Blow-up profiles in quasilinear fully parabolic Keller--Segel systems

Abstract

We examine finite-time blow-up solutions $(u,v)$ to \begin{align} \label{prob:star} \tag{$\star$} \begin{cases} u_t = \nabla \cdot (D(u, v) \nabla u - S(u, v) \nabla v), v_t = Δv - v + u \end{cases} \end{align} in a ball $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, where $D$ and $S$ generalize the functions \begin{align*} D(u, v) = (u+1)^{m-1} \quad \text{and} \quad S(u, v) = u (u+1)^{q-1} \end{align*} with $m,q\in \mathbb{R}$. We show that if $m>\frac{n-2}{n}$ as well as $m-q>-\frac{1}{n}$ and $(u,v)$ is a nonnegative, radially symmetric classical solution to \eqref{prob:star} blowing up at $T_{\text{max}}<\infty$, then there exists a so-called blow-up profile $U:\Omega \setminus \{0\}\to [0,\infty )$ satisfying \begin{align*} u(\cdot, t) \to U \quad \text{in $C_{\text{loc}}^2(\bar{\Omega }\setminus \{0\})$ as $t\nearrow T_{\text{max}}$}. \end{align*} Moreover, for all $\alpha >n$ with \begin{align*} α\gt \frac{n(n-1)}{(m-q)n + 1} \end{align*} we can find $C>0$ such that \begin{align*} U(x) \le C |x|^{-α} \end{align*} for all $x\in \Omega$.

27 pages