Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy

Abstract

We consider viscoelastic wave equations of the Kirchhoff typeutt-M(∥∇u∥22)Δu+∫0tg(t-s)Δu(s)ds+ut=|u|p-1uwith Dirichlet boundary conditions, where∥⋅∥pdenotes the norm in the Lebesgue spaceLp. Under some suitable assumptions ongand the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy, in the sense thatlim⁡t→T*-(∥u(t)∥22+∫0t∥u(s)∥22ds)=∞for some0<T*<+∞.