On the nonlinear wave equation with time periodic potential

TL;DR

This paper demonstrates that adding a nonlinear defocusing term to a wave equation with time periodic potential prevents exponential energy growth, ensuring global existence and polynomial energy bounds, and analyzes the instability of the zero solution.

Contribution

It introduces a nonlinear defocusing interaction that stabilizes solutions of wave equations with time periodic potentials, preventing exponential energy growth.

Findings

Solutions exist globally with polynomial energy bounds.

Zero solution is unstable under certain linear conditions.

Nonlinear interaction stabilizes the wave equation behavior.

Abstract

It is known that for some time periodic potentials $q(t, x) \geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\partial_t^2 u - \Delta_x u + q(t,x)u = 0$ have exponentially increasing energy as $t \to \infty$. We show that if one adds a nonlinear defocusing interaction $|u|^ru, 2\leq r < 4,$ then the solution of the nonlinear wave equation exists for all $t \in {\mathbb R}$ and its energy is polynomially bounded as $t \to \infty$ for every choice of $q$. Moreover, we prove that the zero solution of the nonlinear wave equation is instable if the corresponding linear equation has the property mentioned above.

Equations

$${\partial }_t^{2}u-{\Delta }_{x}u+q(t,x)u=0,u(0,x)=f_1(x),{\partial }_{t}u(0,x)=f_2(x),$$

$$\exists C>0,\exists \alpha >0\mathrm{such}\mathrm{that}\forall t\ge 0,\Vert u(t,\cdot ){\Vert }_{H^1({\mathbb{R}}^3)}\ge C{e}^{\alpha t}.$$

$$V(T,0):\mathcal{H}\ni (f_1(x),f_2(x))⟶(u(T,x),u_t(T,x))\in \mathcal{H}$$

$${\partial }_t^{2}u-{\Delta }_{x}u+q(t,x)u+|u{|}^{r}u=0,u(0,x)=f_1(x),{\partial }_{t}u(0,x)=f_2(x),$$

$$\displaystyle\|\nabla u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}+\|\partial_{t}u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}\leq 2\Bigl{(}X(0)^{\frac{r}{r+2}}+C|t|\Bigr{)}^{\frac{r+2}{2r}},$$

$$\displaystyle\|u(t,\cdot)\|_{L^{2}({\mathbb{R}}^{3})}\leq\|f_{1}\|_{L^{2}}+2|t|\Bigl{(}X(0)^{\frac{r}{r+2}}+C|t|\Bigr{)}^{\frac{r+2}{2r}}\,,$$

$$X(t)=\int_{{\mathbb{R}}^{3}}\big{(}\frac{1}{2}|\partial_{t}u|^{2}+\frac{1}{2}|\nabla_{x}u|^{2}+\frac{1}{2}q|u|^{2}+\frac{1}{r+2}|u|^{r+2}\big{)}dx$$

$$X^{\prime }(t)=\frac{1}{2}\text{mbox}\mathrm{Re}{\int }_{{\mathbb{R}}^3}({\partial }_{t}q)|u{|}^{2}dx$$

$$|X^{\prime }(t)|\le C{X}^{1-\frac{r}{r+2}}(t).$$

$$u(t,x)\in C([0,A],H^2({\mathbb{R}}^3))\cap C^1([0,A],H^1({\mathbb{R}}^3))\cap L_t^{\frac{2r+2}{r-2}}([0,A],L_x^{2r+2}({\mathbb{R}}^3))$$

$$w_{n+1}=\mathcal{F}(w_n),n\ge 0,$$

$${\partial }_t^{2}u-{\Delta }_{x}u+|u{|}^{r}u+\sum _{j=0}^{r-1}{q}_j(t,x)|u{|}^{j}u=0,r=2,3$$

$$H=H_D({\mathbb{R}}^3)\times L^2({\mathbb{R}}^3)$$

$$\|f\|_{0}=\big{(}\|f_{1}\|^{2}_{H_{D}}+\|f_{2}\|^{2}_{L^{2}}\big{)}^{1/2},\quad f=(f_{1},f_{2}).$$

$${\partial }_t^{2}u-{\Delta }_{x}u+q(t,x)u=0,u(s,x)=f_1(x),{\partial }_{t}u(s,x)=f_2(x)$$

