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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 $$|v|> \varepsilon_0> 0$$.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations
##### Keywords:
energy decay estimate; nonlinear dissipative term