On the local-energy decay property for the elastic wave equation with the Neumann boundary condition
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.
uniform local-energy decay property of strong type
scientific article; zbMATH DE number 78561
scientific article