Well-posedness for a class of parabolic equations with singular-degenerate coefficients (2510.20051v1)

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 version of weighted Aubin-Lions compactness theorem for sequences in weighted parabolic Sobolev spaces are established.