Degenerate and Singular Porous Medium Type Equations with Measure Data

Abstract

AbstractWe consider the inhomogeneous porous medium equation $$\partial_t u - \Delta u^m = \mu, \qquad m>\frac{(N-2)_+}{N},$$ and more general equations of porous medium type with a non-negative Radon measure μ on the right-hand side. In a first step, we prove a priori estimates for weak solutions in terms of a linear Riesz potential of the right-hand side measure, which takes exactly the same form as the one for the classical heat equation. Then, we give an optimal criterium for the continuity of weak solutions, again in terms of a Riesz potential. Finally, we prove the existence of non-negative very weak solutions, and show that these constructed very weak solutions satisfy the same estimates. We deal with both the degenerate case \(m>1\), and the singular case \(\frac{(N-2)_+}{N}<m<1\).KeywordsWeak SolutionPotential EstimateNonlinear Parabolic EquationPorous Medium EquationRiesz PotentialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.