A Note on Decay Rates of the Local Energy for Wave Equations with Lipschitz Wavespeeds

TL;DR

This paper improves the understanding of how quickly the local energy decays over time for wave equations with Lipschitz continuous wave speeds, extending previous results that required stronger initial data regularity.

Contribution

It provides a refined decay rate analysis for wave equations with Lipschitz coefficients, relaxing regularity assumptions on initial data compared to prior work.

Findings

Enhanced decay rate estimates for local energy in wave equations with Lipschitz coefficients

Reduced regularity requirements on initial data for decay analysis

Comparison with previous log-order decay results

Abstract

We consider the Cauchy problem for wave equations with variable coefficients in the whole space. We improve the rate of decay of the local energy, which has been recently studied by J. Shapiro, where he derives the log-order decay rates of the local energy under stronger assumptions on the regularity of the initial data.

Equations

$$u_{tt}(t,x)-c(x{)}^2\Delta u(t,x)=0,(t,x)\in (0,\infty )\times {\mathbf{R}}^n,$$

$$u(0,x)=u_0(x),u_t(0,x)=u_1(x),x\in {\mathbf{R}}^n,$$

$$u_0\in H^1({\mathbf{R}}^n),u_1\in L^2({\mathbf{R}}^n),$$

$$u_t=\frac{\partial u}{\partial t},u_{tt}=\frac{{\partial }^{2}u}{\partial t^2},\Delta =\sum _{j=1}^n\frac{{\partial }^2}{\partial x_j^2},x=(x_1,\cdots ,x_n).$$

$$u\in C([0,\infty );H^1({\mathbf{R}}^n))\cap C^1([0,\infty );L^2({\mathbf{R}}^n))=:C_1^n,$$

$$E_u(t)=E_u(0),$$

$$E_{u}(t):=\frac{1}{2}\int_{{\bf R}^{n}}\big{(}\frac{1}{c(x)^{2}}|u_{t}(t,x)|^{2}+|\nabla u(t,x)|^{2}\big{)}dx.$$

$$E_{R}(t):=\frac{1}{2}\int_{|x|\leq R}\big{(}\frac{1}{c(x)^{2}}|u_{t}(t,x)|^{2}+|\nabla u(t,x)|^{2}\big{)}dx.$$

$$B(x,R):=\{y\in {\mathbf{R}}^n:|y-x|<R\}.$$

$$E_{R_2}(t)\le (\frac{C}{\log (2+t)}{)}^2\left(\Vert \nabla u_0{\Vert }_{(H^1({\mathbf{R}}^n){)}^n}^2+\Vert u_1{\Vert }_{H^1({\mathbf{R}}^n)}^2\right).$$

$$L^{p,\gamma }({\mathbf{R}}^n):=\left\{f\in L^p({\mathbf{R}}^n)|\Vert f{\Vert }_{p,\gamma }:={\left({\int }_{{\mathbf{R}}^n}(1+|x{|}^{\gamma })|f(x){|}^{p}dx\right)}^{1/p}<+\infty \right\}.$$

$${\int }_{{\mathbf{R}}^n}(1+|x|)\left(\frac{1}{c(x{)}^2}|u_1(x){|}^2+|\nabla u_0(x){|}^2\right)dx<+\infty ,$$

$$E_R(t)=O(t^{-(1-\eta )})(t\to \infty ),$$

$${\int }_{{\mathbf{R}}^2}(1+|x|)\left(\frac{1}{c(x{)}^2}|u_1(x){|}^2+|\nabla u_0(x){|}^2\right)dx<+\infty ,$$

$${\int }_{{\mathbf{R}}^2}\frac{u_1(x)}{c(x{)}^2}dx=0,$$

$$E_R(t)=O(t^{-(1-\eta )})(t\to \infty ),$$

$${\int }_{\mathbf{R}}(1+|x|)\left(\frac{1}{c(x{)}^2}|u_1(x){|}^2+|\nabla u_0(x){|}^2\right)dx<+\infty ,$$

