A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy

Abstract

In this letter we consider the nonlinear viscoelastic equation u t t − Δ u + ∫ 0 t g ( t − τ ) Δ u ( τ ) d τ + u t = | u | p − 1 u in Ω × ( 0 , ∞ ) with Dirichlet boundary conditions. Under some appropriate assumptions on g and the initial data, a blow-up result with arbitrary positive initial energy is established for 1 < p < n n − 2 (if n = 1 , 2 , then 1 < p < ∞ ).