left(\phi_x^2+\psi_x^2\right)dx\right|d\tau <+\infty.\) A straightforward calculation yields \[\label{phix2t43psix2t} \begin{align} \int_{0}^{+\infty} \left|\frac{d}{dt}\int_{\mathbb{R}} \left(\phi_x^2+\psi_x^2\right)dx\right|d\tau &=2\int_{0}^{+\infty}\left|\int_{\mathbb{R}} \phi_x\phi_{xt}+\psi_x\psi_{xt}dx\right| d\tau\\ &\leq C\int_{0}^{+\infty}\left|\int_{\mathbb{R}} \phi_x\phi_{xt}dx\right|d\tau +C\int_{0}^{+\infty}\left|\int_{\mathbb{R}} \psi_x\psi_{xt}dx\right|d\tau. \end{align}\tag{94}\] Integration by parts and using the first equation of 47 , we have \[\begin{align} \int_{0}^{+\infty}\left|\int_{\mathbb{R}} \phi_x\phi_{xt}dx\right|d\tau &=\int_{0}^{+\infty}\left|\int_{\mathbb{R}} \phi_{xx}\phi_tdx\right|d\tau\\ &=\int_{0}^{+\infty}\left|\int_{\mathbb{R}} \phi_{xx}(\phi\psi+\phi Q_k+N_k\psi)_{x} +\phi_{xx}^2+\phi_{xx}N_{kxx}dx\right|d\tau, \end{align}\] which, in combination with 51 , leads to \[\label{phix2t} \begin{align} \int_{0}^{+\infty}\left|\int_{\mathbb{R}} \phi_x\phi_{xt}\right|d\tau &\leq CE \int_{0}^{+\infty}\int \left(|Q_{kx}|\phi^2+|N_{kt}|\psi^{2}+\phi_x^{2}+\psi_x^{2}\right)\\ &\quad+C\int_{0}^{+\infty} \left\|\phi_{xx}(\tau) \right\|^{2}d\tau +\frac{1}{2}\int_{0}^{+\infty} \left\|N_{kxx}\right\|^{2}d\tau\\ &\leq C (\|(\phi_0,\psi_0)\|_1^2+\varepsilon), \end{align}\tag{95}\] where we have used \(\int_{0}^{+\infty} \left\|N_{kxx}\right\|^{2}d\tau\leq C\varepsilon^3\) from ?? . By the second equation of 47 , we get \[\label{psix2t} \begin{align} \int_{0}^{+\infty}\left|\int_{\mathbb{R}} \psi_x\psi_{xt}dx\right|d\tau &=\int_{0}^{+\infty}\left|\int_{\mathbb{R}} \psi_x\phi_{xx}dx\right|d\tau\\ &\leq C\int_{0}^{+\infty} \left\|\psi_x(\tau)\right\|^{2} +\left\|\phi_{xx}(\tau) \right\|^{2} d\tau\\ &\leq C (\|(\phi_0,\psi_0)\|_1^2+\varepsilon). \end{align}\tag{96}\] Substituting 95 and 96 into 94 leads to \[\begin{align} \int_{0}^{+\infty} \left|\frac{d}{dt}\int_{\mathbb{R}} \left(\phi_x^2+\psi_x^2\right)dx\right|d\tau <\infty. \end{align}\] This together with 93 gives rise to \[\begin{align} \int_{0}^{+\infty}\left[\int_{\mathbb{R}} \left(\phi_x^2+\psi_x^2\right)dx+ \left|\frac{d}{dt}\int_{\mathbb{R}} \left(\phi_x^2+\psi_x^2\right)dx\right|\right]d\tau <+\infty. \end{align}\] Hence \[\lim\limits_{t\rightarrow \infty} \left\|(\phi_x,\psi_x)(\cdot,t)\right\|^2=0.\] It then follows from the Sobolev inequality that \[\label{Sobolev39s32inequality} \begin{align} \sup_{x\in \mathbb{R}} \left|(\phi,\psi)(x,t)\right|^2 &\leq 2 (\left\|\phi(\cdot,t)\right\| \left\|\phi_x(\cdot,t)\right\|+\left\|\psi(\cdot,t)\right\| \left\|\psi_x(\cdot,t)\right\|)\\& \leq C(\left\|\phi_x(\cdot,t)\right\|+\left\|\psi_x(\cdot,t)\right\|)\rightarrow0 \text{ as } t\rightarrow\infty. \end{align}\tag{97}\] This completes the proof of Proposition 1. ◻
