Decay Estimate and Blow-up for a Damped Wave Equation with Supercritical Sources

This article deals with an initial and boundary value problem to the following damped wave equation: u t t − Δ u − ω Δ u t + μ u t = | u | p − 2 u in a bounded domain. An energy decay estimate for the solutions when ω ≥ 0 and μ > − ω λ 1 is obtained by adopting a new method, where λ 1 is the first eigenvalue of the operator − Δ under the homogeneous Dirichlet boundary conditions. Moreover, a blow-up result is proved for solutions with high energy initial data. An estimate of the upper bounded for the lifespan of the solution is showed as well. These results give some answers to the open problems in Gazzola and Squassina (Ann. Inst. Henri Poincaré 23:185–207, 2006 ).