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A class of fourth order wave equations with dissipative and nonlinear strain terms. (English) Zbl 1138.35066

The authors deal with the initial boundary value problem with weak damping term \[ \begin{aligned} u_{tt}+ \Delta^2u+\gamma u_t+ \sum^n_{i=1} {\partial\over\partial x_i} \sigma_i(u_{x_i})= 0,&\quad x\in\Omega,\;t> 0,\\ u(x,0)= u_0(x),\quad u_t(x,0)= u_1(x), &\quad x\in\Omega,\\ u= {\partial u\over\partial\vec n}= 0, &\quad x\in\partial\Omega, \;t\geq 0, \end{aligned} \]
where \(\gamma\geq 0\), \(\Omega\subset\mathbb{R}^d\) is a bounded domain, \({\partial u\over\partial\vec n}\) denotes derivative of \(u\) in outward normal direction of \(\partial\Omega\). The authors introduce a family of potential wells \(\{W_\delta\}\) and outside sets \(\{V_\delta\}\) of \(\{W_\delta\}\). Then by using of them the authors not only obtain the invariant sets and vacuum isolating of solutions, but also give a threshold result of global existence and nonexistence of solutions.

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
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