Proof of Theorem 1. Recalling \((n-N_k,q-Q_k)=(\phi,\psi)\), by Proposition 1, one can see that if the initial value \((n_0,q_0)(x)\) satisfies ?? , then the system 1 2 has a unique global solution satisfying ?? . Furthermore, by Lemma 2-(2) and 97 , \((n,q)\) has the asymptotic behavior ?? . We complete the proof of Theorem 1. ◻
In this section, we study the asymptotic stability of composite rarefaction waves and prove Theorem 2. By decomposing \((n,q)=(N+\Phi,Q+\Psi)\), where \((N,Q)\) is the smooth approximation of the composite rarefaction wave constructed in Section 2, we get the following equation of \((\Phi,\Psi)\) \[\label{rewrite39} \left\{ \begin{align} &\Phi_{t} -(\Phi\Psi+\Phi Q+N\Psi)_{x}=\Phi_{xx}+N_{xx}+g(N,Q)_x,\\ &\Psi_{t}-\Phi_{x}=0, \end{align} \right.\tag{98}\] with initial value \[\label{Phi044Psi0} \begin{align} (\Phi,\Psi)(x,0):=(\Phi_0,\Psi_0)(x) =(n_0-N(x,0),q_0-Q(x,0))\in H^1. \end{align}\tag{99}\] As in the single wave case, we also search for solutions of 98 99 in the function space \(X(0,+\infty)\).
****Proposition** 4** (Global existence). Let \((n_+,q_+)\in RR(n_-,q_-)\) with \(n_+>0\). There exists a constant \(\delta_1>0\) such that if \(\|(\Phi_0,\Psi_0)\|_1\leq \delta_1\), then the system 98 99 has a unique global solution \((\Phi,\Psi)\in X(0,+\infty)\) satisfying \[\label{global32estimates} \begin{align} \sup_{0\leq \tau\leq t} \left\|(\Phi,\Psi)(\cdot,t) \right\|_1^2 +\int_{0}^{t} &\Big(\left\||Q_{x}|^{1/2}\Phi(\cdot,\tau) \right\|^{2} +\left\||N_{t}|^{1/2}\Psi(\cdot,\tau) \right\|^{2}\\&+\left\|\Phi_x(\cdot,\tau) \right\|_1^{2}+\left\|\Psi_x(\cdot,\tau) \right\|^{2}\Big)d\tau\leq C (\|(\Phi_0,\Psi_0)\|_1^2+\varepsilon) \end{align}\qquad{(25)}\] for any \(t >0\).
As in Proposition 2, the local existence of solutions to the system 98 99 is standard. To prove Proposition ?? we only need to establish the following a priori estimate.
****Proposition** 5** (A priori estimate). Let \((n_+,q_+)\in RR(n_-,q_-)\) with \(n_+>0\). Suppose that the system 98 99 has a solution \((\Phi,\Psi)\in X(0,T)\) for some \(T>0\). Then there exists a positive constants \(\chi_1\ll 1\) independent of \(T\) such that if \[\label{assumption39} E(t):=\sup\limits_{0\leq t\leq T}\|(\Phi,\Psi)(\cdot,t)\|_1 \leq \chi_1,\qquad{(26)}\] then the estimate ?? holds for any \(t\in[0,T]\).
To prove Proposition 5, we first derive the \(L^2\) estimate.
