Exponential grow-up rates in a quasilinear Keller–Segel system

Abstract

The chemotaxis system $$\begin{array}{}\text{(}\mathrm{\text{⋆}}\text{)}& \left\{\begin{array}{c}{\mathit{u}}_{\mathit{t}}=\nabla ·\mathrm{(}\mathit{D}\mathrm{(}\mathit{u}\mathrm{)}\nabla \mathit{u}\mathrm{)}-\nabla ·\mathrm{(}\mathit{u}\mathit{S}\mathrm{(}\mathit{u}\mathrm{)}\nabla \mathit{v}\mathrm{)}\mathrm{,}\hfill \\ 0=\mathrm{\Delta }\mathit{v}-\mathit{\mu }+\mathit{u}\mathrm{,}\mathit{\mu }=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}\mathit{u}\mathrm{,}\hfill \end{array}\right.\end{array}$$ is considered in a ball $\mathrm{\Omega }={\mathit{B}}_{\mathit{R}}\mathrm{(}0\mathrm{)}\subset {\mathbb{R}}^{\mathit{n}}$ .

It is shown that if $\mathit{S}\in {\mathit{C}}^2\mathrm{(}[0\mathrm{,}\infty \mathrm{)}\mathrm{)}$ suitably generalizes the prototype given by $$\begin{array}{r}\mathit{S}\mathrm{(}\mathit{\xi }\mathrm{)}={\displaystyle \frac{\mathit{\chi }}{\mathit{\xi }+1}}\mathrm{,}\mathit{\xi }⩾0\mathrm{,}\end{array}$$ with some $\mathit{\chi }\mathrm{>}0$ , and if diffusion is suitably weak in the sense that $0\mathrm{<}\mathit{D}\in {\mathit{C}}^2\mathrm{(}\mathrm{(}0\mathrm{,}\infty \mathrm{)}\mathrm{)}$ is such that there exist ${\mathit{K}}_{\mathit{D}}\mathrm{>}0$ and $\mathit{m}\in \mathrm{(}-\infty \mathrm{,}1-\frac{2}{\mathit{n}}\mathrm{)}$ fulfilling $$\begin{array}{r}\mathit{D}\mathrm{(}\mathit{\xi }\mathrm{)}⩽{\mathit{K}}_{\mathit{D}}{\mathit{\xi }}^{\mathit{m}-1}\text{for all }\mathit{\xi }\mathrm{>}0\mathrm{,}\end{array}$$ then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution $\mathrm{(}\mathit{u}\mathrm{,}\mathit{v}\mathrm{)}$ which blows up in infinite time and satisfies $$\begin{array}{r}{\displaystyle \frac{1}{\mathit{C}}}{\mathit{e}}^{\mathit{\chi }\mathit{t}}⩽{\Vert \mathit{u}\mathrm{(}·\mathrm{,}\mathit{t}\mathrm{)}\Vert }_{{\mathit{L}}^{\infty }\mathrm{(}\mathrm{\Omega }\mathrm{)}}⩽\mathit{C}{\mathit{e}}^{\mathit{\chi }\mathit{t}}\text{for all }\mathit{t}\mathrm{>}0.\end{array}$$ A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of (⋆).