Well-posedness for a modified nonlinear Schrödinger equation modeling the formation of rogue waves

1. Introduction

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{N}\) \((N\geq1)\) with smooth boundary \(\partial\Omega\). We consider the initial-boundary value problem:

where \(p>2\) and \(\nu\) represents the unit outward normal to \(\partial\Omega\). Here, \(g(t)\) is a positive function that represents the kernel of the memory term, which will be specified in Section 2 and \begin{equation*} g\ast\Delta^{2}u(t)=\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau. \end{equation*} The motivation of our work is due to the initial boundary problem of the double dispersive-dissipative wave equation with nonlinear damping and source terms which has been discussed by Di and Shang [1] by considering the existence of global solutions and the asymptotic behavior of global solutions with \(m\geq p\).

In the absence of the dispersive term and the nonlinear damping term, model \(2\) reduces to the following wave equation

Shang [2] studied the well-posedness, asymptotic behavior, and the finite time blow-up of the solutions under some suitable conditions on \(f\) and for \(N=1,2,3\). Zhang and Hu [3] showed the existence and the stability of global weak solutions. Xie and Zhong [4] obtained the existence of global attractors in \(H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)\), where the nonlinear term \(f\) satisfies a critical exponential growth assumption. Xu et al., [5] used the multiplier method to investigate the asymptotic behavior of solutions for (3).

Mellah [6] considered the following initial-boundary value problem

\begin{equation*}\label{W} \left\{ \begin{array}{ll} u_{tt}-\Delta u+\Delta^{2}u-g\ast\Delta^{2}u+u_{t}=|u|^{p-1}u,& x\in\Omega,\ t>0,\\ u=0,\quad \frac{\partial u}{\partial\nu}=0,& x\in\partial\Omega,\ t>0,\\ u(x,0)=u_{0}(x),\quad u_{t}(x,0)=u_{1}(x),& x\in\Omega, \end{array} \right. \end{equation*} in a bounded domain and \(p>1\). He investigated the small data global weak solutions and general decay of solutions, respectively.

Motivated by previous works, it is interesting to prove that problem (1) has a global weak solution assuming small initial data. In addition, we show the general decay of solutions. The global solutions are constructed by means of the Galerkin approximations and the general decay is obtained by employing the technique used in [7].

2. Preliminaries

In this section, we present some materials needed in the proof of our main result. We use the following abbreviations; \(\|\cdot\|_{p}=\|\cdot\|_{L^{p}(\Omega)}\) \((1\leq p\leq+\infty)\) denotes usual \(L^{p}\) norm, \((\cdot,\cdot)\) denotes the \(L^{2}\)-inner product, and consider the Sobolev spaces \(H^{1}_{0}(\Omega)\) and \(H^{2}_{0}(\Omega)\) with their usual scalar products and norms. We also use the embedding \(H^{1}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)\) for \(2< p\leq\frac{2N}{N-2}\) if \(N\geq3\) or \(2< p< \infty\) if \(N=1,2\). In this case, the embedding constant is denoted by \(C_{*}\), that is \( \|u\|_{p}\leq C_{*}\|\nabla u\|_{2}. \) We define \begin{equation*} Q(z)=\frac{1}{2}z^{2}-\frac{C_{*}^{p}}{p}z^{p}. \end{equation*} By the direct computation, we deduce that \(Q\) is increasing in \([0,z_{0}]\), where \( z_{0}=C_{*}^{\frac{p}{2-p}} \) is its unique local maximum.

Next, we give the assumptions for problem (1).

• (G1) The relaxation function \(g:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) is a bounded \(C^{1}\) function such that \begin{equation*} g(0)>0,\quad 0< \eta=1-\int_{0}^{\infty}g(\tau)d\tau\leq1-\int_{0}^{t}g(\tau)d\tau=\eta(t). \end{equation*}
• (G2) There exist positive constants \(\xi_{1}\) and \(\xi_{2}\) such that \begin{equation*} -\xi_{1}g(t)\leq g'(t)\leq-\xi_{2}g(t)\quad \forall t\geq0. \end{equation*}
• (G3) We also assume that \begin{equation*} 2< p\leq \frac{2N}{N-2}\ \ \mbox {if} \ \ N\geq3 \ \ \mbox {and} \ \ \ p>2 \ \ \ \mbox {if} \ \ N=1,2, \end{equation*}

where \(\lambda_{1}\) is the first eigenvalue of the following problem

Remark 1. [8] Assuming \(\lambda_{1}\) is the first eigenvalue of the problem (4), we have

The energy associated with problem (1) is given by for \(u\in H^{2}_{0}(\Omega)\), where \begin{equation*} (g\circ\Delta u)(t)=\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u(t)\|_{2}^{2}d\tau. \end{equation*} Now, we are in a position to state our main results.

