Improved regularity for the inhomogeneous Porous Medium Equation

In this paper, we show that solutions of u(t) - div(m vertical bar u vertical bar(m-1)del u) = f is an element of L-q,L-r, m > 1, are locally of class C-0,C-gamma with gamma = min { nu alpha(-)(0), (2q - n)r -2q/q[mr - (m - 1)] }, where alpha(0) denotes the optimal Holder exponent for solutions of the homogeneous case and nu = 2(2 + (m - 1)alpha(0))(-1). This improves the recent result of Araujo, Maia, Urbano [1]. For the Porous Medium Equation, unlike other known examples, the main difficulty is that the theta-parabolic geometry of the equation and the available regularity do not match. Despite this difficulty, considering gamma as a function of m, our regularity result and the optimal local regularity of the associated homogeneous equation are comparable. (C) 2020 Elsevier Inc. All rights reserved.