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

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 $$- div (χu |\nabla v|^{p-2} \nabla v), \qquad \mbox{ for } \ p \in ( \frac{N}{N-1},2), \qquad N >2 $$ for a positive constant $\chi$ when $v$ satisfies the poisson equation $$-\Delta v=u-\frac{1}{|\Omega |}{\int }_{\Omega }{u}_{0}dx.$$ We study the radially symmetric solutions under the assumption in the initial mass $$\frac{1}{|\Omega |}{\int }_{\Omega }{u}_{0}dx>6.$$ For $\chi$ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.