3. Main results

In this section, we are going to obtain the existence of global weak solutions for problem (1) with the initial conditions \(\|\nabla u_{0}\|_{2}< z_{0}\) and \(E(0)< Q(z_{0})\).

Theorem 1. Assume that \((G1)-(G3)\) hold, and that \(\left\{u_{0},u_{1}\right\}\) belong to \(H^{2}_{0}(\Omega)\times H^{1}_{0}(\Omega)\). Further assume that \(\|\nabla u_{0}\|_{2}< z_{0}\) and \(E(0)< Q(z_{0})\). Then, problem (1) admits a global weak solution, which satisfies \[u\in L^{\infty}(0,\infty;H^{2}_{0}(\Omega)),\quad u_{t}\in L^{\infty}(0,\infty;H^{1}_{0}(\Omega)).\] Moreover, the identity

holds for \(0\leq t< \infty\). Also, for an increasing \(C^{2}\) function \(\zeta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) satisfying and, if \(\|g\|_{L^{1}(0,\infty)}\) is sufficiently small, we have for \(\kappa>0\) \begin{equation*} E(t)\leq E(0)e^{-\kappa \zeta(t)},\quad \forall t\geq0. \end{equation*}

Remark 2. From (8) and \((G2)\), we obtain

Proof of Theorem 1 (Main result)

We divide the proof into two steps. In step 1, we prove the small data global existence of weak solutions by using the Faedo-Galerkin approximation and in step 2, we establish the general decay of energy employing the method used in [7].

Step 1: Global existence of weak solutions

Let \(\left\{\omega_{j}\right\}_{j=1}^{\infty}\) be an orthogonal basis of \(H^{2}_{0}(\Omega)\) with \(\omega_{j}\) being the eigenfunction of the following problem: \begin{equation*} -\Delta \omega_{j}=\lambda_{j}\omega_{j},\quad x\in\Omega,\quad \omega_{j}=0,\quad x\in\partial\Omega. \end{equation*} Let \(V^{n}=\text{Span}\left\{\omega_{1},\omega_{2},\cdot\cdot\cdot,\omega_{n}\right\}\). By the standard method of ODE, we know that there exists only one local solution \begin{equation*} u^{n}(t)=\sum_{j=1}^{n}b^{n}_{j}(t)\omega_{j} \end{equation*} of the Cauchy problem as follows: By the standard theory of ODE system, we prove the existence of solutions of problem (10)-(11) on some interval \([0, t_{n})\), \(0< t_{n}< T\) for arbitrary \(T>0\), then, this solution can be extended to the whole interval \([0,T]\) using the first estimate given below.

A Priori Estimates

Setting \(\omega=u^{n}_{t}(t)\) in (10), we have A direct computation shows that Inserting (13) into (12) and integrating over \([0,t]\subset[0, T]\), we obtain From assumption \((G3)\) and the Sobolev embedding, we have \begin{equation*} \|u^{n}\|^{p}_{p}\leq C_{*}^{p}\|\nabla u^{n}\|^{p}_{2}, \end{equation*} and then we have By using the fact that \begin{equation*} -\int_{0}^{t}(g'\circ\Delta u^{n})(\tau)d\tau+\int_{0}^{t}g(\tau)\|\Delta u^{n}(\tau)\|^{2}_{2}d\tau\geq0, \end{equation*} estimate (15) yields From \(E(0)< \mathcal{Q}(z_{0})\) and (11), it follows that for sufficiently large \(n\). We claim that there exists an integer \(N\) such that Suppose the claim is proved, then \(\mathcal{Q}(\|\nabla u^{n}\|^{2}_{2})\geq0\) and from (16) and (17), for sufficiently large \(n\) and \(0\leq t< \infty\).

