Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions

Abstract

In this paper we consider the following problem utt=Δu+b(x,t)|u|σ(x,t)−2u+f(x,t),x,t∈QT=Ω×(0,T),u(x,0)=u0(x),ut(x,0)=u1(x)≥0,x∈Ω,uΓT=0,ΓT=∂Ω×(0,T) under conditions 0≤b−(t)≤b(x,t)≤b+(t)<∞,1≤σ−(t)≤σ(x,t)≤σ+(t)<∞.We study the existence and uniqueness of weak and strong solutions. We prove the blowup of nonnegative solutions. Also we consider the blowup effect for a more generalized problem {utt=Lu+f(x,t,u)inQT,u(x,0)=u0(x)inΩ,ut(x,0)=u1(x)inΩ,u=0onΓT, where L is the linear elliptic operator Lu=Di(aij(x)Dju+ai(x)u)+a0(x)u, and the right-hand side has the form f(x,t,u)=∑k=1Nbk(x,t)uσk(x,t)−1+∑i=NKci(x,t)∫Ωdi(s,t)uσi(s,t)−1ds, with bk≥0, ci≥0, di≥0.