Stabilization of the total energy for a system of elasticity with localized dissipation
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 \(r\).
scientific article; zbMATH DE number 1839895