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

TL;DR

This paper establishes existence, uniqueness, and qualitative behavior of solutions for a nonlocal fractional p-Laplacian logistic equation, including both elliptic and parabolic cases, highlighting the effects of nonlocality.

Contribution

It provides new results on the existence, uniqueness, and behavior of solutions for a nonlocal fractional p-Laplacian logistic problem, including parabolic dynamics.

Findings

Existence and uniqueness of weak solutions for the elliptic problem.

Behavior of solutions with respect to the parameter mbda.

Results on local and global existence, stabilization, finite time extinction, and blow-up for the parabolic problem.

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.