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Periodic solution and decay for some nonlinear wave equations with sublinear dissipative terms. (English) Zbl 0601.35075

This paper is concerned with nonlinear equations in one space dimension of the form \[ u_{tt}-u_{xx}+\rho (x,u_ t)+\beta (x,u)=f(x,t)\quad on\quad I\times {\mathbb{R}},\quad u|_{\partial I}=0\quad for\quad t\in {\mathbb{R}}, \] where I is an intervall and f(x,t) is an \(\omega\)-periodic function in t. In case \(\rho (x,u_ t)=\nu u_ t\), \(\nu >0\) or \(\rho\) (x,v)v\(\geq c_ 0 | v|^{r+2}-c_ 1\) for some \(r\geq 0\), \(c_ 0,c_ 1>0\) (\(\rho\) (x,v) is superlinear as \(| v| \to \infty)\) this problem has been considered even for higher dimensional equations and more strongly nonlinear equations.
The author here considers the sublinear case (\(\rho\) (x,v)v\(\geq c_ 0 | v|^{2-r}-c_ 1\) for \(0<r<1\), \(c_ 0,c_ 1>0)\) and gives existence theorems and results concerning the energy decay as \(t\to \infty\).
Reviewer: M.Schneider

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L05 Wave equation
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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