Can a chemotaxis-consumption system recover from a measure-type aggregation state in arbitrary dimension?

Abstract:

We consider the chemotaxis-consumption system \[ \begin {cases} u_t &= \Delta u - \chi \nabla \cdot (u\nabla v) \\ v_t &= \Delta v - uv \end {cases} \tag {$\star $} \] in a smooth bounded domain $\Omega \subseteq \mathbb {R}^n$, $n \geq 2$, with parameter $\chi > 0$ and Neumann boundary conditions. It is well known that, for sufficiently smooth nonnegative initial data and under a smallness condition for the initial state of $v$, solutions of the above system never blow up and are even globally bounded. Going in a sense a step further in this paper, we ask the question whether the system can even recover from an initial state that already resembles measure-type blowup. To answer this, we show that, given an arbitrarily large positive Radon measure $u_0$ with $u_0(\overline {\Omega }) > 0$ as the initial data for the first equation and a nonnegative $L^\infty (\Omega )$ function $v_0$ with \[ 0 < \|v_0\|_{L^{\infty }(\Omega )} < \frac {\pi }{\chi } \sqrt {\frac {2}{n}} \] as initial data for the second equation, it is still possible to construct a global classical solution to the above system. Notably, the above condition on the initial data appears to be weaker than those required in all previous works on ($\star$) even in frameworks of smooth initial data.