A supersolution approach to doubly degenerate parabolic equations with weights

2025/12/23 by Daniele Andreucci, Andreucci, Daniele, Anatoli F. Tedeev +2
Engineering · Mathematics · #35K55 35K65 35B40 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.2512.20373

openalex publication_date 2025/12/23 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, where the exponent satisfies the doubling condition. In particular, both the so called logconvex and logconcave cases may be considered. Under the additional natural assumptions we construct supersolutions and subsolutions allowing us to control the precise sharp temporal decay bounds. We apply our results also to an equation with inhomogeneous density, via a suitable variable transformation.

Citations

• The Cauchy problem for doubly degenerate parabolic equations with weights
• Asymptotic properties of solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density on manifolds
• Sobolev-type inequalities on Cartan-Hadamard manifolds and applications to some nonlinear diffusion equations
• Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds
• Fundamental solution and long time behaviour of the Porous Medium Equation in Hyperbolic Space
• Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities

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• Long time behavior of solutions of degenerate parabolic equations with inhomogeneous density on manifolds
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• Weak Harnack estimates for supersolutions to doubly degenerate parabolic equations