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Mathematics 2017, 5(4), 50; doi:10.3390/math5040050

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

Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, Oslo 0316, Norway
Academic Editor: Michel Chipot
Received: 17 September 2017 / Revised: 5 October 2017 / Accepted: 8 October 2017 / Published: 12 October 2017
View Full-Text   |   Download PDF [310 KB, uploaded 12 October 2017]

Abstract

In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation t u div a ( x , t , u ) + λ ( | u | p ( x , t ) 2 u ) = 0 in Ω T , where λ 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. View Full-Text
Keywords: nonlinear parabolic problems; existence theory; variable exponents; stability nonlinear parabolic problems; existence theory; variable exponents; stability
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Erhardt, A.H. The Stability of Parabolic Problems with Nonstandard p(x, t)-Growth. Mathematics 2017, 5, 50.

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