Blow up of solutions for a Parabolic-Elliptic chemotaxis system with gradient dependent chemotactic coefficient

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 “∇v,” i.e. the chemotactic term is given in the form−div(χu|∇v|p−2∇v), for p∈(NN−1,2), N>2for a positive constant χ when v satisfies the poisson equation−Δv=u−1|Ω|∫Ωu0dx.We study the radially symmetric solutions under the assumption in the initial mass1|Ω|∫Ωu0dx>6.For χ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.