Asymptotic behavior of solutions of some nonlinearly damped wave equations on $\mathbb R^N$
Keywords
Semilinear hyperbolic equations, blow-up, nonlinear dissipation, potential well, concavity method, unbounded domains, generalized Sobolev spacesAbstract
We discuss the asymptotic behavior of solutions of the nonlinearly damped wave equation $$ u_{tt} +\delta \vert u_t\vert ^{m-1}u_t -\phi (x)\Delta u = \lambda u\vert u\vert ^{\beta -1}, \quad x \in \mathbb R^n, \ t \geq 0, $$ with the initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3$, $ \delta > 0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{p}(\mathbb R^n)\cap L^{\infty}(\mathbb R^n)$, for some $p$. We prove blow-up of solutions when the source term dominates over the damping, and the initial energy is assumed to be positive. We also discuss global existence energy decay of solutions.Downloads
Published
2001-09-01
How to Cite
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KARACHALIOS, Nikos and STAVRAKAKIS, Nikos M. Asymptotic behavior of solutions of some nonlinearly damped wave equations on $\mathbb R^N$. Topological Methods in Nonlinear Analysis. Online. 1 September 2001. Vol. 18, no. 1, pp. 73 - 87. [Accessed 13 November 2024].
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