Communications on Pure and Applied Analysis (CPAA)

On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type

Pages: 1783 - 1812, Volume 12, Issue 4, July 2013      doi:10.3934/cpaa.2013.12.1783

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Mikhail D. Surnachev - Computational Aeroacoustics Laboratory, Keldysh Institute of Applied Mathematics, Moscow 125047, Russian Federation (email)
Vasily V. Zhikov - Department of Mathematics, Vladimir State University, Vladimir 600000, Russian Federation (email)

Abstract: We prove the existence and uniqueness of global solutions to the Cauchy problem for a class of parabolic equations of the p-Laplace type. In the singular case $p<2$ there are no restrictions on the behaviour of solutions and initial data at infinity. In the degenerate case $p>2$ we impose a restriction on growth of solutions at infinity to obtain global existence and uniqueness. This restriction is given in terms of weighted energy classes with power-like weights.

Keywords:  Cauchy problem, p-Laplacian, existence, uniqueness, energy classes.
Mathematics Subject Classification:  Primary: 35K10, 35K15, 35K65, 35K67, 35K92.

Received: April 2012;      Revised: June 2012;      Available Online: November 2012.