Remarks on the blow-up in linearly damped abstract wave equations

Abstract

We consider a class of abstract nonlinear wave equations with a linear dissipation term. In a previous work, it was proved that if a dissipation coefficient δ>0is sufficiently small, for any positive value of an initial energy, there always exist initial data whose corresponding solution blows up in finite time. In this study, we analyze the same problem for any positive coefficient δ with similar conclusions but with better and broader applications. We compare our results with some others already published in the literature.

Keywords

• Hyperbolic equations
• Generalized Boussinesq equation
• Blow-up
• High energies