$$H\ni f\to U(t,s)f=(u(t,x;s),{\partial }_{t}u(t,x;s))\in H$$

$$\Vert U(t,s)f{\Vert }_0\le C{e}^{\alpha |t-s|}\Vert f{\Vert }_0.$$

$$U(t,s)f-U_0(t-s)f=-{\int }_s^t{U}_0(t-\tau )Q(\tau )U(\tau ,s)fd\tau ,$$

$$U_0(t)=\left(\begin{array}{cc}\cos (t\sqrt{-\Delta })& \frac{\sin (t\sqrt{-\Delta })}{\sqrt{-\Delta }}\\ -\sqrt{-\Delta }\sin (t\sqrt{-\Delta })& \cos (t\sqrt{-\Delta })\end{array}\right),Q(t)=\left(\begin{array}{cc}0& 0\\ q(t,x)& 0\end{array}\right).$$

$$\Vert U(t,s)f-U_0(t-s)f{\Vert }_{H^2({\mathbb{R}}^3)\times H^1({\mathbb{R}}^3)}\le C\Vert U(t,s)f{\Vert }_0.$$

$${\mathrm{supp}}_x(U(t,s)f-U_0(t-s)f)\subset \{|x|\le \rho +|t-s|\}.$$

$$\|f\|_{1}=\big{(}\|f_{1}\|^{2}_{H^{1}({\mathbb{R}}^{3})}+\|f_{2}\|^{2}_{L^{2}({\mathbb{R}}^{3})}\big{)}^{1/2},\quad\|f_{1}\|^{2}_{H^{1}({\mathbb{R}}^{3})}=\|\nabla_{x}f_{1}\|^{2}_{L^{2}({\mathbb{R}}^{3})}+\|f_{1}\|^{2}_{L^{2}({\mathbb{R}}^{3})}.$$

$$\Vert U_0(t)f{\Vert }_1\le C(1+|t|)\Vert f{\Vert }_1,\forall t\in \mathbb{R},$$

$$\frac{d}{dt}\int_{{\mathbb{R}}^{3}}\big{(}|\partial_{t}v(t,x;s)|^{2}+|\nabla_{x}v(t,x;s)|^{2}+|v(t,x;s)|^{2}\big{)}dx=-2\mbox{\rm Re}\int_{{\mathbb{R}}^{3}}qv\overline{\partial_{t}v}dx+2\mbox{\rm Re}\int_{{\mathbb{R}}^{3}}v\overline{\partial_{t}v}dx$$

$$\frac{d}{dt}\Vert (v(t,x;s),{\partial }_{t}v(t,x;s)){\Vert }_1^2\le C_1\Vert (v(t,x;s),{\partial }_{t}v(t,x;s)){\Vert }_1^2$$

$$\Vert (v(t,x;s),{\partial }_{t}v(t,x;s)){\Vert }_1\le C_2{e}^{\beta |t-s|}\Vert f{\Vert }_1,s\le t\le s+t_0,\beta \ge 0.$$

$$\mathcal{H}\ni f\mapsto V(t,s)f=(v(t,x;s),{\partial }_{t}v(t,x;s))\in \mathcal{H}$$

$$U(t,s)\circ U(s,r)=U(t,r),U(s,s)=\mathrm{Id},U(t+T,s+T)=U(t,s),t,s,r\in \mathbb{R}.$$

$$(V(T,0)-zI)^{-1}=(U_{0}(T)-zI)^{-1}-(U_{0}(T)-zI)^{-1}\big{(}V(T,0)-U_{0}(T)\big{)}(V(T,0)-zI)^{-1}$$

$$\big{[}I+(U_{0}(T)-zI)^{-1}\big{(}V(T,0)-U_{0}(T)\big{)}\big{]}(V(T,0)-zI)^{-1}=(U_{0}(T)-zI)^{-1}.$$

$${\partial }_t^{2}u-{\Delta }_{x}u+q(t){\chi }^{\delta }(x)u=0,t\ge 0,|x|<L$$

Full text

111MSC [2010]: Primary 35L71, Secondary 35L15

1. Introduction

Our goal in this paper is to show that a defocusing nonlinear interaction may improve, in a certain sense, the long time properties of the solutions of the wave equation with a time periodic potential.

Consider the Cauchy problem for the following potential perturbation of the classical wave equation in the Euclidean space ${\mathbb{R}}^{3}$

[TABLE]

where $0\leq q(t,x)\in C^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{3})$ is periodic in time $t$ with period $T>0$ and has a compact support with respect to $x$ included in $\{x\in{\mathbb{R}}^{3}:|x|\leq\rho\},$ for some positive $\rho$. It is easy to show that the Cauchy problem (1.1) is globally well-posed in ${\mathcal{H}}=H^{1}({\mathbb{R}}^{3})\times L^{2}({\mathbb{R}}^{3})$. The analysis of the long time behavior of the solution of (1.1) may be quite intricate (see e.g. [6, 1]). A slight adaptation of the arguments presented in [1] leads the following result.

