On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms. (English) Zbl 0962.35025

From the introduction: Let \(\Omega\subset \mathbb{R}^N\) be a bounded domain with smooth boundary \(\partial\Omega\). We are concerned with the initial-boundary value problem \[ u_{tt}- M(\|\nabla u(t)\|^2_2)\Delta u+ \delta|u_t|^{p- 1}u_t= \mu|u|^{q- 1}u,\quad t\geq 0,\quad x\in\Omega, \]
\[ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x),\quad x\in\Omega, \]
\[ u(x,t)|_{\partial\Omega}= 0,\quad t\geq 0, \] where \(M(s)\) is a positive \(C^1\)-class function for \(s\geq 0\) satisfying \(M(s)\geq m_0>0\), \(\delta> 0\) and \(\mu\in\mathbb{R}\) are given constants.
Our purpose in this paper is to give a global solvability in the class \(H^2\times H^1_0\) and energy decay without the smallness of the initial data in a certain sense. We shall use the method of M. Nakao [J. Differ. Equations 98, No. 2, 299-327 (1992; Zbl 0762.35013)]. The nonlinear damping term \(\delta|u_t|^{p- 1}u_t\) will play an important role in deriving the decay estimate.


35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
45K05 Integro-partial differential equations


Zbl 0762.35013
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