Logistic elliptic and parabolic problem for the fractional $p$-Laplacian

Abstract

In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-\Delta)_p^s u_\lambda=\lambda u_\lambda^q - b(x)u_\lambda^r \quad \text{in} \;\Omega,\\ u_\lambda=0 \quad \text{in} \; ( \mathbb{R}^d \backslash \Omega), \\ u_\lambda>0 \text{ in} \; \Omega. \end{cases}\ \end{equation*} We also prove behavior of $u_\lambda$ with respect to $\lambda,$ underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up.