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 = \Delta v - 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 $m \gt \frac{n-2}{n}$ as well as $m-q \gt -\frac1n$ and $(u, v)$ is a nonnegative, radially symmetric classical solution to \eqref{prob:star} blowing up at $T_{\textrm{max}} \lt \infty$, then there exists a so-called blow-up profile $U \colon \Omega \setminus \{0\} \to [0, \infty)$ satisfying \begin{align*} u(\cdot, t) \to U \quad \text{in $C_{\textrm{loc}}^2(\bar \Omega \setminus \{0\})$ as $t \nearrow T_{\textrm{max}}$}. \end{align*} Moreover, for all $\alpha \gt n$ with \begin{align*} \alpha \gt \frac{n(n-1)}{(m-q)n + 1} \end{align*} we can find $C \gt 0$ such that \begin{align*} U(x) \le C |x|^{-\alpha} \end{align*} for all $x \in \Omega$.