On Solutions to Nonlinear Reaction–Diffusion–Convection Equations with Degenerate Diffusion

Abstract

This paper consists of three parts. In Section 2, the Cauchy problem for general reaction-convection equations with a special diffusion term G(u)=um in multidimensional space is studied and Hölder estimates of weak solutions with explicit Hölder exponents are obtained by applying the maximum principle. In Section 3, for any nondecreasing smooth function G, the sharp regularity estimate G(u)∈C(1) up to the boundaries for the radial solution u of the general equation of Newtonian filtration is obtained by applying the maximum principle with the Minty's device. A direct by-product is the sharp regularity estimate of the temperature to the classical two-phase Stefan model. In Section 4, the Hölder continuity of weak solutions of the initial-boundary value problem for general nonlinear reaction–diffusion–convection equations is considered. Under the critical condition on the diffusion function G:meas{u:G′(u)=g(u)=0}=0, we obtain a Hölder continuous solution u and the sharp regularity estimate G(u)∈C(1) up to the boundaries. Our proof is based on the maximum principle.