Theorem 1**.**

There exist $q$ and $(f_{1},f_{2})\in{\mathcal{H}}$ such that the solution of (1.1) satisfies :

[TABLE]

The above result has been established in [1] for the Cauchy problem with initial data in $H=H_{D}({\mathbb{R}}^{3})\times L^{2}({\mathbb{R}}^{3})$. In fact we show that the propagator of (1.1)

[TABLE]

has an eigenvalue $y,|y|>1$ which implies (1.2).

Our purpose is to show that adding a nonlinear perturbation to (1.1) forbids the existence of solutions satisfying (1.2). Consider therefore the following Cauchy problem

[TABLE]

where $2\leq r<4$. We have the following statement.

Theorem 2**.**

For any choice of $q$ the Cauchy problem (1.3) is globally well-posed in ${\mathcal{H}}$. Moreover, for every $(f_{1},f_{2})\in{\mathcal{H}}$ there exists a constant $C>0$ such that for every $t\in{\mathbb{R}}$, the solution of (1.3) satisfies the polynomial bound

[TABLE]

where

[TABLE]

and $C>0$ depends only on $q$ and $r$.

By global well-posedness we mean the existence, the uniqueness and the continuous dependence with respect to the data. The proof of Theorem 2 is based on the equality

[TABLE]

and the estimate

[TABLE]

It is classical to expect that the result of Theorem 1 implies the instability of the zero solution of (1.3). More precisely, we have the following instability result.

Theorem 3**.**

With $q$ as in Theorem 1 the following holds true. There is $\eta>0$ such that for every $\delta>0$ there exists $(f_{1},f_{2})\in{\mathcal{H}}$ , $\|(f_{1},f_{2})\|_{{\mathcal{H}}}<\delta$ and there exists $n=n(\delta)>0$ such that the solution of (1.3) satisfies $\|(u(nT,\cdot),\partial_{t}u(nT,\cdot)\|_{{\mathcal{H}}}>\eta.$

We are not aware of any nontrivial choice of $(f_{1},f_{2})\in{\mathcal{H}}$ such that the solution $u(t,x)$ of (1.3) and $u_{t}(t,x)$ remain uniformly bounded in ${\mathcal{H}}$ for all $t\geq 0.$ The paper is organized as follows. In the next section, we prove Theorem 1. The third section is devoted to the proof of Theorem 2. First we obtain a local existence and uniqueness result on intervals $[s,s+\tau]$ with $\tau=c(1+\|(f_{1},f_{2})\|_{{\mathcal{H}}})^{-\gamma}$ with constants $c>0$ and $\gamma>0$ independent on $f$. Next we establish (1.4) for solutions

[TABLE]

and finally, by a local approximation in small intervals we justify (1.4) for every fixed $A>0$ and $0\leq t\leq A$. In the fourth section, we prove Theorem 3 passing to a system

[TABLE]

where ${\mathcal{F}}={\mathcal{U}}(0,T)$ is the propagator of the nonlinear equation. In the fifth section we discuss the generalizations concerning the nonlinear equations

[TABLE]

with time-periodic functions $q_{j}(t+T_{j},x)=q_{j}(t,x)\geq 0,\>j=0,1,r-1$ having compact support with respect to $x$.

2. Proof of Theorem 1

2.1. The linear wave equation with time periodic potential

Let $H_{D}({\mathbb{R}}^{3})$ be the closure of $C_{0}^{\infty}({\mathbb{R}}^{3})$ with respect to the norm $\|f\|_{H_{D}}=\|\nabla_{x}f\|_{L^{2}({\mathbb{R}}^{3})}$. Define the (energy) space

[TABLE]

with norm

[TABLE]

Let $u(t,x;s)$ be the solution of the Cauchy problem

[TABLE]

with $f=(f_{1},f_{2})\in H$. Therefore the operator

[TABLE]

is called the propagator (monodromy operator) of (2.1) and there exist $C>0$ and $\alpha\geq 0$ so that

[TABLE]

Let $U_{0}(t-s)f=(u_{0}(t,x;s),\partial_{t}u_{0}(t,x;s))$, where $u_{0}$ solves $\partial_{t}^{2}u_{0}-\Delta_{x}u_{0}=0$ with initial data $f$ for $t=s.$ Then we have

