Global existence and blow-up of solutions for a parabolic equation involving the fractional p(x)-Laplacian

Abstract

In this paper, we consider a nonlocal diffusion equation involving the fractional $p\left(x\right)$-Laplacian with nonlinearities of variable exponent type. Employing the subdifferential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.

Keywords:

• Fractional $p\left(x\right)$-Laplacian
• global existence
• blow-up
• potential well

2010 Mathematics Subject Classifications:

• 35R11
• 35B40
• 35K57
• 35B41

Acknowledgments

The author would like to thank Professor Claudianor Alves for his suggestions and fruitful discussions. The author would like to thank the anonymous referee for the careful reading of the paper and for his/her valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author.