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

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 $t\to \infty$.

Keywords

• p-fractional operator
• existence and regularity of weak solutions
• asymptotic behavior of global solutions
• stabilization