Global generalized solutions to a nonlinear Keller-Segel equation with singular sensitivity

Abstract

We consider the chemotaxis system \begin{eqnarray*} \begin{cases} \begin{array}{lll} \medskip u_t =\Delta u^m - \nabla(\frac{u}{v}\nabla v),&{} x\in\Omega,\ t>0, \medskip v_t =\Delta v -uv,&{}x\in\Omega,\ t>0, \medskip \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu}=0,&{}x\in\partial\Omega,\ t>0, \medskip u(x,0)=u_0(x),\ v(x,0)=v_0(x), &{}x\in\Omega, \end{array} \end{cases} \end{eqnarray*} in a smooth bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq2$. In this work it is shown that for all reasonably regular initial data $u_0\geq0$ and $v_0>0$, the corresponding Neumann initial-boundary value problem possesses a global generalized solution provided that $m>1+\frac{n-2}{2n}$.