Existence of solutions to some degenerate parabolic equation associated with the pp-Laplacian in the critical case

This paper is concerned with the initial–boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t)ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t) with initial data u0∈Lr(Ω)u0∈Lr(Ω). Akagi (2007) [1] established the existence of local (in time) solutions to this problem in the case r>N(q−p)/pr>N(q−p)/p; however, the critical case r=N(q−p)/pr=N(q−p)/p has been left as an open problem. In this paper, even in the critical case r=N(q−p)/pr=N(q−p)/p, the existence of solutions to the problem is established under a certain restriction on u0u0. The key to our proof is Tartar’s inequality, which enables us to derive desired convergences of approximate solutions to the problem from the compactness of the embedding W01,p(Ω)⊂L2(Ω). Incidentally, any smoothness is not imposed on ∂Ω∂Ω at all while a smooth boundary is needed in  [1].