Finite time blow-up results for the damped wave equations with arbitrary initial energy in an inhomogeneous medium

Details Finite time blow-up results for the damped wave equations with arbitrary initial energy in an inhomogeneous medium Finite time blow-up results for the damped wave equations with arbitrary initial energy in an inhomogeneous medium

2007-02-07

arxiv.org/abs/math/0702190v1

Mathematics

Analysis of PDEs

15pages

Scientific paper

In this paper we consider the long time behavior of solutions of the initial value problem for the damped wave equation of the form \begin{eqnarray*} u_{tt}-\rho(x)^{-1}\Delta u+u_t+m^2u=f(u) \end{eqnarray*} with some $\rho(x)$ and $f(u)$ on the whole space $\R^n$ ($n\geq 3$). For the low initial energy case, which is the non-positive initial energy, based on concavity argument we prove the blow up result. As for the high initial energy case, we give out sufficient conditions of the initial datum such that the corresponding solution blows up in finite time.

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