Lemma 9. Let the assumptions of Proposition 5 hold. If \(\chi_1\ll1\), then there exists a constant \(C>0\) such that \[\label{L239} \begin{align} &\|\Phi(\cdot,t)\|^2 +\|\Psi(\cdot,t)\|^2 +\int_0^t\int_{\mathbb{R}}|Q_{x}|\phi^2dxd\tau +\int_0^t\int_{\mathbb{R}}|N_{t}|\psi^2dxd\tau +\int_0^t\|\Phi_x(\cdot,\tau)\|^2d\tau\\ &\leq C(\|\Phi_0\|^2 +\|\Psi_0\|^2 +\varepsilon) \;\text{ for }t\in[0,T]. \end{align}\qquad{(27)}\]
Proof. We multiply the first equation of 98 by \(\Phi\) and the second one by \(\Psi N\), sum them up and integrate it over \(\mathbb{R}\times [0,t]\) to have \[\label{Phi243Psi2} \begin{align} &\left.\left(\int_{\mathbb{R}}\frac{\Phi^2}{2} +\frac{\Psi^2}{2} Ndx\right)\right|_0^t -\frac{1}{2}\int_0^t\int_{\mathbb{R}}Q_{x}\Phi^2dxd\tau -\frac{1}{2}\int_0^t\int_{\mathbb{R}}N_{t}\Psi^2dxd\tau +\int_0^t\int_{\mathbb{R}}\Phi_x^2dxd\tau\\ &=\int_0^t\int_{\mathbb{R}}\Phi N_{xx}dxd\tau +\int_0^t\int_{\mathbb{R}}\Phi g(N,Q)_xdxd\tau +\int_0^t\int_{\mathbb{R}}\frac{\Phi^2}{2} \Psi_xdxd\tau. \end{align}\tag{100}\] To estimate the last term of 100 , we rewrite the first equation of 98 as \[\label{Psix} \begin{align} \Psi_x=(N+\Phi)^{-1}[\Phi_t- (Q+\Psi)\Phi_{x}-N_x\Psi-\Phi Q_x -\Phi_{xx}-N_{xx}-g(N,Q)_x]. \end{align}\tag{101}\] It then follows that \[\label{Psix39} \begin{align} \int_0^t\int_{\mathbb{R}}\frac{\Phi^2}{2} \Psi_xdxd\tau &=\int_0^t\int_{\mathbb{R}}(N+\Phi)^{-1} \frac{\Phi^2}{2}\left[\Phi_t- Q\Phi_{x}-\Psi\Phi_{x}-(N_x\Psi+\Phi Q_x) -\Phi_{xx}\right.\\ &~~~~\left.-N_{xx}-g(N,Q)_x\right]dxd\tau. \end{align}\tag{102}\]
We next estimate each term on the RHS of 102 . Integrating by parts gives \[\label{4469} \begin{align} \int_0^t\int_{\mathbb{R}} \frac{\Phi^2}{2}(N+\Phi)^{-1}\Phi_t =\int_{\mathbb{R}} \left.(N+\Phi)^{-1}\frac{\Phi^3}{6}\right| _{\tau=0}^{\tau=t}dx -\int_0^t\int_{\mathbb{R}} \partial_t\left((N+\Phi)^{-1}\right) \frac{\Phi^3}{6}. \end{align}\tag{103}\] As in 51 , by ?? and the Sobolev embedding theorem, we have \[\label{44610} \sup\limits_{0\leq \tau\leq t}(\|\Phi(\cdot,\tau)\|_{L^\infty}+\|\Psi(\cdot,\tau)\|_{L^\infty})\leq CE(t).\tag{104}\] By the fact that \(\frac{\partial N}{\partial x}<0\), we get \[N(x,t)+\Phi(x,t)>\frac{n_+}{2}, \text{ if } E(t)\ll1.\] It then follows \[\label{J1461} \begin{align} \left|\int_{\mathbb{R}} \left.(N+\Phi)^{-1}\frac{\Phi^3}{6}\right| _{\tau=0}^{\tau=t}dx\right| \leq CE(t)\|\Phi(t)\|^2+CE(0)\|\Phi_0\|^2. \end{align}\tag{105}\] A direct calculation yields \[\label{J1462} \begin{align} \left|\int_0^t\int_{\mathbb{R}} \partial_t\left((N+\Phi)^{-1}\right) \frac{\Phi^3}{6}\right| \leq C \int_0^t\int_{\mathbb{R}}|N_t||\Phi|^3 +\left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}\Phi_t\frac{\Phi^3}{6}\right|. \end{align}\tag{106}\] Owing to Lemma 4-(4), it holds that \[\label{J1462461} \begin{align} \int_0^t\int_{\mathbb{R}}|N_t||\Phi|^3dxd\tau \leq CE(t)\int_0^t\int_{\mathbb{R}} |Q_x|\Phi^2dxd\tau. \end{align}\tag{107}\] Using the first equation of 98 , we get \[\label{J1462462} \begin{align} \left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}\Phi_t\frac{\Phi^3}{6}\right| \leq&\left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}(\Phi\Psi+\Phi Q+N\Psi+\Phi_x)_{x} \frac{\Phi^3}{6}\right|\\ &+\left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}(N_{xx}+g(N,Q)_x) \frac{\phi^3}{6}\right|. \end{align}\tag{108}\] By 51 , we derive \[\label{J1462462463} \begin{align} \left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}(N_{xx}+g(N,Q)_x) \frac{\phi^3}{6}\right| \leq CE(t)\int_0^t\int_{\mathbb{R}} |\Phi (N_{xx}+g(N,Q)_x)|dxd\tau. \end{align}\tag{109}\] Integrating by parts gives rise to \[\label{J146246246139} \begin{align} &\left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}(\Phi\Psi+\Phi Q+N\Psi+\Phi_x)_{x} \frac{\Phi^3}{6}\right|\\ &=\left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-3}(N_x+\Phi_x) (\Phi\Psi+\Phi Q+N\Psi+\Phi_x) \frac{\Phi^3}{3}\right.\\ &~~~\left.+\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}(\Phi\Psi+\Phi Q+N\Psi+\Phi_x) \frac{\Phi^2}{2}\Phi_x\right|\\ &\leq C\int_0^t\int_{\mathbb{R}} \left|N_x(\Phi\Psi+\Phi Q+N\Psi+\Phi) \Phi^3\right| +C\int_0^t\int_{\mathbb{R}} \left|\Phi_x\Phi^4\Psi+\Phi_x\Phi^4Q+ \Phi_xN\Psi\Phi^3\right|\\ &~~~+C\int_0^t\int_{\mathbb{R}} \left|\Phi_x\Phi^3\Psi+\Phi_x\Phi^3Q+ \Phi_xN\Psi\Phi^2+\Phi_x^2\Phi^2+\Phi_x^2\Phi^3 \right|. \end{align}\tag{110}\] By Lemma 4-(4), 51 and Young’s inequality, we have \[\label{J1462462461461} \begin{align} \int_0^t\int_{\mathbb{R}} \left|N_x(\Phi\Psi+\Phi Q+N\Psi+\Phi_x) \Phi^3\right| \leq CE(t) \int_0^t\int_{\mathbb{R}}(|Q_x|\Phi^2 +\Phi_x^2). \end{align}\tag{111}\] As in 63 64 , by the Sobolev inequality we get \[\label{J1462462461462} \begin{align} \int_0^t\int_{\mathbb{R}} \left|\Phi_x\Phi^4\Psi+\Phi_x\Phi^4Q+ \Phi_xN\Psi\Phi^3\right| \leq CE(t) \int_0^t\int \Phi_x^2, \end{align}\tag{112}\] and \[\label{J1462462461463} \begin{align} \int_0^t\int_{\mathbb{R}} \left|\Phi_x\Phi^3\Psi+\Phi_x\Phi^3Q+ \Phi_xN\Psi\Phi^2+\Phi_x^2\Phi^2+\Phi_x^2\Phi^3 \right| \leq CE(t) \int_0^t\int \Phi_x^2. \end{align}\tag{113}\] Substituting 111 113 into 110 and combining the result with 109 , we obtain \[\begin{align} \left|\int_0^t\int_{\mathbb{R}} (N+\Phi)^{-2}\Phi_t\frac{\Phi^3}{6}\right| \leq CE(t) \int_0^t\int(|Q_x|\Phi^2 +\Phi_x^2+|\Phi (N_{xx}+g(N,Q)_x)|). \end{align}\] This together with 103 107 leads to \[\label{J1} \begin{align} \int_0^t\int_{\mathbb{R}} \frac{\Phi^2}{2}(N+\Phi)^{-1}\Phi_t \leq &CE(t)\|\Phi(t)\|^2+CE(0)\|\Phi_0\|^2 +CE(t) \int_0^t\int(|Q_x|\Phi^2 +\Phi_x^2)\\ &+CE(t)\int_0^t\int|\Phi (N_{xx}+g(N,Q)_x)|. \end{align}\tag{114}\] Integrating by parts, by Lemma 4-(4), we find \[\label{J239} \begin{align} -\int_0^t\int_{\mathbb{R}}(N+\Phi)^{-1} \frac{\Phi^2}{2} Q\Phi_{x}dxd\tau &=\int_0^t\int_{\mathbb{R}} \partial_x \left((N+\Phi)^{-1}Q\right) \frac{\Phi^3}{6}dxd\tau\\ &\leq C \int_0^t\int_{\mathbb{R}}|Q_x| |\Phi|^3dxd\tau +C\int_0^t \|\Phi\|^2_{L^\infty}\|\Phi\|\|\Phi_x\|d\tau\\ &\leq C \int_0^t\int_{\mathbb{R}}|Q_x| |\Phi|^3dxd\tau +C\int_0^t \|\Phi\|^2\|\Phi_x\|^2d\tau\\ &\leq CE(t) \int_0^t\int(|Q_x|\Phi^2 +\Phi_x^2), \end{align}\tag{115}\] where we have used Sobolev inequality in the second inequality. Similarly, by Lemma 4-(4), ?? and 51 , we arrive at \[\label{J3-J5} -\int_0^t\int_{\mathbb{R}}(N+\Phi)^{-1} \frac{\Phi^2}{2} \Psi\Phi_{x}dxd\tau \leq CE(t) \int_0^t\int\Phi_x^2,\tag{116}\] \[-\int_0^t\int_{\mathbb{R}}(N+\Phi)^{-1} \frac{\Phi^2}{2} (N_x\Psi+\Phi Q_x)dxd\tau \leq CE(t) \int_0^t\int_{\mathbb{R}}|Q_x|\Phi^2dxd\tau,\] and \[\begin{align} -\int_0^t\int_{\mathbb{R}}(N+\Phi)^{-1} \frac{\Phi^2}{2}\Phi_{xx}dxd\tau &=\int_0^t\int_{\mathbb{R}} \partial_x \left((N+\Phi)^{-1}\frac{\Phi^2}{2}\right) \Phi_xdxd\tau \\& \leq CE(t) \int_0^t\int_{\mathbb{R}}(|Q_x|\Phi^2+\Phi_x^2).\end{align}\] Noting \(N(x,t)+\Phi(x,t)>\frac{n_+}{2}\), by 51 , we have \[\label{J6-J7} \begin{align} -\int_0^t\int_{\mathbb{R}}(N+\Phi)^{-1} \frac{\Phi^2}{2}(N_{xx}+g(N,Q)_x)dxd\tau &\leq C\int_0^t\int_{\mathbb{R}} |\Phi|^2|(N_{xx}+g(N,Q)_x)| dxd\tau\\ &\leq CE(t)\int_0^t\int_{\mathbb{R}}|\Phi (N_{xx}+g(N,Q)_x)|dxd\tau. \end{align}\tag{117}\] Substituting 114 117 into 101 , one has \[\begin{align} \left|\int_0^t\int_{\mathbb{R}}\frac{\Phi^2}{2} \Psi_xdxd\tau\right| \leq &CE(t)\|\Phi(t)\|^2+CE(0)\|\Phi_0\|^2 +CE(t) \int_0^t\int_{\mathbb{R}}(|Q_x|\Phi^2+\Phi_x^2)\\ &+CE(t)\int_0^t\int_{\mathbb{R}}|\Phi (N_{xx}+g(N,Q)_x)|dxd\tau. \end{align}\] It then follows from 100 that \[\label{446139} \begin{align} &\|\Phi(t)\|^2 +\|\Psi(t)\|^2 +\int_0^t\int_{\mathbb{R}}|Q_{x}|\Phi^2+\int_0^t\int_{\mathbb{R}}|N_{t}|\psi^2+\int_0^t\|\Phi_x(\tau)\|^2\\ &\leq \|\Phi_0\|^2 +\|\Psi_0\|^2 +C\int_0^t\int_{\mathbb{R}}|\Phi N_{xx}|dxd\tau +C\int_0^t\int_{\mathbb{R}}|\Phi g(N,Q)_x|dxd\tau\\& \leq \|\Phi_0\|^2 +\|\psi_0\|^2 +C\int_0^t \|\Phi\|(\|N_{xx}\|+\|g(N,Q)_x\|)d\tau, \end{align}\tag{118}\] provided that \(E(t)\) is suitably small. Taking in Lemma 5 \[\begin{align} F(t)\triangleq \|\Phi_0\|^2 +\|\psi_0\|^2+C\int_0^t \|\Phi\|(\|N_{xx}\|+\|g(N,Q)_x\|)d\tau ,\;\beta(t)\triangleq \|N_{xx}\|+\|g(N,Q)_x\|, \end{align}\] by ?? and ?? , we obtain the estimate ?? . The proof is complete. ◻
We next derive the estimate for \(\|\Psi_x\|\) .
