On the $L^{2}$-critical nonlinear Schrodinger equation with an inhomogeneous damping term

TL;DR

This paper investigates the $L^{2}$-critical nonlinear Schrödinger equation with an inhomogeneous damping term, demonstrating the existence of global solutions and determining minimal blow-up times for certain initial data.

Contribution

It establishes the existence of global solutions and quantifies minimal blow-up times for the inhomogeneous damped nonlinear Schrödinger equation.

Findings

Existence of initial data leading to global solutions.

Determination of minimal blow-up time for some initial data.

Analysis of the effects of inhomogeneous damping on solution behavior.

Abstract

We consider the $L^2$-critical nonlinear Schrodinger equation with an inhomogeneous damping term. We prove that there exists an initial data such that the corresponding solution is global in $H^1(R^d)$ and we give the minimal time of the blow up for some initial data.

Equations

$$\{\begin{array}{l}i{u}_t+\Delta u+|u{|}^{\frac{4}{d}}u+ia(x)u=0,(t,x)\in [0,\infty [\times {\mathbb{R}}^d,d=1,2,3,4.\\ u(0)=u_0\in H^1({\mathbb{R}}^d)\end{array}$$

$$\{\begin{array}{l}i{u}_t+\Delta u+|u{|}^{\frac{4}{d}}u=0\\ u(0)=u_0\in H^1({\mathbb{R}}^d)\end{array}$$

$$\Delta Q+Q|Q{|}^{\frac{4}{d}}=Q.$$

$$\frac{d}{dt}\Vert u(t){\Vert }_{L^2}^2=-\int a(x)|u(t,x){|}^{2}dx,t\in [0,T),$$

$$\frac{d}{dt}E(u(t))=-\int a(x)\nabla u(t,x)dx+\int a(x)u(t,x{)}^{\frac{4}{d}+2}-\Re \int (\nabla u.\nabla a)\overline{u}dx$$

$$\frac{d}{dt}P(u(t))=-2\int a(x)\Im (\nabla u\overline{u})dx,t\in [0,T).$$

$$\underset{n\to +\infty }{\mathrm{limsup}}{\Vert \nabla v_n\Vert }_{L^2({\mathbb{R}}^d)}\le Mand\underset{n\to +\infty }{\mathrm{limsup}}{\Vert v_n\Vert }_{L^{\frac{4}{d}+2}}\ge m.$$

$$v_n(\cdot +x_n)\rightharpoonup Vweakly,$$

$$\underset{t\to T}{\mathrm{limsup}}{\Vert u(t)\Vert }_{L^2(|x-x(t)|<w(t))}\ge {\Vert Q\Vert }_{L^2({\mathbb{R}}^d)}.$$

$$\underset{t_k\to T}{\lim }\frac{{\int }_0^{t_k}F(\tau )d\tau }{F(t_k)}=0.$$

$$E(u(t))=E(u_0)-{\int }_0^{t}H(u(\tau ))d\tau ,t\in [0,T[.$$

$${\Vert u\Vert }_{L^{2+\frac{4}{d}}}^{2+\frac{4}{d}}\le C\Vert \nabla u{\Vert }_{L^2}^2\Vert u{\Vert }_{L^2}^{\frac{4}{d}},$$

$$\Vert u_0{\Vert }_{L^2}{e}^{-\Vert a{\Vert }_{L^{\infty }}t}\le \Vert u{\Vert }_{L^2}\le \Vert u_0{\Vert }_{L^2}{e}^{\Vert a{\Vert }_{L^{\infty }}t}$$

$$\left|H(u(t))\right|$$

$$\le \Vert a{\Vert }_{L^{\infty }}{\Vert \nabla u(t)\Vert }_{L^2({\mathbb{R}}^d)}^2+C\Vert a{\Vert }_{L^{\infty }}{e}^{\Vert a{\Vert }_{L^{\infty }}t}\Vert \nabla u{\Vert }_{L^2}^2\Vert u_0{\Vert }_{L^2}^{\frac{4}{d}}$$

$$+e^{\Vert a{\Vert }_{L^{\infty }}t}\Vert \nabla a{\Vert }_{L^{\infty }}\Vert u_0{\Vert }_{L^2}\Vert \nabla u{\Vert }_{L^2}$$

$$\left|H(u(t))\right|\le C(\Vert a{\Vert }_{H^1},\Vert u_0{\Vert }_{L^2})e^{\Vert a{\Vert }_{L^{\infty }}t}\Vert \nabla u{\Vert }_{L^2}^2$$

