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A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations. (English) Zbl 1023.35076

Most of the paper deals with the initial-boundary value problem \(u_{tt}-\alpha \Delta u + g(u_t)=f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), \(u(x,0)=i_0\), \(u_t(x,0)=v_0\), \(x\in \Omega,\) where \(\alpha>0\), \(\Omega\subset \mathbb R^n\) is a bounded domain with sufficiently smooth boundary, \(g(u_t)=\delta u_t|u_t|^{\lambda-2}\), \(\delta\geq 0\), \(\lambda>2\), \(f(u)=\mu u|u|^{r-2}\), \(\mu>0\), \(r>0.\) The nondissipative (\(\delta\neq 0\)) and the dissipative cases are distinguished. Necessary and sufficient conditions for existence of global and nonglobal solutions are derived. The author uses the concepts of stable and unstable sets. The qualitative behaviour of forward and backward solutions for the equation without dissipation is analyzed. The characterization of blow-up and the asymptotic behaviour is presented. The sufficient conditions for existence of nonglobal solutions are extended for the nonlinear Kirchhoff equation \(u_{tt}-M(\|\nabla u \|_2^2)\Delta u + g(u_t)=f(u)\) in \(\Omega\) with \(\|.\|_2\) the norm in \(L^2(\Omega)\), \(M(s^2)=\alpha+\beta s^{2\gamma}\), \(\alpha\geq 0\), \(\beta\geq 0\), \(\alpha+\beta>0\), \(\gamma\geq 1.\)

MSC:

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