Lemma 10. Let the assumptions of Proposition 5 hold. If \(\chi_1\ll1\), there exists a constant \(C>0\) such that \[\label{H146139} \begin{align} \|\Psi_x(\cdot,t)\|^2 +\int_0^t\| \Psi_x(\cdot,\tau)\|^2 \leq &C(\|\Phi_0\|^2 +\|\Psi_0\|^2 +\|\Psi_{0x}\|^2 +\varepsilon) \;\text{ for } t\in[0,T]. \end{align}\qquad{(28)}\]
Proof. Substituting \(\Psi_t=\Phi_x\) into the first equation of 98 yields \[\label{Psitx} \begin{align} \Psi_{tx}=\Phi_t -(\Phi\Psi+\Phi Q+N\Psi)_x-N_{xx}-g(N,Q)_x. \end{align}\tag{119}\] Multiplying 119 by \(\Psi_x\), and integrating the result, noting \[\begin{align} \Phi_t\Psi_x&=(\Phi\Psi_x)_t-\Phi\Psi_{xt} =(\Phi\Psi_x)_t-\Phi\Phi_{xx} =(\Phi\Psi_x)_t-(\Phi\Phi_x)_x+\Phi_x^2, \end{align}\] we have by integration by parts, \[\label{Psix2} \begin{align} &\left.\left(\int_{\mathbb{R}}\frac{\Psi_x^2}{2} - \Phi\Psi_x dx\right)\right|_0^t +\int_0^t\int_{\mathbb{R}} N\Psi_x^2dxd\tau\\ =&\int_0^t\int_{\mathbb{R}}\Phi_x^2dxd\tau -\int_0^t\int_{\mathbb{R}}\Psi_{x}(\Phi Q)_{x} dxd\tau -\int_0^t\int_{\mathbb{R}}\psi_{x}(\Phi\Psi)_{x} dxd\tau\\ &+\int_0^t\int_{\mathbb{R}}\frac{\Psi^2}{2}N_{xx} dxd\tau -\int_0^t\int_{\mathbb{R}}\Psi_{x}N_{xx} dxd\tau -\int_0^t\int_{\mathbb{R}}\Psi_{x}g(N,Q)_x dxd\tau. \end{align}\tag{120}\] As in the proof of Lemma 7, we obtain \[\label{Psix239} \begin{align} &\|\Psi_x(t)\|^2 +\int_0^t\|\Psi_x(\tau)\|^2\\ &\leq C(\|\phi_0\|^2 +\|\Psi_0\|^2+\|\Psi_{0x}\|^2) +C E\varepsilon +\int_0^t\int_{\mathbb{R}} |\Psi_x(N_{xx}+g(N,Q)_x )| dxd\tau\\ &\leq C(\|\phi_0\|^2 +\|\Psi_0\|^2+\|\Psi_{0x}\|^2) +C \varepsilon +C\int_0^t \|\Psi_x\|(\|N_{xx}\|+\|g(N,Q)_x\|)dxd\tau, \end{align}\tag{121}\] provided that \(E(t)\ll1\) is suitably small. By Lemma 5, ?? and ?? , we obtain the estimate ?? . ◻
Finally, we establish the estimate for \(\|\Phi_x\|\) .