$$E_R(t)=O(t^{-(1-\eta )})(t\to \infty ),$$

$$\mathcal{F}(f)(\xi ):=\hat{f}(\xi ):=(\frac{1}{2\pi }{)}^{\frac{n}{2}}{\int }_{{\mathbf{R}}^n}{e}^{-ix\cdot \xi }f(x)dx$$

$$(f,g):={\int }_{{\mathbf{R}}^n}f(x)g(x)dx,f,g\in L^2({\mathbf{R}}^n).$$

$$t{E}_u(t)=\frac{n-1}{2}(\frac{1}{c(\cdot {)}^2}{u}_1,u_0)+(\frac{1}{c(\cdot {)}^2}{u}_1,x\cdot \nabla u_0)$$

$$-\frac{n-1}{2}(\frac{1}{c(\cdot {)}^2}{u}_t(t,\cdot ),u(t,\cdot ))-(\frac{1}{c(\cdot {)}^2}{u}_t(t,\cdot ),x\cdot \nabla u(t,\cdot ))$$

$$+{\int }_0^t{\int }_{{\mathbf{R}}^n}\frac{1}{c(x{)}^3}(x\cdot \nabla c(x))|u_s(s,x){|}^{2}dxds(t\ge 0).$$

$$M(u):=t{u}_t+x\cdot \nabla u+\frac{n-1}{2}u,$$

$$\|u(t,\cdot)\|\leq C\|c^{-1}\|_{\infty}^{2}\big{(}\|u_{1}\|+\|u_{1}\|_{1}\big{)}+C\|c^{-1}u_{0}\|,$$

$${\int }_{{\mathbf{R}}^n}\frac{u_1(x)}{c(x{)}^2}dx=0,$$

$$\|u(t,\cdot)\|\leq C\|c^{-1}\|_{\infty}^{2}\big{(}\|u_{1}\|+\|u_{1}\|_{1,\gamma}\big{)}+C\|\bf c^{-1}u_{0}\|,$$

$${\int }_{{\mathbf{R}}^n}f(x)dx=0$$

$${\int }_{{\mathbf{R}}^n}\frac{|\hat{f}(\xi ){|}^2}{|\xi {|}^{2\theta }}d\xi \le C(\Vert f{\Vert }_{1,\gamma }^2+\Vert f{\Vert }^2)$$

$$\displaystyle{\int_{{\bf R}^{n}}}\displaystyle{\frac{|\hat{f}(\xi)|^{2}}{|\xi|^{2\theta}}}d\xi\leq C\big{(}\;\|f\|_{1,\gamma}^{2}+\|f\|^{2}+\left|\int_{{\bf R}^{n}}f(x)dx\right|^{2}\big{)}$$

Full text

1 Introduction

000Keywords and Phrases: Wave equation; Lipschitz wavespeed; local energy; algebraic decay, non-compactly supported initial data, weighted energy estimate.0002010 Mathematics Subject Classification. Primary 35L05; Secondary 35B40, 35L30.

We consider in this work the Cauchy problem associated to the wave equation with variable coefficient in ${\bf R}^{n}$ ($n\geq 1$) as follow

[TABLE]

[TABLE]

where $(u_{0},u_{1})$ are initial data chosen as

[TABLE]

and the function $c:{\bf R}^{n}\to{\bf R}$ satisfies the two assumptions below:

(A-1) $c(x)>0$ ($x\in{\bf R}^{n}$), $c,c^{-1}\in L^{\infty}({\bf R}^{n})$, $\nabla c\in(L^{\infty}({\bf R}^{n}))^{n}$,

(A-2) there exists a constant $L>0$ such that $c(x)=1$ for $|x|>L$.

In particular, the condition (A-1) implies $c\in C^{0,1}({\bf R}^{n})$ (see e.g., [1, Theorem IX.12]). Here, we have set

[TABLE]

Note that solutions and/or functions considered in this paper are all real valued except for some parts concerning the Fourier transform.

