Large time behavior for inhomogeneous damped wave equations with nonlinear memory

Abstract

We investigate the large time behavior for the inhomogeneous damped wave equation with nonlinear memory φtt (t, ω) − ∆φ(t, ω) + φt (t, ω) =1 ∫t Γ(1−ρ) 0(t − σ)−ρ |φ(σ, ω)|q dσ + µ(ω), t > 0, ω ∈ RN imposing the condition (φ(0, ω), φt (0, ω)) = (φ0 (ω), φ1 (ω)) in RN, where N ≥ 1, q > 1, 0 < ρ < 1, φi ∈ L1loc(RN), i = 0, 1, µ ∈ L1loc(RN) and µ ̸≡ 0. Namely, it is shown that, if φ0, φ1 ≥ 0, ∫ µ ∈ L1 (RN) and µ(ω) dω > 0, then for all q > 1, the considered problem has no global RNweak solution.