Does strong repulsion lead to smooth solutions in a repulsion-attraction chemotaxis system even when starting with highly irregular initial data?

Abstract

It has been well established that, in attraction-repulsion Keller–Segel systems of the form$ \begin{equation*} \left\{ \begin{aligned} u_t & = \Delta u - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w), \\ \tau v_t & = \Delta v + \alpha u - \beta v, \\ \tau w_t & = \Delta w + \gamma u - \delta w \end{aligned} \right. \end{equation*} $in a smooth bounded domain $ \Omega \subseteq \mathbb{R}^n $, $ n\in\mathbb{N} $, with Neumann boundary conditions and parameters $ \chi, \xi \geq 0 $, $ \alpha, \beta, \gamma, \delta > 0 $ and $ \tau \in \{0, 1\} $, finite-time blow-up can be ruled out in many scenarios given sufficiently smooth initial data if the repulsive chemotaxis is sufficiently stronger than its attractive counterpart. In this paper, we will go - in a sense - a step further than this by studying the same system with initial data that could already be understood as being in a blown-up state (e.g. a positive Radon measure for the first solution component) and then ask the question whether sufficiently strong repulsion has enough of a regularizing effect to lead to the existence of a smooth solution, which is still connected to said initial data in a sensible fashion. Regarding this, we in fact establish that the construction of such a solution is possible in the two-dimensional parabolic-parabolic system and the two- and three-dimensional parabolic-elliptic system under appropriate assumptions on the interaction of repulsion and attraction as well as the initial data.