## On the local-energy decay property for the elastic wave equation with the Neumann boundary condition.(English)Zbl 0795.35061

Let $$\Omega$$ be an exterior domain in $$\mathbb{R}^ n$$ $$(n\geq 3)$$ with smooth and compact boundary $$\Gamma$$. We consider the isotropic elastic wave equation with the Neumann boundary condition $\begin{cases} (A(\partial_ x)- \partial_ t^ 2) u(t,x)=0 &\text{in } \mathbb{R}\times\Omega,\\ N(\partial_ x) u(t,x)=0 &\text{on } \mathbb{R}\times\Gamma,\\ u(0,x)= f_ 0(x),\;\partial_ t u(0,x)= f_ 1(x) \quad &\text{on } \Omega. \end{cases}\tag{1}$ Here $$A(\partial_ x)$$ and $$N(\partial_ x)$$ are of the forms $$A(\partial_ x)u= \mu \Delta u+ (\lambda+\mu) \text{grad(div } u)$$, $$(N(\partial_ x) u)_ i= \sum_{j=1}^ n \nu_ j (x)\sigma_{ij}(u)|_ \Gamma$$ $$(i=1,2,\dots,n)$$, where $$\nu(x)$$ is the unit outer-normal vector to $$\Omega$$ at $$x\in\Gamma$$. In the above, $$\sigma_{ij}(u)= \lambda(\text{div } u)\delta_{ij}+ 2\mu\varepsilon_{ij}(u)$$ is the stress tensor for the isotropic elastic materials, where $$\varepsilon_{ij}(u)= (1/2) (\partial_{x_ i} u_ j+ \partial_{x_ j} u_ i)$$ is the strain tensor. We assume that the Lamé constants $$\lambda$$ and $$\mu$$ are independent of the variables $$t$$ and $$x$$ and satisfy $$\lambda+ {2\over n}\mu>0$$ and $$\mu>0$$. For a domain $$D\subset \mathbb{R}^ n$$ we define the local energy $${\mathbf E}(u,D,t)$$ for the solution of (1) in $$D\cap \Omega$$ as ${\mathbf E}(u,D,t)= {\textstyle{1\over 2}} \int_{D\cap\Omega} \bigl\{ \lambda| \text{div } u(t,x)|^ 2+\mu \sum_{i,j=1}^ n | \varepsilon_{ij} (u(t,x))|^ 2+ | \partial_ t u(t,x)|^ 2 \bigr\}dx.$ We say that problem (1) has the uniform local-energy decay property of strong type when, for any bounded domains $$D$$ and $$D_ 0$$, there exists a bounded, continuous, and nonnegative-valued function $$p(t)$$ defined on $$[0,\infty)$$ satisfying $\int_ 0^ \infty p(t)^{1/2} dt<\infty \qquad \text{and} \qquad \int_ 0^ \infty \int_ s^ \infty p(t)^{1/2} dt ds<\infty$ such that $${\mathbf E}(u,D,t)\leq p(t) {\mathbf E}(u,\Omega,0)$$ for any $$t\geq 0$$ holds for any solution of (1) with initial data $$f_ 0, f_ 1\in C_ 0^ \infty (D_ 0\cap \overline{\Omega})$$. The main theorem in the present paper is the following one.
Theorem. Problem (1) does not have the uniform local-energy decay property of strong type.

### MSC:

 35L20 Initial-boundary value problems for second-order hyperbolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 74B99 Elastic materials 74H99 Dynamical problems in solid mechanics
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### References:

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