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Energy decay for the wave equation with a nonlinear weak dissipation. (English) Zbl 0848.35076
The author considers the decay property of the solutions of the mixed problem for the wave equation with the nonlinear dissipative term \[ u_{tt}- \Delta u+ \varphi(u_t)= 0\quad \text{in} \quad \Omega\times (0, \infty),\tag{1} \]
\[ u(x, 0)= u_0(x),\quad u_t(x, 0)= u_1(x)\quad \text{and} \quad u|_{\partial \Omega}= 0,\tag{2} \] where \(\Omega\in C^2\) is a domain in \(\mathbb{R}^N\). The function \(\rho: \mathbb{R}\to \mathbb{R}\) is continuous and nondecreasing on \(\mathbb{R}\) and satisfies some growth conditions. The initial functions \((u_0, u_1)\) belong to the space \((H^2\cap H^1_0)\times H^1_0\). The main aim of the present paper is to derive a precise decay rate of the energy \(E(t)= 1/2\{|u_t(t)|^2+ |\nabla u(t)|^2\}\) for a unique solution of (1), (2). These results generalize previous results of the author, quoted in the reviewed paper, and they can be used for the case \(\rho(v)= k_0\text{ sgn } v\) for \(|v|> \varepsilon_0> 0\).

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations