POINTWISE ESTIMATES OF SOLUTIONS TO WEIGHTED POROUS MEDIUM AND FAST DIFFUSION EQUATIONS VIA WEIGHTED RIESZ POTENTIALS

Abstract

Problems related to the study of the properties of solutions of partial differential equations have attracted the attention of many authors in recent decades. The main qualitative properties of solutions of homogeneous linear elliptic equations of the second-order divergent type with measurable coefficients without lower-order terms are already known from the results of De Giorgi, Nash, and Moser. These results are generalized by Serrin, Ladyzhenska and Uraltseva, Aronson and Serrin, and Trudinger for wide classes of elliptic and parabolic equations with lower-order terms from the corresponding $ L^{q} $-classes. Analogous results for evolution equations with $ p \,-$Laplacian appeared much later. The first significant transition to the $ p \,-$Laplace equation with the measure $~\mu~$ in the right-hand side was achieved by Kilpelainen and Maly. They established point estimates of the solutions in terms of the nonlinear Wolff potential. These results were later extended by \linebreak Trudinger and Wang and Laboutin to nonlinear and subelliptic quasilinear equations. Irregularly elliptic and inhomogeneous parabolic equations without/or with singular lower terms have been studied for a long time. The first results in this direction were obtained by Fabes, Kenig and Separioni and Gutierrez for a weighted linear elliptic equation with weight representing $ A_{2} $ of the Mackenhaupt class. In recent decades, there has been a growing interest in parabolic and elliptic equations due to their application in modeling nonlinear physical processes occurring in heterogeneous media. Also, these equations are interesting because a general qualitative theory has not been constructed for them. Among the researchers who obtained the first significant results, we note Di Benedetto E., Bogelein V., Ivanov A. V., Duzaar F., Gianazza U., Vespri V..