On a class of fully nonlinear parabolic equations-Reference-Cited by-同舟云学术

On a class of fully nonlinear parabolic equations

Author:

Antontsev Stanislav¹,Shmarev Sergey²

Affiliation:

1. Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk, Russia; and CMAF-CIO, University of Lisbon, Portugal

2. Departamento de Matemáticas, Universidad de Oviedo, c/Calvo Sotelo s/n, Oviedo 33007, Spain

Abstract

Abstract We study the homogeneous Dirichlet problem for the fully nonlinear equation

u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,}

with the parameters

{m>1}

{\sigma>1}

and

{d\geq 0}

. At the points where

{\Delta u=0}

, the equation degenerates if

{m>2}

, or becomes singular if

{m\in(1,2)}

. We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as

{t\to\infty}

. Sufficient conditions for exponential or power decay of

{\|\nabla u(t)\|_{2,\Omega}}

are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

Funder

Russian Science Foundation

Ministerio de Ciencia e Innovación

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Publisher

Walter de Gruyter GmbH

Subject

Analysis

Link

Reference50 articles.

2. On abstract Barenblatt equations;Differ. Equ. Appl.,2011

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