[TABLE]

where

[TABLE]

Using the relation (2.3) and the compact support of $q$ allows us to obtain the estimate

[TABLE]

Moreover the support property of $q$ also yields

[TABLE]

Consequently $U(t,s)$ is a compact perturbation of the unitary operator $U_{0}(t-s).$

Now consider the space ${\mathcal{H}}=H^{1}({\mathbb{R}}^{3})\times L^{2}({\mathbb{R}}^{3})\subset H$ with norm

[TABLE]

The map $U_{0}(t)$ is not unitary in ${\mathcal{H}}.$ However, one easily checks that

[TABLE]

with a constant $C>0$ independent of $t$. Consequently, the spectral radius of the operator $U_{0}(T):{\mathcal{H}}\to{\mathcal{H}}$ is not greater than 1.

By using (2.3), it is easy to show by a fixed point theorem that for small $t_{0}>0$ and $s\leq t\leq s+t_{0}$ we have a local solution $(v(t,x;s),\partial_{t}v(t,x;s))\in{\mathcal{H}}$ of the Cauchy problem (2.1) with initial data $f\in{\mathcal{H}}.$ For this solution one deduces

[TABLE]

which yields

[TABLE]

with a constant $C_{1}>0$ independent of $f$ and $s$. The last inequality implies an estimate

[TABLE]

By a standard argument this leads to a global existence of a solution of (2.1). Introduce the propagator

[TABLE]

corresponding to the Cauchy problem (1.1) with initial data $f\in{\mathcal{H}}.$ As in Section 5 in [6] it is easy to see that we have the following properties

[TABLE]

The same properties hold for the propagator $V(t,s)$. In particular, $V(T,0)=V((k+1)T,kT)$, $k\in{\mathbb{N}}$ and $V(nT,0)=(V(T,0))^{n}.$

As above notice that $V(t,s)-U_{0}(t-s)$ is a compact operator in ${\mathcal{L}}({\mathcal{H}})$. For $|z|\gg 1$ we have

[TABLE]

hence

[TABLE]

Set K(z)=I+(U_{0}(T)-zI)^{-1}\big{(}V(T,0)-U_{0}(T)\big{)}. For $|z|$ large enough $K(z)$ is invertible. By the analytic Fredholm theorem for $|z|\geq 1+\delta>1$ the operator $K(z)$ is invertible outside a discreet set and the inverse $K(z)^{-1}$ is a meromorphic operator-valued function. Consequently, the operator $V(T,0)\in{\mathcal{L}}({\mathcal{H}})$ has in the open domain $|z|>1$ a discreet set of eigenvalues with finite multiplicities which could accumulate only to the circle $|z|=1$.

2.2. Extending the result of [1] to ${\mathcal{H}}$

In [1] it was proved that there are potentials $q(t,x)\geq 0$ for which the operator $U(T,0):H\to H$ has an eigenvalue $z,|z|>1$. In this paper we deal with the operator $V(T,0):{\mathcal{H}}\to{\mathcal{H}}$ and it is not clear if the eigenfunction $\psi\in H$ with eigenvalues $z$ constructed in [1] belongs to ${\mathcal{H}}.$

Below we make some modifications on the argument of [1] in order to show that for the potential constructed in [1] the corresponding operator $V(T,0):{\mathcal{H}}\to{\mathcal{H}}$ has an eigenvalue $y,|y|>1.$ For convenience we will use the notations in [1] and we recall some of them. The potential in [1] has the form $V^{\epsilon}(t,x):=b^{\epsilon}(x)+q(t)\chi^{\delta}(x)$ with $\epsilon>0,$ where $b^{\epsilon}(x)\in C_{0}^{\infty}({\mathbb{R}}^{3})$ is supported in $\{0<L\leq|x|\leq L+1\}$ and equal to $1/\epsilon$ for $\{L+\epsilon\leq|x|\leq L+1-\epsilon\}$, $\chi^{\delta}(x)\geq 0$ is a smooth function with support in $|x|<L$ and equal to 1 for $|x|\leq L-\delta<L.$ Finally, $q(t)\geq 0$ is a periodic smooth function with period $T>0$. The number $L$ is related to the interval of instability of the Hill operator associated with $q(t)$. The number $\delta>0$ is fixed sufficiently small and the propagator $K^{\delta}(T)$ related to the equation

