MiS Preprint Repository

The degenerate Keller-Segel type system $$\begin{array}{r}\{\begin{array}{rcll}{u}_t& =& \mathrm{\nabla }\cdot (u^{m-1}\mathrm{\nabla }u)-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v),& x\in \mathrm{\Omega },t>0,\\ 0& =& \mathrm{\Delta }v-\mu +u,{\int }_{\mathrm{\Omega }}v=0,\mu =\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}u,& x\in \mathrm{\Omega },t>0,\end{array}\end{array}$$ is considered in balls $\mathrm{\Omega }=B_R(0)\subset R^n$ with $n\ge 1$, $R>0$ and $m>1$.\abs Our main results reveal that throughout the entire degeneracy range $m\in (1,\mathrm{\infty })$, the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact subset of $\mathrm{\Omega }$, 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 $u_0\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ is nonnegative and radially symmetric with $\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{u}_0=\mu$ and $$\begin{array}{r}\frac{1}{|B_r(0)|}{\int }_{B_r(0)}{u}_0\ge \frac{\mu }{{\theta }^n}\text{for all }r\in (0,\theta R),\end{array}$$ then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution $(u,v)$, extensible up to a maximal time $T_{max}\in (0,\mathrm{\infty }]$ and satisfying $\underset{t\nearrow T_{max}}{lim}\Vert u(\cdot ,t){\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}=\mathrm{\infty }$ if $T_{max}<\mathrm{\infty }$, which has the additional property that $$\begin{array}{r}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u(\cdot ,t)\subset {\bar{B}}_{\sigma R}(0)\text{for all }t\in (0,T_{max}).\end{array}$$ In particular, this conclusion is seen to be valid whenever $u_0$ is radially nonincreasing with $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}{u}_0\subset {\bar{B}}_{\theta R}(0)$.