A nonlocal logistic equation with nonlinear advection term

Abstract

In this paper, we study a nonlocal logistic equation with nonlinear advection term \begin{equation}\label{Pp} \begin{cases} \displaystyle -\Delta u+\overrightarrow {\alpha}(x)\cdot {\nabla (u^p)} =\left(\lambda-\int_{\Omega}K(x,y)u^{\gamma}(y)dy \right)u &\mbox{in }\Omega,\\ u=0 &\mbox{ on }\partial\Omega, \end{cases} \tag{\rom{P}$_p$} \end{equation} where $\Omega\subset\R^N$, $N\geq1$, is a bounded domain with smooth boundary, $\overline{\alpha}(x)=(\alpha_1(x),\dots,\alpha_N(x))$ is a flow satisfying suitable condition, $\gamma> 0$, $p\geq1$, $\lambda\in\R$ and $K\colon \Omega\times\Omega\rightarrow\R$ is a nonnegative function with $K\in L^{\infty}(\Omega\times\Omega)$ and verifying other conditions that will be detailed below. It is very important to note that, this equation is not the classic logistic equation due to the inclusion of the term $\overline{\alpha}(x)\cdot \nabla (u^p)$, moreover, the inclusion of the integral nonlocal term on the right-hand side makes the problem closer to a real world situation.