Stabilization of the total energy for a system of elasticity with localized dissipation.(English)Zbl 1031.35094

The paper studies the decay property for a class of elastic waves in a bounded domain $$\Omega\subset \mathbb{R}^3$$. The system of equations with state $$u(x, t)= (u_1,u_2,u_3)$$ is described by $\begin{gathered} u_{tt}- a^2\Delta u- (b^2- a^2)\nabla\text{ div }u+ q(x) u_t= 0,\\ u(x,0)= f(x)\in [H^2(\Omega)\cap H^1_0(\Omega)]^3,\quad u_t(x, 0)= g(x)\in [H^1_0(\Omega)]^3,\\ u(x, t)= 0,\quad x\in\Gamma= \partial\Omega,\end{gathered}$ where the constants $$a$$, $$b$$ are such that $$0< a< b$$, and $$q\geq 0$$ means the dissipation term. For any $$x_0\in\mathbb{R}^3$$, set $$\Gamma_1= \{x\in\Gamma: (x- x_0)\cdot\eta(x)> 0\}$$, where $$\eta(x)$$ denotes the unit outward normal on $$\Gamma$$. Let $$\omega\in \overline\Omega$$ be a neighborhood of $$\Gamma_1$$. The assumption on $$q(x)$$ is such that $q\in L^r(\Omega),\quad \int_\omega {dx\over q(x)^p}< \infty,\quad\text{and }q(x)= 0\quad\text{in }\Omega\setminus\omega$ for some constants $$p\geq 1$$ and $$r\geq {1\over 4}(9+ \sqrt{33})$$. Thus the dissipation occurs locally only near the boundary $$\Gamma$$. Let $$E(t)$$ denote the total energy of the system. Then $$E(t)$$ is a nonincreasing function. In fact, it is shown that $$E'(t)= -\int_\omega q(x)|u_t|^2 dx\leq 0$$. Based mainly on the multiplier technique, it is then shown that $$E(t)$$ decays polynomially as $$t\to\infty$$, where the decay rate is determined by $$p$$ and $$r$$.
Reviewer: T.Nambu (Kobe)

MSC:

 35L55 Higher-order hyperbolic systems 93D15 Stabilization of systems by feedback 35B35 Stability in context of PDEs 93C20 Control/observation systems governed by partial differential equations 35Q72 Other PDE from mechanics (MSC2000) 35B45 A priori estimates in context of PDEs