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Zbl 1098.35033
Szomolay, Barbara
Decay of solutions of some degenerate hyperbolic equations of Kirchhoff type. (English)
[J] Commentat. Math. Univ. Carol. 44, No. 1, 71-84 (2003). ISSN 0010-2628

The existence of global solution of the initial-boundary value problem to the equation $u_{tt}+\gamma u_{t}-m(\Vert \nabla u\Vert ^2)\triangle u+\delta \vert u\vert ^{\alpha }u=f,\ t\in (0,\infty ),\ x\in \Omega \subset \Bbb R^{n}$, where $\gamma ,\delta >0$, $\alpha \geq 0$, $m(r)\geq 0$ is proved for small data. If $f$ decays exponentially to zero for $t\rightarrow \infty $, every solution of the problem tends to zero. The decay is exponential for non-degenerate case, i.e. $m(r) \geq m_0 > 0$. The results are extended to the equations with $\triangle ^2u$.
[Marie Kopáčková (Praha)]

MSC 2000:: *35B40 Asymptotic behavior of solutions of PDE
35L80 Hyperbolic equations of degenerate type
35L20 Second order hyperbolic equations, boundary value problems
45K05 Integro-partial differential equations

Keywords: asymptotic behavior of solution; hyperbolic partial differential equation of degenerate type; exponential decay

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