Lemma 11. Let the assumptions of Proposition 5 hold. If \(\chi_1\ll1\), then there exists a constant \(C>0\) such that \[\label{H391462} \begin{align} \|\Phi_x(\cdot,t)\|^2 +\int_0^t\|\Phi_{xx}(\cdot,\tau)\|^2 \leq &C(\|\Phi_0\|_1^2 +\|\Psi_0\|_1^2 +\varepsilon) \text{ for } t\in[0,T]. \end{align}\qquad{(29)}\]
Proof. Multiplying the first equation of 98 by \(-\Phi_{xx}\) and integrating the result, we have \[\label{Phixx} \begin{align} &\left.\int_{\mathbb{R}}\frac{\Phi_x^2}{2} dx\right|_0^t +\int_0^t\int_{\mathbb{R}} \Phi_{xx}^2dxd\tau\\ =&-\int_0^t\int_{\mathbb{R}} \Phi_{xx}\left[(\Phi\Psi+\Phi Q+N\Psi)_{x} +N_{xx}+g(N,Q)_x \right]dxd\tau\\ \leq &\frac{1}{2}\int_0^t\int_{\mathbb{R}} \Phi_{xx}^2 dxd\tau +\frac{1}{2}\int_0^t\int_{\mathbb{R}} |N_{xx}|^2dxd\tau +C\int_0^t\int_{\mathbb{R}} (|Q||\Phi_x|^2+|Q_{x}|^2|\Phi|^2+|N_{x}|^2|\Psi|^2 \\&\quad+|N||\Psi_x|^2 +|\Phi_x|^2|\Psi|^2+|\Phi|^2|\Psi_x|^2 )dxd\tau +\frac{1}{2}\int_0^t\int_{\mathbb{R}} |g(N,Q)_x|^2dxd\tau. \end{align}\tag{122}\] As in the proof of Lemma 8, we get \[\label{H} \begin{align} &\int_0^t\int_{\mathbb{R}} |Q_{x}|^2|\Phi|^2dxd\tau \leq C\int_{\mathbb{R}} |Q_{x}||\Phi|^2dxd\tau,\\ &\int_0^t\int_{\mathbb{R}} |N_{x}|^2|\Psi|^2dxd\tau \leq CE \int_0^t \|\Psi_x\|^2d\tau+CE\varepsilon,\\ &\int_0^t\int_{\mathbb{R}} |\Phi_x|^2|\Psi|^2dxd\tau \leq CE^2\int_0^t \|\Phi_x\|^2d\tau,\\ &\int_0^t\int_{\mathbb{R}} |\Phi|^2|\psi_x|^2dxd\tau \leq CE^2\int_0^t \|\Psi_x\|^2d\tau. \end{align}\tag{123}\] By ?? and \(\theta>\frac{3}{2}\), the last term of 122 satisfies \[\label{gx2} \begin{align} \frac{1}{2}\int_0^t\int_{\mathbb{R}} |g(N,Q)_x|^2dxd\tau \leq C_{p\theta}\int_0^t \varepsilon^3 (1+(\varepsilon \tau)^2)^{-2\theta/3}d\tau \leq C_{p\theta}\varepsilon^2. \end{align}\tag{124}\] Substituting 123 and 124 into 122 , by Lemmas 9 and 10, we obtain the desired estimate ?? and finish the proof of Lemma 11. ◻
Proof of Proposition 4. The a priori estimate ?? guarantees that \(E(t)\) is small for all \(t>0\) if \(E(0)\) is small enough. Thus, as in the proof of Proposition 1, applying the standard extension procedure, one can obtain the global well-posedness of system 98 99 in \(X(0,\infty)\). By the estimate ?? and the equation 98 , applying the same argument as that of Proposition 1, we have \[\begin{align} \int_{0}^{+\infty}\left[\int_{\mathbb{R}} \left(\Phi_x^2+\Psi_x^2\right)dx+ \left|\frac{d}{dt}\int_{\mathbb{R}} \left(\Phi_x^2+\Psi_x^2\right)dx\right|\right]d\tau <+\infty. \end{align}\] Hence \[\lim\limits_{t\rightarrow \infty} \left\|(\Phi_x,\Psi_x)(\cdot,t)\right\|^2=0.\] It then follows from the Sobolev inequality that \[\label{44635} \begin{align} \sup\limits_{x\in \mathbb{R}} \left|(\Phi,\Psi)(x,t)\right|^2&\leq C\max\{\left\|\Phi(\cdot,t)\right\|\left\|\Phi_x(\cdot,t)\right\|,\left\|\Psi(\cdot,t)\right\|\left\|\Psi_x(\cdot,t)\right\|\}\\&\leq C\max\{\left\|\Phi_x(\cdot,t)\right\|,\left\|\Psi_x(\cdot,t)\right\|\}\rightarrow0 \text{ as }t\rightarrow\infty. \end{align}\tag{125}\] This completes the proof of Proposition 4. ◻
We are ready to finish the proof of Theorem 2.
Proof of Theorem 2. Since \((n-N,q-Q)=(\Phi,\Psi)\), if \((n_0-N_0,q_0-Q_0)(x)\) is small, then by Proposition 4, the system 1 2 has a unique global solution satisfying ?? . Furthermore, by 41 and 125 , \((n,q)\) has the asymptotic behavior ?? . We complete the proof of Theorem 2. ◻
This work is supported by the National Natural Science Foundation of China (No. 12371216).