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

$$-\mathit{div}(\chi u|\nabla v{|}^{p-2}\nabla v),\text{for}\mathrm{ }p\in (\frac{N}{N-1},2),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}N>2$$

for a positive constant χ when v satisfies the poisson equation

$$-\Delta v=u-\frac{1}{\left|\Omega \right|}{\int }_{\Omega }{u}_{0}dx.$$

We study the radially symmetric solutions under the assumption in the initial mass

$$\frac{1}{\left|\Omega \right|}{\int }_{\Omega }{u}_{0}dx>6.$$

For χ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.

Keywords:

• Blow up of solutions
• chemotaxis
• comparison principle

Acknowledgment

The author wants to thank to the anonymous reviewers and also to professor Michael Winkler, for their helpful comments and suggestions.

Additional information

Funding

The author was supported by Ministerio de Ciencia e Innovación, Spain, under grant number MTM2017-83391-P.