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

2007/02/07 by Yanjin Wang, Wang, Yanjin
Computer Science · Engineering · Mathematics · #35L15 #35Q72 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.math/0702190

openalex publication_date 2007/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the long time behavior of solutions of the initial value problem for the damped wave equation of the form u_tt-ρ(x)^-1Δu+u_t+m²u=f(u) with some ρ(x) and f(u) on the whole space \Rⁿ (n≥ 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.

Related

• Finite time blow up for wave equations with strong damping in an exterior domain
• Blow up of solutions for semilinear wave equations with noneffective damping
• Initial boundary value problem for a strongly damped wave equation with a general nonlinearity
• Finite time blow up for wave equations with strong damping in an exterior domain
• Blow-up of solutions to semilinear wave equations with a time-dependent strong damping