Well-posedness for a class of parabolic equations with singular-degenerate coefficients

TL;DR

This paper establishes existence, uniqueness, and regularity results for a class of linear parabolic equations with coefficients that may be degenerate or singular, using weighted Sobolev spaces and the level-set method.

Contribution

It introduces new weighted Sobolev spaces and proves well-posedness for parabolic equations with singular-degenerate coefficients under smallness conditions.

Findings

Existence and uniqueness of weak solutions in weighted Sobolev spaces.

Regularity estimates for solutions with degenerate or singular coefficients.

Development of weighted inequalities and a weighted Aubin-Lions compactness theorem.

Abstract

This paper studies a class of linear parabolic equations with measurable coefficients in divergence form whose volumetric heat capacity coefficients are assumed to be in some Muckenhoupt class of weights. As such, the coefficients can be degenerate, singular, or both degenerate and singular. A class of weighted parabolic cylinders with a non-homogeneous quasi-distance function, and a class of weighted parabolic Sobolev spaces intrinsically suitable for the class of equations are introduced. Under some smallness assumptions on the mean oscillations of the coefficients, regularity estimates, existence, and uniqueness of weak solutions in the weighted Sobolev spaces are proved. To achieve the results, we apply the level-set method introduced by Caffarelli and Peral. Several weighted inequalities and a weighted Aubin-Lions compactness theorem for sequences in weighted parabolic Sobolev spaces are established.