On the continuity of solutions of the nonhomogeneous evolution p(x,t)-Laplace equation

We study the continuity of solutions of the homogeneous Dirichlet problem for the equation ut−div|∇u|p(x,t)−2∇u=fin QT=Ω×(0,T],where Ω⊂Rn, n≥2, is a domain with the boundary ∂Ω∈C2+α and p(x,t) is a given function such that 2nn+20. Moreover, if f is Lipschitz-continuous in t, p≡p(x), ∇p∈L∞(Ω) and either p(x)≥p−>2, or 2nn+2