Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions

The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div (a(x, t, u)|u|�(x, t)|�u|p(x, t)-2 with given variable exponents �(x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type. DOI: https://doi.org/10.3934/cpaa.2013.12.1527 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-85929 Published Version Originally published at: Antontsev, Stansislav Nikolaevich; Chipot, Michel C (2013). Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Communications on Pure and Applied Analysis, 12(4):1527-1546. DOI: https://doi.org/10.3934/cpaa.2013.12.1527 COMMUNICATIONS ON doi:10.3934/cpaa.2013.12.1527 PURE AND APPLIED ANALYSIS Volume 12, Number 4, July 2013 pp. 1527–1546 UNIQUENESS AND COMPARISON THEOREMS FOR SOLUTIONS OF DOUBLY NONLINEAR PARABOLIC EQUATIONS WITH NONSTANDARD GROWTH CONDITIONS Stanislav Antontsev CMAF, University of Lisbon, Portugal Michel Chipot University of Zurich, Switzerland Sergey Shmarev University of Oviedo, Spain (To the memory of Professor I. V. Skrypnik) Abstract. The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div ( a(x, t, u)|u|α(x,t)|∇u|p(x,t)−2∇u ) + f(x, t) with given variable exponents α(x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type. The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div ( a(x, t, u)|u|α(x,t)|∇u|p(x,t)−2∇u ) + f(x, t) with given variable exponents α(x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.