Proof of the claim

Suppose that (18) false, then for each \(n>N\), there exists \(t\in[0,t_{n})\) such that \(\|\nabla u^{n}(t)\|_{2}\geq z_{0}\). Note that from \(\|\nabla u_{0}\|_{2}< z_{0}\) and (11) there exists \(N_{0}\) such that \begin{equation*} \|\nabla u^{n}(0)\|_{2}< z_{0}\quad \forall n>N_{0}. \end{equation*} Then by continuity there exits a first \(\widetilde{t_{n}}\in[0,t_{n})\) such that from where \begin{equation*} \mathcal{Q}(\|\nabla u^{n}(t)\|_{2})\geq0 \quad \forall t\in[0,\widetilde{t_{n}}]. \end{equation*} From \(E(0)< \mathcal{Q}(z_{0})\) and (19), there exists \(N>N_{0}\) and \(\gamma\in(0,z_{0})\) such that \begin{eqnarray*} 0&\leq&\frac{1}{2}\|u^{n}_{t}(t)\|^{2}_{2}+\frac{1}{2}\|\nabla u^{n}_{t}(t)\|^{2}_{2}+\frac{\eta(t)}{2}\|\Delta u^{n}(t)\|^{2}_{2}+\frac{1}{2}(g\circ\Delta u^{n})(t)+\mathcal{Q}(\|\nabla u^{n}(t)\|^{2}_{2})\nonumber\\ &\leq&\mathcal{Q}(\gamma)\quad \forall\;\;\; t\in[0,\widetilde{t_{n}}]\quad \forall n>N. \end{eqnarray*} The monotonicity of \(\mathcal{Q}\) in \([0,z_{0}]\) implies that \begin{equation*} 0\leq\|\nabla u^{n}(t)\|^{2}_{2}\leq\gamma< z_{0}\quad \forall t\in[0,\widetilde{t_{n}}], \end{equation*} in particular, \(\|\nabla u^{n}(t)\|^{2}_{2}< z_{0}\), which is a contradiction to (20). From (19), we have Using Sobolev inequality, (5) and (21), it follows that Moreover, by (25), we get Therefore, there exist \(u\), \(\chi\) and a subsequence still denotes \(\left\{u_{n}\right\}\) such that Besides, from Lions-Aubin Lemma we also have and consequently, making use of the Lemma 1.3 in [9], we deduce Thus, we obtain that \(u\) is a global weak of problem (1). In order to prove (7), we use the mean value theorem, we see that there exists \(0< \theta_{n}< 1\) such that \begin{eqnarray*} \|u^{n}\|_{p}^{p}-\|u\|_{p}^{p} &\leq& p\left|\int_{\Omega}|u+\theta_{n}u^{n}|^{p-2}(u+\theta_{n}u^{n})(u^{n}-u)dx\right|\nonumber\\ &\leq& p\|u+\theta_{n}u^{n}\|_{p}^{p-1}\|u^{n}-u\|_{p}\nonumber\\ &\leq&c\|u^{n}-u\|_{p}\rightarrow 0 \ \ \mbox {as} \ \ \ n\rightarrow +\infty, \end{eqnarray*} and for each fixed \(t>0\), we obtain \begin{eqnarray*} |(g\circ\Delta u)(t)-(g\circ\Delta u^{n})(t)|&=&\left|\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u(t)\|^{2}_{2}d\tau-\int_{0}^{t}g(t-\tau)\|\Delta u^{n}(\tau)-\Delta u^{n}(t)\|^{2}_{2}d\tau\right|\nonumber\\ &\leq& \int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u^{n}(\tau)\|_{2}\|\Delta u(\tau)+\Delta u^{n}(\tau)\|_{2}d\tau\nonumber\\ &&+\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u^{n}(\tau)\|_{2}d\tau\|\Delta u(t)+\Delta u^{n}(t)\|_{2}\nonumber\\ &&+\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)+\Delta u^{n}(\tau)\|_{2}d\tau\|\Delta u(t)-\Delta u^{n}(t)\|_{2}\nonumber\\ &&+\int_{0}^{t}g(\tau)d\tau\|\Delta u(t)+\Delta u^{n}(t)\|_{2}\|\Delta u(t)-\Delta u^{n}(t)\|_{2}\nonumber\\ &\leq&c\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u^{n}(\tau)\|_{2}d\tau\\ &&+c\int_{0}^{t}g(\tau)d\tau\|\Delta u(t)-\Delta u^{n}(t)\|_{2}\rightarrow 0 \ \ \mbox {as} \ \ \ n\rightarrow +\infty. \end{eqnarray*} Thus, we have \begin{equation*} \lim_{n\rightarrow +\infty}\|u^{n}\|_{p}^{p}=\|u\|_{p}^{p},\quad \lim_{n\rightarrow +\infty}(g\circ\Delta u^{n})(t)=(g\circ\Delta u)(t). \end{equation*} From (11), it follows that \(E^{n}(0)\rightarrow E(0)\) as \(n\rightarrow+\infty\). Finally, taking \(n\rightarrow +\infty\) in (14), we deduce that the energy identity (7) holds for \(0\leq t< \infty \).