Considering the previous assumptions on the initial data and $c(x)$ it is known that the problem (1.1)-(1.2) has a unique weak solution

[TABLE]

satisfying the energy conservation property:

[TABLE]

where the total energy $E_{u}(t)$ to the equation (1.1) is defined by

[TABLE]

Furthermore, the local energy $E_{R}(t)$ on the zone $\{|x|\leq R\}$ ($R>0$) corresponding to the solution $u(t,x)$ of (1.1)-(1.2) is defined by

[TABLE]

Also, we set

[TABLE]

Our main concern of this paper is to obtain a local energy decay estimate with an algebraic decay order. For related important results concerning the local energy decay, one can cite several celebrated papers due to Morawetz [8], Lax-Phillips [7], Morawetz-Ralston-Strauss [9], Ralston [11], Vainberg [14], and the references therein.

By the way, concerning local energy decay results, quite recently Shapiro [12] announces the following interesting result. It should be mentioned that the decay rate of (1.4) below was first obtained by Burq [2] for smooth perturbations of the Laplacian outside an obstacle.

Theorem 1.1

(Shapiro [12])* Let $n\geq 2$, and assume (A-1) and (A-2). Suppose that the supports of $u_{0}$ and $u_{1}$ are contained in $B(0,R_{1})$, and $\nabla u_{0}\in(H^{1}({\bf R}^{n}))^{n}$ and $u_{1}\in H^{1}({\bf R}^{n})$. Then for any $R_{2}>0$, there exists $C>0$ such that the solution $u$ to (1.1)-(1.2) satisfies for $t\geq 0$,*

[TABLE]

Our observation is that Shapiro [12] imposes rather stronger hypothesis on the regularity of the initial data such as

(I) the supports of initial data are compact, and as a result $[u_{0},u_{1}]\in H^{2}({\bf R}^{n})\times H^{1}({\bf R}^{n})$.

Furthermore, in a sense,

(II) the obtained decay order $(\log t)^{-2}$ of the local energy seems to be rather slow.

In this paper, under weaker regularity assumptions on the initial data to modify (I), one obtains faster algebraic decay rate which improves (II) in the case when the coefficient $c(x)$ and the parameter $L$ have a special relation.

Our method is based on the so-called Morawetz identity [8], so we never use the spectral analysis like resolvent estimates. In order to state our results, we introduce the following weighted functional spaces.

[TABLE]

Our main results read as follows.

Theorem 1.2

Let $n\geq 3$, and assume (A-1) and (A-2). If the initial data $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times(L^{2}({\bf R}^{n})\cap L^{1}({\bf R}^{n}))$ further satisfies

[TABLE]

then the unique solution $u\in C_{1}^{n}$ to problem (1.1)-(1.2)* satisfies*

[TABLE]

for each $R>L$ provided that $\eta:=2L\|\frac{1}{c(\cdot)}\|_{\infty}\|\nabla c\|_{\infty}\in[0,1)$.

Theorem 1.3

Let $n=2$, and assume (A-1) and (A-2). Let $\gamma\in(0,1]$. If $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times(L^{2}({\bf R}^{n})\cap L^{1,\gamma}({\bf R}^{n}))$ further satisfies

[TABLE]

and

[TABLE]

then the unique solution $u\in C_{1}^{2}$ to problem (1.1)-(1.2)* satisfies*

[TABLE]

for each $R>L$ provided that $\eta:=2L\|\frac{1}{c(\cdot)}\|_{\infty}\|\nabla c\|_{\infty}\in[0,1)$.

Theorem 1.4

Let $n=1$, and assume (A-1) and (A-2). If $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times L^{2}({\bf R}^{n})$ further satisfies

[TABLE]

then the unique solution $u\in C_{1}^{1}$ to problem (1.1)-(1.2)* satisfies*

[TABLE]

for each $R>L$ provided that $\eta:=2L\|\frac{1}{c(\cdot)}\|_{\infty}\|\nabla c\|_{\infty}\in[0,1)$.