$$\underset{k\to \infty }{\lim }\frac{{\int }_0^{t_k}H(u(\tau ))d\tau }{{\Vert \nabla u(t_k)\Vert }_{L^2({\mathbb{R}}^d)}^2}=0.$$

$$\rho (t)=\frac{{\Vert \nabla Q\Vert }_{L^2({\mathbb{R}}^d)}}{{\Vert \nabla u(t)\Vert }_{L^2({\mathbb{R}}^d)}}\text{and}v(t,x)={\rho }^{\frac{d}{2}}u(t,\rho x)$$

$${\Vert v_k\Vert }_{L^2({\mathbb{R}}^d)}\le e^{\Vert a{\Vert }_{L^{\infty }}T}{\Vert u_0\Vert }_{L^2({\mathbb{R}}^d)}\text{and}{\Vert \nabla v_k\Vert }_{L^2({\mathbb{R}}^d)}={\Vert \nabla Q\Vert }_{L^2({\mathbb{R}}^d)}.$$

$$E(v_k)={\rho }_k^{2}E(u_0)-{\rho }_k^2{\int }_0^{t_k}H(u(\tau ))d\tau \to 0,$$

$${\Vert v_k\Vert }_{L^{\frac{4}{d}+2}}^{\frac{4}{d}+2}\to \frac{d+2}{d}{\Vert \nabla Q\Vert }_{L^2({\mathbb{R}}^d)}^2.$$

$$m^{\frac{4}{d}+2}=\frac{d+2}{d}{\Vert \nabla Q\Vert }_{L^2({\mathbb{R}}^d)}^2\text{and}M={\Vert \nabla Q\Vert }_{L^2({\mathbb{R}}^d)},$$

$${\rho }_k^{\frac{d}{2}}u(t_k,{\rho }_k\cdot +x_k)\rightharpoonup V\in H^1\text{weakly}.$$

$$\underset{n\to +\infty }{\mathrm{liminf}}{\int }_{B(0,A)}{\rho }_n^d|u(t_n,{\rho }_{n}x+x_n){|}^{2}dx\ge {\int }_{B(0,A)}|V{|}^{2}dx,$$

$$\underset{n\to +\infty }{\mathrm{liminf}}\underset{y\in {\mathbb{R}}^d}{\sup }{\int }_{|x-y|\le w(t_n)}|u(t_n,x){|}^{2}dx\ge {\int }_{|x|\le A}|V{|}^{2}dx.$$

$$\underset{t\to T}{\mathrm{liminf}}\underset{y\in {\mathbb{R}}^d}{\sup }{\int }_{|x-y|\le w(t)}|u(t,x){|}^{2}dx\ge \int Q^{2}dx.$$

Full text

Résumé. On considere l’equation de Schrodinger nonlinéaire $L^{2}$-critique avec un terme d’amortissement non homogène. On montre qu’ils existent des données initiales tels que la solution est globale dans $H^{1}(\mathbb{R}^{d})$ et on donne le temps minimal d’explosion pour quelques données initiales.

1. Section francaise abrégée

On montre dans cette note qu’ils existent des données initiales tels que la solution est globale dans $H^{1}(\mathbb{R}^{d})$ et on donne le temps minimal d’explosion pour quelques données initiales, le resultat sera obtenue en montrant qu’un phénomène de $L^{2}$ concentration aura lieu proche de point d’explosion.

Notre résultat principal concernant l’equation (2.1)est donné dans le théorème suivant:

Theorem 1.1**.**

Soit $u_{0}\in H^{1}(\mathbb{R}^{d})$ et $d=1,2,3,4$:

1. (1)

Si $a(x)>0$ et $\|u_{0}\|_{L^{2}}\leq\|Q\|_{L^{2}}$, alors la solution de (2.1) est globale dans $H^{1}(\mathbb{R}^{d})$. 2. (2)

Si $a(x)$ est de signe quelconque et s’il existent des données initiales $u_{0}$ avec $\|u_{0}\|_{L^{2}}\leq\|Q\|_{L^{2}}$ tels que la solution de (2.1) explose en temps fini $T$ alors $T>\frac{1}{\|a\|_{L^{\infty}}}\log(\frac{\|Q\|_{L^{2}}}{\|u_{0}\|_{L^{2}}})$.

