On the existence of solutions for a parabolic-elliptic chemotaxis model with flux limitation and logistic source

In this paper we study the existence of solutions of a parabolic-elliptic system of partial differential equations describing the behaviour of a biological species $u$ and a chemical stimulus $v$ in a bounded and regular domain $\mathrm{\Omega }$ of ${\mathbb{R}}^N$. The equation for $u$ is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as $-\chi div(u\mathrm{\mid }\mathrm{\nabla }\psi {\mathrm{\mid }}^{p-2}\mathrm{\nabla }v)$, for $p>1$. The chemical substance distribution $v$ satisfies the elliptic equation $-\mathrm{\Delta }v+v=u$. The evolution of $u$ is also determined by a logistic type growth term $\mu u(1-u)$. The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for $p<3\mathrm{/}2$ and any $N\ge 2$.