Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Blow-up profiles in quasilinear fully parabolic Keller--Segel systems   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 $x \in \Omega$.

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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org