[TABLE]

with Dirichlet boundary conditions on $|x|=L$ has an eigenvalue $z_{1},|z_{1}|>1$ with eigenfunction $\varphi\in H^{1}_{0}(|x|\leq L)$, that is $K^{\delta}(T)\varphi=z_{1}\varphi.$ Let $S^{\epsilon}(T):H\to H$ be the propagator corresponding to the Cauchy problem for the equation

[TABLE]

and let $W^{\epsilon}(T):{\mathcal{H}}\to{\mathcal{H}}$ be the propagator for the same problem with initial data in ${\mathcal{H}}.$ The problem is to show that for $\epsilon>0$ sufficiently small $W^{\epsilon}(T)$ has an eigenvalues $y,|y|>1$ (Here $S^{\epsilon}(T),W^{\epsilon}(T)$ correspond to our notations $U(T,0),V(T,0)$ and these operators have domains $H$ and ${\mathcal{H}}$, respectively).

Extend $\varphi$ as 0 outside $|x|\geq L$ and denote the new function $\varphi\in{\mathcal{H}}$ again by $\varphi.$ Let

[TABLE]

be a circle with center $z_{1}$ such that $K^{\delta}(T)-zI$ is analytic on $\gamma$ and $z_{1}$ is the only eigenvalue of $K^{\delta}(T)$ in $|z-z_{1}|\leq\eta.$ If $W^{\epsilon}(T)$ has an eigenvalues on $\gamma$ the problem is solved. Assume that $W^{\epsilon}(T)$ has no eigenvalues on $\gamma$. It is easy to see that

[TABLE]

Indeed,

[TABLE]

and

[TABLE]

Our purpose is to study

[TABLE]

where $(.,,)_{{\mathcal{H}}}$ denotes the scalar product in ${\mathcal{H}}$ and $(.,.)_{H}$ denotes the scalar product in $H$. It was proved in [1] that for $z\in\gamma$ one has the weak convergence in $H$

[TABLE]

so

[TABLE]

Here we have used the fact that $\varphi=0$ for $|x|>L.$ Let $\varphi=(\varphi_{1},\varphi_{2}).$ We claim that as $\epsilon\to 0$ we have

[TABLE]

To prove this write

[TABLE]

The main point is the following

Lemma 1**.**

We have $\psi\in H_{D}({\mathbb{R}}^{3}).$

Proof.

Since

[TABLE]

we can apply the Hardy-Littlewood-Sobolev inequality. More precisely, by using Theorem 4.3 of [5] with $n=3,\>\lambda=2,\>r=2,\>p=6/5$, we obtain that

[TABLE]

Now using that $\varphi_{1}(x)$ is with compact support and the Hölder inequality, we obtain that

[TABLE]

This completes the proof of Lemma 1. ∎

Therefore

[TABLE]

[TABLE]

which proves the claim (2.4). Consequently,

[TABLE]

Moreover, Proposition 4.2 in [1] says that with a constant $C_{0}>0$ we have uniformly for $z\in\gamma$ the norm estimate

[TABLE]

Since

[TABLE]

the sequence $(\varphi,(W^{\epsilon}(T)-zI)^{-1}\varphi)_{{\mathcal{H}}}$ is bounded for $z\in\gamma$. Repeating the argument of Section 5 in [1], one deduces

[TABLE]

This completes the proof that for small $\epsilon$ the operator $W^{\epsilon}(T)$ has an eigenvalue $y,|y|>1.$

3. Proof of Theorem 2

3.1. Local well-posedness

Consider the linear problem

[TABLE]

By using the argument in [6], one may show that the solution of (3.1) satisfies the same local in time Strichartz estimates as in the case $q=0$. Notice that for these local Strichartz estimates we don’t need a global control of the local energy and we can establish them without a condition on the cut-off resolvent $\varphi(V(T,0)-z)^{-1}\varphi$. More precisely for every finite $a>0$ and $f=(f_{1},f_{2})\in{\mathcal{H}},F\in L^{1}([s,s+a];L^{2}({\mathbb{R}}^{3}))$ we have that the solution of (3.1) satisfies

[TABLE]

provided $\frac{1}{p}+\frac{3}{q}=\frac{1}{2}$, $p>2$ (the constant $C(a)$ in (3.2) depends on $a$, $p$ and $q(t,x)$). Moreover, if $(f_{1},f_{2})\in H^{2}({\mathbb{R}}^{3})\times H^{1}({\mathbb{R}}^{3})$ and $F\in L^{1}([s,s+a];H^{1}({\mathbb{R}}^{3}))$, we have

[TABLE]

A standard application of (3.2), (3.3) is the following local well-posedness result for the non