Equivalence between radial solutions of different parabolic gradient-diffusion equations and applications

TL;DR

This paper demonstrates an equivalence between a generalized parabolic gradient-diffusion equation and the standard p-parabolic equation in a fictitious space dimension for radial solutions, enabling explicit solutions and asymptotic analysis.

Contribution

It establishes a novel equivalence for radial solutions, allowing explicit solutions and detailed asymptotic behavior analysis for generalized parabolic equations.

Findings

Explicit solutions of Barenblatt type derived

Asymptotic behavior of solutions characterized

Parabolic Harnack's inequality established

Abstract

We consider a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that has been proposed in stochastic game theory. We establish an equivalence between this equation and the standard $p$-parabolic equation posed in a fictitious space dimension, valid for radially symmetric solutions. This allows us to find suitable explicit solutions for example of Barenblatt type, and as a consequence we settle the exact asymptotic behaviour of the Cauchy problem even for nonradial data. We also establish the asymptotic behaviour in a bounded domain. Moreover, we use the explicit solutions to establish the parabolic Harnack's inequality.