A supersolution approach to doubly degenerate parabolic equations with weights

TL;DR

The paper analyzes a Cauchy problem for a doubly degenerate parabolic equation with a space-dependent exponential weight $f(x)=e^{g(|x|)}$, under doubling-type growth for $g$ and the parameter range $1<p<N$, $p+m-3>0$. It develops self-similar barrier functions (supersolutions and subsolutions) to obtain sharp temporal decay rates in $L^{\infty}$ and finite speed of propagation, including explicit bounds tied to $g^{-1}(\log t)$. The authors also extend the barrier method to an inhomogeneous-density setting via a radial transformation, deriving a transformed density $\rho(s)$ with behavior $\rho(s)\sim s^{-p}$ up to logarithmic factors and showing the barrier framework persists under this change of variables. A detailed analysis of the transformation is provided in Section 5 and its subsection 5.1, which constructs the auxiliary radial map $\hat r(s)$ and proves its asymptotics, ensuring the transformed problem inherits the same qualitative barrier estimates. Overall, the work introduces a robust barrier-based approach for sharp decay and propagation results in non-power nonlinearities and connects weighted parabolic problems to inhomogeneous-density reformulations on manifolds and related geometric settings.

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.

Theorems & Definitions (21)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proofs of Theorems \ref{['t:sup_sup']} and \ref{['t:sub_sub']}
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1