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MiS Preprint

4/2022

Taxis-driven persistent localization in a degenerate Keller-Segel system

Angela Stevens and Michael Winkler

Abstract

The degenerate Keller-Segel type system \begin{eqnarray*} \left\{ \begin{array}{rcll} u_t &=& \nabla \cdot (u^{m-1}\nabla u) - \nabla\cdot (u\nabla v), \qquad & x\in\Omega, \ t>0, \\[1mm] 0 &=& \Delta v-\mu+u, \qquad \int_\Omega v=0, \quad \mu=\frac{1}{|\Omega|} \int_\Omega u, \qquad & x\in\Omega, \ t>0, \end{array} \right. \end{eqnarray*} is considered in balls $\Omega=B_R(0)\subset R^n$ with $n\ge 1$, $R>0$ and $m>1$.\abs Our main results reveal that throughout the entire degeneracy range $m\in (1,\infty)$, the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact subset of $\Omega$, no matter whether solutions remain bounded or blow up. More precisely, it is shown that for arbitrary $\mu>0, \sigma \in (0,1)$ and $\theta\in (0,\sigma)$ one can find $R_\star=R_\star(n,m,\mu,\sigma,\theta)>0$ such that if $R\ge R_\star$ and $u_0\in L^\infty(\Omega)$ is nonnegative and radially symmetric with $\frac{1}{|\Omega|} \int_\Omega u_0=\mu$ and \begin{eqnarray*} \frac{1}{|B_r(0)|} \int_{B_r(0)} u_0 \ge \frac{\mu}{\theta^n} \qquad \mbox{for all } r\in (0,\theta R), \end{eqnarray*} then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution $(u,v)$, extensible up to a maximal time $T_{max}\in (0,\infty]$ and satisfying $\lim_{t\nearrow T_{max}} \|u(\cdot,t)\|_{L^\infty(\Omega)} =\infty$ if $T_{max}<\infty$, which has the additional property that \begin{eqnarray*} {\rm supp} \, u(\cdot,t) \subset \overline{B}_{\sigma R}(0) \qquad \mbox{for all } t\in (0,T_{max}). \end{eqnarray*} In particular, this conclusion is seen to be valid whenever $u_0$ is radially nonincreasing with ${\rm supp} \, u_0 \subset \overline{B}_{\theta R}(0)$.