On nonlinear diffusion problems with strong degeneracy

In this paper, we study the “triply” degenerate problem: b(v)t−Δg(v)+divΦ(v)=f on Q:=(0,T)×Ω, b(v(0,⋅))=b(v0) on Ω and “g(v)=g(a) on some part of the boundary (0,T)×∂Ω,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions b,g are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition b(0)=g(0)=0 and the range condition R(b+g)=R. Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111–139].