Step 2: General decay of the energy

Here, we prove the energy decay estimate of the global solutions obtained in the previous section. To obtain the decay result, we use the following lemmas which are of crucial importance in the proof.

Lemma 1. Let \(u\in L^{\infty}(0,\infty;H^{2}_{0}(\Omega))\) with \(u_{t}\in L^{\infty}(0,\infty;H^{1}_{0}(\Omega))\) be the solution of (1) and \(E(0)< \mathcal{Q}(z_{0})\), \(\|\nabla u_{0}\|_{2}< z_{0}\), then we have

where \(C_{1}=\frac{1}{2}(1+B^{2})\), \(C_{2}=\frac{1}{2}(1+\lambda_{1}^{-1})\) and \(B\) is the optimal constant satisfying the Poincare inequality \(\|u_{t}\|_{2}\leq B\|\nabla u_{t}\|_{2}\).

Proof. From \(E(0)< \mathcal{Q}(z_{0})\) and \(\|\nabla u_{0}\|_{2}< z_{0}\), we can obtain \(\mathcal{Q}(\|\nabla u(t)\|_{2})\geq0\) for \(0\leq t< \infty\). Thus we have \begin{eqnarray*} E(t)&=&\frac{1}{2}\|u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t)+\frac{1}{2}\left(1-\int_{0}^{t}g(\tau)d\tau\right)\|\Delta u\|^{2}_{2}-\frac{1}{p}\|u\|_{p}^{p}\nonumber\\ &\geq&\frac{1}{2}\|u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u_{t}\|^{2}_{2}+\frac{\eta}{2}\|\Delta u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t)+\mathcal{Q}(\|\nabla u(t)\|_{2})\\&\geq&0, \end{eqnarray*} and \begin{eqnarray*} E(t)&\leq&\frac{1}{2}\|u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u_{t}\|^{2}_{2}+\frac{1}{2}\|\Delta u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t)+\frac{1}{2}\|\nabla u\|^{2}_{2}\nonumber\\ &\leq& \frac{1}{2}B^{2}\|\nabla u_{t}\|_{2}^{2}+\frac{1}{2}\|\nabla u_{t}\|_{2}^{2}+\frac{1}{2}\lambda_{1}^{-1}\|\Delta u\|^{2}_{2}+\frac{1}{2}\|\Delta u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t). \end{eqnarray*} Let \(C_{1}=\frac{1}{2}(1+B^{2})\) and \(C_{2}=\frac{1}{2}(1+\lambda_{1}^{-1})\), then we have (32).

Lemma 2. The energy \(E(t)\) satisfies

Proof. From (9), we have

From assumptions \((G2)\) and since \(\int_{0}^{t}g'(\tau)d\tau=g(t)-g(0)\), we obtain Then, Combining (34) and (35) our conclusion holds. Multiplying (33) by \(e^{\kappa\zeta(t)}\) \((\kappa>0)\) and using (32), we have Using the fact that \(\zeta_{t}\) is decreasing by (8), we conclude that Choosing \(\|g\|_{L^{1}(0,\infty)}\) sufficiently small so that \[g(0)-\xi_{1}\|g\|_{L^{1}(0,\infty)}=K>0\] and defining \[\kappa_{0}=\min\left\{\frac{1}{C_{1}\zeta_{t}(0)},\frac{\xi_{2}}{\zeta_{t}(0)},\frac{K}{2C_{2}\zeta_{t}(0)}\right\},\] we conclude by taking \(\kappa\in(0, \kappa_{0}]\) in (37) that Integrating (38) over \((0,t)\), it follows that

Example 1. For \(\zeta(t)=t+\frac{t}{t+1}\), we can get the exponential decay rate \( E(t)\leq E(0)e^{-\kappa t},\quad \forall t\geq0. \) For \(\zeta(t)=ln(1+t)\), we can get polynomial decay rate \( E(t)\leq E(0)(1+t)^{-\kappa },\quad \forall t\geq0. \)

Conflicts of Interests

''The author declares no conflict of interest.''