On the double degenerate equation ut−diva(x,t,u,∇u)=b(x,t,u,∇u)

We show that a locally bounded nonnegative weak solution of the general double degenerate parabolic equation ut−diva(x,t,u,∇u)=b(x,t,u,∇u), satisfying the structure conditions a(x,t,u,ū)⋅ū≥ϕ(|u|)|ū|2−φ0(x,t),|a(x,t,u,ū)|≤ϕ(|u|)|ū|+ϕ12(|u|)φ1(x,t),|b(x,t,u,ū)|≤ϕ(|u|)|ū|+φ2(x,t), being ϕϕ a continuous function vanishing in u=0u=0 and u=1u=1, is locally continuous, generalizing therefore the study of the local regularity theory for the saturation in the flow of two immiscible fluids in a porous medium presented in DiBenedetto et al. (in press) [16].