We study the existence and the asymptotic behavior of global solutions of the damped wave equation [Formula: see text] where a ∈ ℝ, α >1, t > 0, x ∈ ℝn, n = 1,2,3, with initial condition (u (0), ut (0)) = (φ,ψ). For α > 2 and α > 1+2 / n, we prove the existence of mild global solutions for small initial data with low regularity and which are not in L1(ℝn). Under the additional hypothesis, (2 < α < 5, when n = 3), we prove that some of these solutions are asymptotic to the self-similar solutions of the associated semi-linear heat equation [Formula: see text] with homogeneous slowly decreasing initial data behaving like c|x|-2 / (α-1) as |x|→ ∞.
We investigate the large time behavior for the inhomogeneous damped wave equation with nonlinear memory ϕtt(t,ω)−Δϕ(t,ω)+ϕt(t,ω)=1Γ(1−ρ)∫0t(t−σ)−ρ|ϕ(σ,ω)|qdσ+μ(ω),t>0, ω∈RN imposing the condition (ϕ(0,ω),ϕt(0,ω))=(ϕ0(ω),ϕ1(ω))inRN, where N≥1, q>1, 0<ρ<1, ϕi∈Lloc1(RN), i=0,1, μ∈Lloc1(RN) and μ≢0. Namely, it is shown that, if ϕ0,ϕ1≥0, μ∈L1(RN) and ∫RNμ(ω)dω>0, then for all q>1, the considered problem has no global weak solution.
Large Time Behavior for Inhomogeneous Damped Wave Equations with Nonlinear Memory