Remark 1.1

Our gain is that the $n=1$ dimensional case is included in our results, and we do not assume any compactness of the supports of initial data, and weaker regularity assumptions such as $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times L^{2}({\bf R}^{n})$ are imposed. Instead of stronger regularity as in [12] we have to pay a price to assume various weighted conditions on the initial data in some functional spaces, and the parameter $\eta$ must be chosen to satisfy $\eta\in[0,1)$. This condition on $\eta$ is crucial in this paper. In this connection, if $c(x)=1$ for all $x\in{\bf R}^{n}$, then $\|\nabla c\|_{\infty}=0$, so that $\eta=0$, and in this case the obtained results remind us of those of [5] studied in an exterior domain with a star-shaped compliment set (star-shaped obstacle). **

Remark 1.2

For example, if $L>0$ is small, $\displaystyle{\inf_{x\in{\bf R}^{n}}}c(x)$ is sufficiently far from [math], and $\|\nabla c\|_{\infty}$ is small, then we can realize the hypothesis $\eta\in[0,1)$. The smallness of $L$ implies $-c(x)^{2}\Delta=-\Delta$ for $x\in{\bf R}^{n}\setminus B(0,\varepsilon)$ with small $\varepsilon>0$. Note that $\displaystyle{\inf_{x\in{\bf R}^{n}}}c(x)>0$ under the assumption (A-1).**

This paper is organized as follows. In section 2 after preparing several propositions and lemmas we shall prove Theorems 1.2, 1.3 and 1.4 at a stroke. The key tool is already prepared in [6].

Notation. Throughout this paper, $\|\cdot\|_{q}$ stands for the usual $L^{q}({\bf R}^{n})$-norm. For simplicity of notation, in particular, we use $\|\cdot\|$ instead of $\|\cdot\|_{2}$. Furthermore, we denote $\|\cdot\|_{H^{1}}$ as the usual $H^{1}$-norm. On the other hand, we denote the Fourier transform of $f(x)$ by

[TABLE]

as usual with $i:=\sqrt{-1}$. As the $L^{2}$-inner product, one employs the following notation:

[TABLE]

2 $L^{2}$-bounds of solutions

In order to prove the previous theorems one first prepare in this section the so-called Morawetz identity. This is our starting point.

Proposition 2.1

Let $n\geq 1$. Under the assumption (A-1), the (unique) weak solution $u\in C_{1}^{n}$ to problem (1.1)-(1.2)* satisfies*

[TABLE]

[TABLE]

[TABLE]

The proof of the Morawetz identity can be derived first to the smooth solution with $u(t,x)$ for initial data with compact support, say $[u_{0},u_{1}]\in C_{0}^{\infty}({\bf R}^{n})\times C_{0}^{\infty}({\bf R}^{n})$, by relying on the multiplier

[TABLE]

the finite speed of propagation property, integration by parts, and then by the density arguments. The final identity can be established to the desired weak solution $u\in C_{1}^{n}$. Note that $c\in C({\bf R}^{n})\cap L^{\infty}({\bf R}^{n})$ under the assumption (A-1) (cf. [1, Theorem IX.12]).

As a second work, we derive several $L^{2}$-bounds of solutions under non-compact support conditions on the initial data. For this purpose, we rely on an improvement version of an original idea established in [4] because we can now use the Fourier transform appropriately to obtain them. We have the following significant propositions. These results will be used when one estimates the term $(\frac{1}{c(\cdot)^{2}}u_{t}(t,\cdot),u(t,\cdot))$ in Proposition 2.1.

Proposition 2.2

Let $n\geq 3$. If $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times(L^{2}({\bf R}^{n})\cap L^{1}({\bf R}^{n}))$, then the unique solution $u\in C_{1}^{n}$ to problem (1.1)-(1.2)* satisfies*

[TABLE]

with some constant $C>0$.

Proposition 2.3

Let $n=2$ and $\gamma\in(0,1]$. If $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times(L^{2}({\bf R}^{n})\cap L^{1,\gamma}({\bf R}^{n}))$ further satisfies

[TABLE]

then the unique solution $u\in C_{1}^{2}$ to problem (1.1)-(1.2)* satisfies*

[TABLE]

with some constant $C>0$.

In the course of proofs of Propositions 2.2 and 2.3, the next inequality concerning the Fourier image of the Riesz potential plays an crucial role. This comes from [3, (ii) of Proposition 2.1].