2. Introduction

In this paper, we study the Cauchy problem for the $L^{2}$-critical nonlinear Schrödinger equations:

[TABLE]

with a real inhomogeneous damping term a $\in C^{1}(\mathbb{R}^{d},\mathbb{R})\cap W^{1,\infty}(\mathbb{R}^{d},\mathbb{R})$ and initial data $u(0)=u_{0}\in H^{1}(\mathbb{R}^{d})$.Equation (2.1) arises in several areas of nonlinear optics and plasma physics. The inhomogenous damping term corresponds to an electromagnetic wave absorved by an inhomogenous medium. (cf [1],[2])

It is known that the Cauchy problem for (2.1) is locally well-posed in $H^{1}(\mathbb{R}^{d})$(see Kato[8] and also Cazenave[4]): For any $u_{0}\in H^{1}(\mathbb{R}^{d})$, there exist $T\in(0,\infty]$ and a unique solution $u(t)$ of (2.1) with $u(0)=u_{0}$ such that $u\in C([0,T);H^{1}(\mathbb{R}^{d}))$. Moreover, T is the maximal existence time of the solution $u(t)$ in the sense that if $T<\infty$ then $\displaystyle{\lim_{t\rightarrow T}{\|u(t)\|_{H^{1}(\mathbb{R}^{d})}}}=\infty$.

Dias and figueira [6] studied the supercritical case($|u|^{p}u$ with $p>\frac{4}{d}$) and showed that blow-up in finite time can occur, using the virial method. In [5], Correia was studied the equation in dimension one, and he proved the existence of blowup phenomena in the energy space $H^{1}$

Let us notice that for $a=0$ (2.1) becomes the $L^{2}$-critical nonlinear Schrödinger equation:

[TABLE]

Special solutions play a fundamental role for the description of the dynamics of (2.2). They are the solitary waves of the form $u(t,x)=\exp(it)Q(x)$, where $Q$ solves:

[TABLE]

Let $u$ be a solution of (2.1), we define the following quantities:

$L^{2}$norm : $\left\|u(t)\right\|_{L^{2}}.$

Energy : $E(u(t))=\frac{1}{2}\|\nabla u\|_{L^{2}}^{2}-\frac{d}{4+2d}\|u\|_{L^{\frac{4}{d}+2}}^{\frac{4}{d}+2}.$

Kinetic momentum : $P(u(t))=Im(\displaystyle{\int}\nabla u\overline{u}(t,x)).$

It is easy to prove that if $u$ is a solution of (2.1) then :

[TABLE]

[TABLE]

and

[TABLE]

Remark 2.1**.**

Remark that if $a(x)>0,\forall x\in\mathbb{R}^{d}$, then $\|u(t)\|_{L^{2}}<\|u_{0}\|_{L^{2}}$, for all $t\in[0,T[$.

Let us now our results:

Theorem 2.1**.**

1. Let $u_{0}\in H^{1}(\mathbb{R}^{d})$ with $d=1,2,3,4$, then:
2. (1)

If $a(x)>0$ and $\|u_{0}\|_{L^{2}}\leq\|Q\|_{L^{2}}$, then the corresponding solution of (2.1) is global in $H^{1}$. 3. (2)

If $a(x)$ has an arbitrary sign and if there exists an initial data $u_{0}\in H^{1}$ with $\|u_{0}\|<\|Q\|_{L^{2}}$ such that the corresponding solution blows up at finite time $T$, then $T>\frac{1}{\|a\|_{\infty}}\log(\frac{\|Q\|_{L^{2}}}{\|u_{0}\|_{L^{2}}})$.

3. $L^{2}$-concentration

In this section, we prove theorem 2.1 by extending the proof of the $L^{2}$-concentration phenomen, proved by Ohta and Todorova [10] in the radial case, to the non radial case.

Hmidi and Keraani showed in [7] the $L^{2}$-concentration for the equation (2.2) without the hypothese of radiality, using the following theorem:

Theorem 3.1**.**

Let $(v_{n})_{n}$ be a bounded family of $H^{1}(\mathbb{R}^{d})$, such that:

[TABLE]

Then, there exists $(x_{n})_{n}\subset\mathbb{R}^{d}$ such that:

[TABLE]

with $\left\|V\right\|_{L^{2}(\mathbb{R}^{d})}\geq(\frac{d}{d+4})^{\frac{d}{4}}\frac{m^{\frac{d}{2}+1}+1}{M^{\frac{d}{2}}}\left\|Q\right\|_{L^{2}(\mathbb{R}^{d})}$.

