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Regularity results for nonlinear parabolic problems in variable exponent Sobolev space with degenerate coercivity

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Abstract

In this paper, based on the theory of variable exponent Sobolev space, we study the existence and regularity results for a class of nonlinear parabolic problems with degenerate coercivity and variable exponents whose model problem is

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{\partial u}{\partial t}-\displaystyle {\text{ div } \left( \frac{\vert \nabla u\vert ^{p(x)-2}\nabla u}{(1+\vert u\vert )^{\theta (p(x)-1)}}\right) =-\text{ div } f }&{} \text{ in } Q=\Omega \times (0,T)\\ u=0 &{} \text{ on } \Gamma =\partial \Omega \times (0,T)\\ u(x,0)=0 &{} \text{ in } \Omega ,\\ \end{array} \right. \end{aligned}$$

where \(\Omega\) is a bounded, open subset of \({\mathbb {R}}^N\) \((N\ge 2),\) and the measurable function f belongs to \((L^{m(.)}(0,T,L^{r(.)}(\Omega )))^N.\) Our results are a natural generalization of some existing ones in the context of constant exponents. In order to get existence and regularity of solutions we use an approximation procedure.

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  1. Department of Mathematics and Informatics, National School of Applied Sciences Khouribga, Sultan Moulay Slimane University, Khouribga, 25000, Morocco

    Abdellatif Bouhal & Youssef El Hadfi

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  1. Abdellatif Bouhal
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Bouhal, A., El Hadfi, Y. Regularity results for nonlinear parabolic problems in variable exponent Sobolev space with degenerate coercivity. J Elliptic Parabol Equ 8, 1013–1040 (2022). https://doi.org/10.1007/s41808-022-00184-7

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