Electron. J. Differential Equations, Vol. 2018 (2018), No. 44, pp. 1-20.

Existence and global behavior of solutions to fractional p-Laplacian parabolic problems

Jacques Giacomoni, Sweta Tiwari

Abstract:
First, we discuss the existence, the uniqueness and the regularity of the weak solution to the following parabolic equation involving the fractional p-Laplacian,
$$\displaylines{ u_t+(-\Delta)_{p}^su +g(x,u)= f(x,u)\quad \text{in } Q_T:=\Omega\times (0,T), \cr u = 0 \quad \text{in } \mathbb{R}^N\setminus\Omega\times(0,T),\cr u(x,0) =u_0(x)\quad \text{in }\mathbb{R}^N. }$$
Next, we deal with the asymptotic behavior of global weak solutions. Precisely, we prove under additional assumptions on f and g that global solutions converge to the unique stationary solution as $t\to \infty$.

Submitted July 29, 2017. Published February 8, 2018.
Math Subject Classifications: 35K59, 35K55, 35B40.
Key Words: p-fractional operator; existence and regularity of weak solutions; asymptotic behavior of global solutions; stabilization.

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Jacques Giacomoni
Université de Pau et des Pays de l'Adour
CNRS, LMAP (UMR 5142), Bat. Ipra
avenue de l'Université, Pau, France
email: jacques.giacomoni@univ-pau.fr
Sweta Tiwari
Department of Mathematics
IIT Guwahati, India
email: swetatiwari@iitg.ernet.in

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