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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
##### Keywords:
Kirchhoff equations
Full Text:
##### References:
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