Proposition 2.4

Let $[n,\gamma,\theta]$ satisfy $n\geq 1$, $\gamma\in[0,1]$ and $\theta\in[0,\gamma+\displaystyle{\frac{n}{2}})$. Then, for all $f\in L^{2}({\bf R}^{n})\cap L^{1,\gamma}({\bf R}^{n})$ satisfying

[TABLE]

it is true that

[TABLE]

with some constant $C=C_{n,\theta,\gamma}>0$.

While, the following result is well-known (cf., B. Muckenhoupt [10, Theorem 1])

Proposition 2.5

Let $[n,\gamma,\theta]$ satisfy $n\geq 1$, $\gamma\in[0,1]$ and $\theta\in[0,\displaystyle{\frac{n}{2}})$. Then, for all $f\in L^{2}({\bf R}^{n})\cap L^{1,\gamma}({\bf R}^{n})$ it is true that

[TABLE]

with some constant $C=C_{n,\theta,\gamma}>0$.

Proof of Propositions 2.2 and 2.3.  Let us prove Propositions 2.2 and 2.3 at a stroke. We use the idea from [4] in the Fourier space. Let $v(t,x):=\displaystyle{\int_{0}^{t}}u(s,x)ds$. Then, the function $v$ satisfies

[TABLE]

[TABLE]

Multiplying both sides of (2.1) by $v_{t}$, and integrating over $[0,t]$ we derive that

[TABLE]

where

[TABLE]

Note that $w\in L^{1}({\bf R}^{n})\cap L^{2}({\bf R}^{n})$ because of the assumption (A-1) and the condition that $u_{1}\in L^{1}\cap L^{2}$. By using the Plancherel theorem, the last term of (2.3) can be estimated as follows. For any $\varepsilon>0$ there exists a constant $C_{\varepsilon}>0$ such that for all $t\geq 0$ one has

[TABLE]

[TABLE]

[TABLE]

Thus, one has from (2.3)

[TABLE]

Now, when $n\geq 3$, by using Proposition 2.5 with $\gamma=0$, one has

[TABLE]

[TABLE]

In the case when $n=2$, by relying on Proposition 2.4 with $\gamma\in(0,1]$ for $n=2$, one can have

[TABLE]

where one has just used the assumption $\displaystyle{\int_{{\bf R}^{n}}}w(x)dx=0$ in Proposition 2.3. Therefore, it follows from (2.4), (2.5) or (2.6) that

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

The desired estimates of the Propositions 2.2 and 2.3 can be derived because of $v_{t}=u$ by taking $\varepsilon>0$ small enough. $\Box$

The next step is to handle with the term $(\frac{1}{c(\cdot)^{2}}u_{t}(t,\cdot),x\cdot\nabla u(t,\cdot))$ which appears in Morawetz identity. To do that we prepare an important weighted energy estimate below. For this purpose we define a weight function $\psi:[0,\infty)\times{\bf R}^{n}\to{\bf R}$ given by

[TABLE]

The following properties of the function $\psi\in C^{1}([0,\infty)\times{\bf R}^{n})$ are a direct calculation

[TABLE]

[TABLE]

and

[TABLE]

The equation (2.8) is the so-called Eikonal equation to the free wave equation

[TABLE]

How to choose $\psi(t,x)$ has its origin in [5]. Based on the (modified) weighted energy estimates originated in [13], one can get the following lemma.

Lemma 2.1

Let $n\geq 1$ and assume (A-2). If the initial data $[u_{0},u_{1}]\in H^{1}({\bf R}^{n})\times L^{2}({\bf R}^{n})$ further satisfies

[TABLE]

for each $R>L$, then the solution $u\in C_{1}^{n}$ to the problem (1.1)-(1.2) satisfies

[TABLE]

where the constant $C=2+L>0$ .

Since one can not apply the style itself of the weighted function $\psi(t,x)$ to the equation (1.1) with variable coefficient $c(x)$, one has to introduce a new auxiliary function $\phi(t)$ of $\psi(t,x)$ defined by

[TABLE]

Note that $\phi\in C^{1}([0,\infty))$, and

[TABLE]

It is easy to check that $\psi(t,x)$ and $\phi(t)$ satisfy the following identities.

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

where

[TABLE]

Proof of Lemma 2.1. The proof can be proceeded similarly to [6, Lemma 3.3] with a constant $r_{0}$ replaced by $L$. However, for making this paper self-contained we shall draw its full proof.

