Topological Methods in Nonlinear Analysis

Asymptotic behavior of solutions of some nonlinearly damped wave equations on $\mathbb R^N$

Nikos Karachalios and Nikos Stavrakakis

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Abstract

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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 18, Number 1 (2001), 73-87.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1471876533

Mathematical Reviews number (MathSciNet)
MR1888495

Zentralblatt MATH identifier
1199.35229

Citation

Karachalios, Nikos; Stavrakakis, Nikos. Asymptotic behavior of solutions of some nonlinearly damped wave equations on $\mathbb R^N$. Topol. Methods Nonlinear Anal. 18 (2001), no. 1, 73--87. https://projecteuclid.org/euclid.tmna/1471876533


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