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

Nikos Karachalios, Nikos M. Stavrakakis

DOI: http://dx.doi.org/10.12775/TMNA.2001.024

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.

Keywords


Semilinear hyperbolic equations; blow-up; nonlinear dissipation; potential well; concavity method; unbounded domains; generalized Sobolev spaces

Full Text:

FULL TEXT

Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism