Blow up of solutions for a Parabolic-Elliptic Chemotaxis System with gradient dependent chemotactic coefficient

TL;DR

This paper investigates conditions under which solutions to a chemotaxis PDE system with gradient-dependent chemotactic coefficient blow up in finite time, focusing on radially symmetric solutions in a bounded domain.

Contribution

It establishes finite-time blow-up results for a nonlinear chemotaxis system with gradient-dependent chemotactic sensitivity, extending understanding of solution behavior in such models.

Findings

Solutions blow up in finite time for large chemotactic sensitivity.

Blow-up occurs under certain initial mass conditions.

Radially symmetric solutions exhibit finite-time singularities.

Abstract

We consider a Parabolic-Elliptic system of PDE's with a chemotactic term in a $N$-dimensional unit ball describing the behavior of the density of a biological species "$u$" and a chemical stimulus "$v$". The system includes a nonlinear chemotactic coefficient depending of ``$\nabla v$", i.e. the chemotactic term is given in the form $$- div (\chi u |\nabla v|^{p-2} \nabla v), \qquad \mbox{ for } \ p \in ( \frac{N}{N-1},2), \qquad N >2 $$ for a positive constant $\chi$ when $v$ satisfies the poisson equation $$- \Delta v = u - \frac{1}{|\Omega|} \int_{\Omega} u_0dx.$$ We study the radially symmetric solutions under the assumption in the initial mass $$ \frac{1}{|\Omega|} \int_{\Omega} u_0dx>6.$$ For $\chi$ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.