Continuity and Harnack inequality for solutions of the weight parabolic equation with p-Laplasian contained by the singular lower-order term

Abstract

Problems related to the study of the properties of solutions of differential equations in partial derivatives have attracted the attention of many authors over the past few decades. Many works in recent years have been devoted to the study of the $ p \,-$Laplacian equation, which simulates non-Newtonian elastic filtration and describes many physical processes. When $ p = 1 $ it has a geometric meaning (the equation of the minimum surface) and its study is fundamentally different from the theory for $ p> 1. $ In the case of $ p <{2} $ possible complete attenuation of the solution in a finite time, and when $ p> {2} $ we have a compact solution carrier. This effect is observed on the example of known ``Barenblatt's solutions''. It seems that the question of Harnack's inequality for the solutions of parabolic equations with $ p \,-$Laplacian, which contain a singular junction, still remains open. In this paper, we made the first attempt to study the Harnack inequality for solutions of degenerate equations with $ p \,-$Laplacian perturbed singularity of the lower-order terms. We extend the result to the case of weight parabolic equations with $ p \,-$Laplacian. The proof is based on the corresponding modifications of the de Giorgi iterative technique after the adaptation of the Kilpel\"{a}inen -- Mal\'{y} technique to parabolic equations in combination with the ideas of Liskevich and Skrypnik. This article is a continuation of our study of that devoted to the qualitative properties of the solutions of the parabolic equation with $ p \,-$Laplacian. We summarize the known results in the case of an equation that contains the weights of the Muckenhoupt classes and the singular lower-order terms. Using the weight nonlinear parabolic potential of Wolf of the right-hand side of the equation, we prove a Lemma of De Giorgi type, continuity of solutions, Harnack inequality, show the performance of the Expansion of Positivity.