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On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. (English) Zbl 0878.35081
The author considers the Cauchy problem for the strongly damped Kirchhoff equation of degenerate type \[ u_{tt}-\left(\int_{\Omega}|\nabla u|^2 dx\right)^{2\gamma} \Delta u -\Delta u_t=|u|^{\alpha}u, \eqno (*) \] where \(\gamma \geq1\), and \(\Omega\) is a bounded domain in \({\mathbb{R}}^N\), and proves the following result: If the initial data \(u_0\equiv u(0,x)\in H^2(\Omega)\cap H^1_0(\Omega)\) and \(u_1\equiv u_t(0,x)\in H^1_0(\Omega)\) are sufficiently small and \(\int |\nabla u_0|^{2(\gamma+1)}dx > \int |u_0|^{\alpha+2}dx\), then the problem has a global solution provided \(2\gamma <\alpha \leq 4/(N-2)\) if \(N> 3\), or \(2\gamma <\alpha\) if \(N=1,2\).

MSC:
35L80 Degenerate hyperbolic equations
35L75 Higher-order nonlinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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