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 $$- div (\chi u |\nabla v|^{p-2} \nabla v), \qquad \mbox{ for } \ p \in ( \frac{N}{N-1},2), \qquad N >2 $$ 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.