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Non-local Diffusion Equations Involving the Fractional \(p(\cdot )\)-Laplacian

  • Elard J. HurtadoEmail author
Article
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Abstract

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional \(p(\cdot )\)-Laplacian with variable exponents, which is a fractional version of the nonhomogeneous \(p(\cdot )\)-Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem \(({\mathscr {P}}_{1})\) in a bounded domain \(\varOmega \) of \({\mathbb {R}}^N\) and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional \(p(\cdot )\)-Laplacian operator generates a (nonlinear) submarkovian semigroup on \(L^{2}(\varOmega ).\) In the second part of the paper we establish the existence of global attractors for problem \(({\mathscr {P}}_{2})\) under certain conditions in the potential \({\mathbb {V}}.\) Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent.

Keywords

Fractional p(x)-Laplace operator Asymptotic behavior of solutions Diffusion equations Attractors 

Mathematics Subject Classification

35R11 35B40 35K57 35B41 

Notes

Acknowledgements

The author would like to thanks Professor Juan Rocha Barriga and Professor Lauren Maria Mezzomo Bonaldo for their suggestions and fruitful discussions. The author would like to thank the anonymous referee by the careful reading and all of his/her valuable comments.

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Authors and Affiliations

  1. 1.Departamento de Matemática e Computação, Faculdade de Ciências e TecnologiaUNESP - Universidade Estadual PaulistaPresidente PrudenteBrazil

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