Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source

Abstract

We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic-elliptic chemotaxis-growth systems of the form \begin{align*} u_t &= \Delta u - \nabla\cdot(u\nabla v) + \kappa(x)u-\mu(x)u^2, 0 &= \Delta v - v + u \end{align*} in smoothly bounded domains $\Omega\subset\mathbb{R}^2$. Assuming that the coefficient functions satisfy $\kappa,\mu\in C^0(\overline{\Omega})$ with $\mu\geq0$ we prove that finite-time blow-up of the classical solution can only occur in points where $\mu$ is zero, i.e.\ that the blow-up set $\mathcal{B}$ is contained in \begin{align*} \big\{x\in\overline{\Omega}\mid\mu(x)=0\big\}. \end{align*} Moreover, we show that whenever $\mu(x_0)>0$ for some $x_0\in\overline{\Omega}$, then one can find an open neighbourhood $U$ of $x_0$ in $\overline{\Omega}$ such that $u$ remains bounded in $U$ throughout evolution.