On solutions to nonlinear reaction-diffusion-convection equations with degenerate diffusion

This paper consists of three parts. In Section 2, the Cauchy problem for ge neral reaction-convection equations with a special diffusion term G(u) = u( m) in multi-dimensional space is studied and Holder estimates of weak solut ions with explicit Holder exponents are obtained by applying the maximum pr inciple. In Section 3, for any nondecreasing smooth function G, the sharp r egularity estimate G(u) is an element of C-(1) up to the boundaries for rad ial solution u of the general equation of Newtonian filtration is obtained by applying the maximum principle with the Minty's device. A direct by-prod uct is the sharp regularity estimate of the temperature to the classical tw o-phase Stefan model. In Section 4, the Holder continuity of weak solutions of the initial-boundary value problem for general nonlinear reaction-diffu sion-convection equations is considered. Under the critical condition on th e diffusion function G: meas [u: G'(u) - g(u) = 0] = 0, we obtain a Holder continuous solution u and the sharp regularity estimate G(u) is an element of C-(1) up to the boundaries. Our proof is based on the maximum principle. (C) 2001 Academic Press.