Young measure solutions for the wave equation with p(x,t)p(x,t)-Laplacian: Existence and blow-up

We consider the Dirichlet problem utt=Lu+f(x,t),(x,t)∈QT=Ω×(0,T),Lu=div(a(x,t)|∇u|p(x,t)−2∇u)+b(x,t)|u|σ(x,t)−2u,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u|ΓT=0,ΓT=∂Ω×(0,T), where the coefficients a(x,t),b(x,t),f(x,t)a(x,t),b(x,t),f(x,t) and the exponents of nonlinearities p(x,t),σ(x,t)p(x,t),σ(x,t) are given functions. We prove local and global existence and blow-up of Young measure solutions. We construct Young measure solutions as the limit of the sequence of solutions of the regularized equations utt=Lu+div(ε∇ut)+f(x,t).