Degenerate parabolic equations in divergence form: fundamental solution and Gaussian bounds

2025/03/10 by Baadi, Khalid · 2 citations
#35K10 #35K15 Secondary: 35K08 #35K65 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 35A08

paper · doi:10.48550/arxiv.2503.07569

Abstract

In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial A₂-weight. We prove that having a generalized fundamental solution with upper Gaussian bounds is equivalent to Moser's L²-L^∞ estimates for local weak solutions. In the special case of real coefficients, Moser's L²-L^∞ estimates are known, which provide an easier proof of Gaussian upper bounds, and a known Harnack inequality is then used to derive Gaussian lower bounds.

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• On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms
• On Hardy-Littlewood-Sobolev estimates for degenerate Laplacians

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