On the decay rate of local energy for the elastic wave equation
The paper is concerned with the rate of local energy decay of solutions to the linear elastic wave equation with the Dirichlet or the Neumann boundary condition in the exterior domain \(\Omega\) in \(\mathbb{R}^ n\) \((n \geq 3)\). More precisely, the equation considered in the paper is the following: \[ \begin{cases} \bigl( \partial^ 2_ t - A (\partial_ x) \bigr) u(t,x) = 0 \quad & \text{in } \mathbb{R} \times \Omega, \\ B (\partial_ x) u(t,x) = 0 \quad & \text{on } \mathbb{R} \times \partial \Omega, \\ u(0,x) = f_ 1(x),\;u_ t(0,x) = f_ 2(x) \quad & \text{in } \Omega. \end{cases} \tag{1} \] Here, \(A (\partial_ x) u = \sum^ n_{i,j = 1} a_{ij} \partial_ ju\), \(a_{ij} = (a_{ipjq})\) are \(n \times n\) constants matrices, \[ B (\partial_ x)u = u |_{\partial \Omega} \text{ (Dirichlet), or } = \sum^ n_{i,j = 1} \nu_ i (x)a_{ij} \partial_ ju |_{\partial \Omega} \text{ (Neumann)}, \] and \(\nu (x) = {^ t(\nu_ 1}(x), \dots, \nu_ n (x))\) is the unit outer normal to \(\partial \Omega\) at \(x \in \partial \Omega\). \(a_{ij}\) satisfy the following assumptions: \(a_{ipjq} = a_{pijq} = a_{jqip}\); \[ \sum^ n_{i,p,j,q = 1} \varepsilon_{jq} \overline {\varepsilon_{ip}} \geq \delta_ 1 \sum^ n_{i,p = 1} | \varepsilon |^ 2; \] \[ A (\xi) = \sum^ n_{i,j = 1} a_{ij} \xi_ i \xi_ j \text{ has } d \text{ characteristic roots of constant multiplicity for any } \xi \in \mathbb{R}^ n - \{0\}, \] where \((\varepsilon_ {ij})\) is any \(n \times n\) symmetric matrix and \(\delta_ 1\) is some positive constant independent of \((\varepsilon_{ij})\). Let \(u(t,x)\) be a solution of (1) with initial data \(f(x) = (f_ 1(x), f_ 2(x))\) and define the solution map \(U(t)\) by the formula: \(U(t)f = (u(t, \cdot), u_ t(t, \cdot))\). Then, the following theorem is the main result. Theorem 1. Assume that the space dimension \(n\) is even and \(\geq 4\). Assume that there exist a function \(p \in C ([0, \infty))\) and a constant \(\widetilde {\rho} > 0\) satisfying the conditions: \[ \lim_{t \to \infty} p(t) = 0,\;\| U(t)f \|_{H (\Omega_{\rho + \tilde \rho})} \leq p(t) \| f \| {^ \forall t} \geq 0,\;f \in H^ \rho, \] where \(\| \cdot \| = \| \cdot \|_{H (\Omega)}\), \[ \| f \|^ 2_{H (D)} = {1 \over 2} \int_ D \left\{ \sum^ n_{i,p,j,q = 1} a_{ipjq} \partial_ j f_{1q} (x) \overline {\partial_ i f_{1p} (x)} + | f_ 2 (x) |^ 2 \right\} dx, \] \(\Omega_ q = \{x \in \Omega | | x | < q\}\), and \(H^ \rho = \{f | \| f \| < \infty, \text{ supp} f \subset \overline {\Omega_ \rho}\}\). Then, there exists a constant \(C = C (\rho) > 0\) such that \[ \| U(t)f \|_{H (\Omega_ \rho)} \leq C(1 + t)^{- (n - 1)} \| f \| \forall t \geq 0,\;\forall f \in H^ \rho. \tag{2} \] When the space dimension \(n\) is odd and \(\geq 3\), the corresponding theorem can be proved by the argument due to \textit{P. D. Lax}, \textit{C. S. Morawetz} and \textit{R. S. Phillips} [Commun. Pure Appl. Math. 16, 477-486 (1963; Zbl 0161.018)], replacing \((1 + t)^{- (n - 1)}\) by \(e^{- ct}\) with \(c > 0\), because \textit{Y. Shibata} and \textit{H. Soga} [Publ. Res. Inst. Math. Sci. 25, No. 6, 861-887 (1989; Zbl 0714.35066)], formulated the scattering theory of (1) analogous to the Lax and Phillips famous theory for the D'Alembertian and we know that the Cauchy problem for the operator \(\partial^ 2_ t - A (\partial_ x)\) has Huygens' principle in the odd space dimension case. In the case of the isotropic elastic wave equation, that is \(a_{ip jq} = \lambda \delta_{ip} \delta_{jq} + \mu (\delta_{ij} \delta_{pq} + \delta_{iq} \;\delta_{jp})\) with \(\lambda + (2/n) \mu > 0\) and \(\mu > 0\), there are several informations more than the general case. In the Neumann boundary condition case, since so-called Rayleigh surface wave propagates along the boundary, we can not expect any rate of decay. In fact, in the paper the following result is the corollary of Theorem 1: Corollary 2. In the case of the isotropic elastic wave equation with Neumann boundary condition, the problem (1) does not have the uniform local energy decay property. Note that it follows from \textit{H. Iwashita} and \textit{Y. Shibata}, [Glas. Math., III. Ser. 23(43), No. 2, 291-313 (1988; Zbl 0696.35120)] that the local energy of solutions always decays, but the problem is the rate of decay. Corollary 3. In the case of the isotropic elastic wave equation with the Dirichlet boundary condition, if the obstacle \(\mathbb{R}^ n - \Omega\) is star-shaped, then we have the estimate (2). The corollary 3 was already proved by \textit{B. V. Kapitonov} [Sib. Math. J. 28, No. 3, 444-457 (1987; Zbl 0645.35059)] and Iwashita and Shibata by completely different methods.
scientific article; zbMATH DE number 591441