The Stability of Parabolic Problems with Nonstandard p ( x, t )-Growth

André H. Erhardt
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André H. Erhardt: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, Oslo 0316, Norway

Mathematics, 2017, vol. 5, issue 4, 1-14

Abstract: In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation ? t u ? div a ( x , t , ? u ) + ? ( | u | p ( x , t ) ? 2 u ) = 0 in ? T , where ? ? 0 and ? t u denote the partial derivative of u with respect to the time variable t , while ? u denotes the one with respect to the space variable x . Moreover, the vector-field a ( x , t , · ) satisfies certain nonstandard p ( x , t ) -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values.

Keywords: nonlinear parabolic problems; existence theory; variable exponents; stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2017
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