It follows from (2.7) and (2.12) that

[TABLE]

[TABLE]

Integrating (2.15) over $[0,t]\times({\bf R}^{n}\setminus B(0,L))$ one can get

[TABLE]

[TABLE]

where one has just used the assumption (A-2). By applying (2.8) one obtains

[TABLE]

[TABLE]

On the other hand, by integrating (2.13) over $[0,t]\times B(0,L)$, because of (2.11) one can get

[TABLE]

[TABLE]

Now, since $\phi(t)=\psi(t,x)$ on the sphere $|x|=L$, it follows from the divergence formula that

[TABLE]

Thus, by summing up (2.16) and (2.17) one can arrived at the desired estimate

[TABLE]

where one has used the fact that $R>L$ and $\phi(t)>0$. $\Box$

Based on Lemma 2.1, one can also obtain the following lemma. This is re-stated version of [6, Lemma 3.4] with $r_{0}$ replaced by $L$. Note that from the assumption on $c(x)$, one has

[TABLE]

Lemma 2.2

Let $R>L$, $t>R$, and $c(x)$ satisfies the assumptions (A-1) and (A-2). Then it is true that

[TABLE]

where $C>0$ is a constant independent from initial data.

Remark 2.1

When one checks the proof of Lemma 2.2 above, one has to use the assumption (A-2) such that $c(x)=1$ for $x\in{\bf R}^{n}$ satisfying $|x|>L$, and in this case one notices that (see (2.14))

[TABLE]

3 Proof of Theorems 1.2, 1.3 and 1.4

Now, in this section, we are in a position to prove Theorems 1.2, 1.3 and 1.4 based on Lemmas 2.1 and 2.2.

Note that it suffices to check only the case for $n\geq 3$, and the case for $n=2$ is similar by using Proposition 2.3 in place of Proposition 2.2. Note also that for the case of $n=1$, one does not need to use Proposition 2.3 because of the existence of the coefficient $\displaystyle{\frac{n-1}{2}}$ in Proposition 2.1.

Let $R>L$. We first start with Proposition 2.1 under the assumption (A-2):

[TABLE]

[TABLE]

where

[TABLE]

We observe that $J_{0}^{2}$ is not necessarily positive.

Note that under the assumption on the regularity imposed on the initial data, one can check that (see (2.10))

[TABLE]

[TABLE]

Then, it follows from the Schwarz inequality, (A-1) and (A-2) that

[TABLE]

[TABLE]

Since

[TABLE]

it follows from (3.1) and (3.2) that

[TABLE]

[TABLE]

where

[TABLE]

This parameter $\eta$ is quite important.

While, by Lemma 2.2 and (3.3) one can obtain

[TABLE]

[TABLE]

which implies

[TABLE]

Here, let us estimate the term

[TABLE]

by relying on Propositions 2.2 and 2.3.

We estimate only the case for $n\geq 3$. Based on Proposition 2.2, (A-1) and the Schwarz inequality we can derive

[TABLE]

[TABLE]

[TABLE]

Since $2^{-1}\|\frac{1}{c(\cdot)}u_{t}(t,\cdot)\|^{2}\leq E_{u}(t)=E_{u}(0)$ (see (1.3)), one has

[TABLE]

Because of (3.5) and (3.6) one can arrive at the significant inequality of the Gronwall type:

[TABLE]

[TABLE]

where

[TABLE]

Now, let us solve the integral inequality (3.7) under the assumption $\eta\in[0,1)$. This is rather standard. For completeness we write its full proof.

To do this we consider the function

[TABLE]

Then, it follows from (3.7) that

[TABLE]

Integrating over $[t_{0},t]$ with large $t_{0}\gg 1$, one can get

[TABLE]

By (3.7) and (3.8) one can obtain the desired estimate

[TABLE]

This completes the proof of Theorems 1.2, 1.3 and 1.4. $\Box$

Acknowledgement. The work of the second author (R. IKEHATA) was supported in part by Grant-in-Aid for Scientific Research (C)15K04958 of JSPS.