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

2018/01/02 by Parviainen, Mikko, Vázquez, Juan Luis · 3 citations
#35B40 #35K55 #35K65 #35K67 #49L25 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.00613

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.

Cited by

• Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations
• Harnack inequality for degenerate fully nonlinear parabolic equations
• Boundary Hölder gradient estimates for parabolic p-Laplace type equations

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• Boundary regularity for a general nonlinear parabolic equation in non-divergence form
• Second order Sobolev regularity results for the generalized p-parabolic equation