Now we have the following theorem:

Theorem 3.2**.**

Assume that $u_{0}\in H^{1}(\mathbb{R}^{d})$ , and suppose that the solution of (2.1) with $u(0)=u_{0}$ blows up in finite time $T\in(0,+\infty)$. Then, for any function $w(t)$ satisfying $w(t)\left\|\nabla u(t)\right\|_{L^{2}(\mathbb{R}^{d})}\rightarrow\infty$ as $t\rightarrow T$, there exists $x(t)\in\mathbb{R}^{d}$ such that, up to a subsequence,

[TABLE]

To show this theorem we shall need the following lemma:

Lemma 3.1**.**

Let $T\in(0,+\infty)$, and assume that a function $F:[0,T)\longmapsto(0,+\infty)$ is continuous, and $\lim_{t\rightarrow T}F(t)=+\infty$. Then, there exists a sequence $(t_{k})_{k}$ such that $t_{k}\rightarrow T$and

[TABLE]

For the proof see [10].

Proof of Theorem 3.2:

Suppose that there exist an initial data $u_{0}$ in $H^{1}$ with $\|u_{0}\|\leq\|Q\|_{L^{2}}$ such that the corresponding solution blows up at finite time $T$.

By the energy identity $(\ref{derivee de lenergie})$, we have

[TABLE]

Where $H(u(t))=\displaystyle{-\int a(x)|\nabla u(t,x)|^{2}dx+\int a(x)|u(t,x)|^{\frac{4}{d}+2}-\Re\int(\nabla u.\nabla a)\overline{u}dx}$. Let us recall the Gagliardo-Nirenberg inequality:

[TABLE]

Note that (2.4) gives that:

[TABLE]

Now using (3.10) and (3.11) we obtain that:

[TABLE]

for all $t\in[0,T[$. Then

[TABLE]

Moreover, we have $\displaystyle{\lim_{t\rightarrow T}\left\|\nabla u(t)\right\|_{L^{2}(\mathbb{R}^{d})}}=+\infty$, thus by Lemma 3.1, there exists a sequence $(t_{k})_{k}$ such that $t_{k}\rightarrow T$ and

[TABLE]

Let

[TABLE]

and $\rho_{k}=\rho(t_{k}),v_{k}=v(t_{k},.)$. The family $(v_{k})_{k}$ satisfies

[TABLE]

By (3.9) and (3.12), we have

[TABLE]

which yields

[TABLE]

The family $(v_{k})_{k}$ satisfies the hypotheses of Theorem 3.1 with

[TABLE]

thus there exists a family $(x_{k})_{k}\subset\mathbb{R}^{d}$ and a profile $V\in H^{1}(\mathbb{R}^{d})$ with $\left\|V\right\|_{L^{2}(\mathbb{R}^{d})}\geq\left\|Q\right\|_{L^{2}(\mathbb{R}^{d})}$, such that,

[TABLE]

Using (3.15), $\forall A\geq 0$

[TABLE]

but $\lim_{n\to+\infty}\frac{w(t_{n})}{\rho_{n}}=+\infty$  thus $\frac{w(t_{n})}{\rho_{n}}>A$, $\rho_{n}A<w(t_{n})$. This gives immediately:

[TABLE]

This it is true for all $A>0$ thus :

[TABLE]

Finally we obtain:

1. (1)

If $a(x)>0$, then the norm $L^{2}$ is strictly decreasing, with (3.16) in hand we obtain that:$\displaystyle{\|u_{0}\|_{L^{2}}^{2}>\|Q\|_{L^{2}}^{2}}$ 2. (2)

If the sign of $a$ is arbitrary, (3.16) gives that $e^{2\|a\|_{L^{\infty}}T}\|u_{0}\|_{L^{2}}^{2}\geq\|Q\|^{2}_{L^{2}}$.

**Proof of Theorem 2.1:

**Now if $a(x)>0$ $\forall x\in\mathbb{R}^{d}$, and $u_{0}$ be an initial data such that $\|u_{0}\|\leq\|Q\|$, we obtain a contradiction, that means that the solution is global in $H^{1}$, and this gives the proof of part 1 of the theorem.

If a has an arbitrary sign, and if $u$ blows up with initial data $u_{0}$ with $\|u_{0}\|_{L^{2}}<\|Q\|_{L^{2}}$ at finite time $T$, we obtain that $T>\frac{1}{\|a\|_{L^{\infty}}}\log(\frac{\|Q\|_{L^{2}}}{\|u_{0}\|_{L^{2}}})$, this gives the proof of the second part of the theorem.