Prevention of infinite-time blowup by slightly super-linear degradation in a Keller-Segel system with density-suppressed motility (Q6576901)

The authors study the chemotaxis system with nonlinear motility and superlinear degradation \N\[\N\begin{cases} \Nu_t = \Delta \left(u e^{-v} \right) -u f(u), &(x,t) \in \Omega \times (0,\infty),\\\N\tau v_t = \Delta v - v +u, &(x,t) \in \Omega \times (0,\infty),\\\N\end{cases}\N\] \Nendowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0 \in C (\overline{\Omega})\), \(\tau v_0 \in W^{1,\infty}(\Omega)\) such that \(u_0 \not\equiv 0\). Moreover, it is assumed that \(\Omega \subset \mathbb{R}^N\) is a bounded domain with smooth boundary with \(N \in \{2,3\}\) and \(f \in C^1 ([0,\infty))\) satisfies \N\[\N\lim\limits_{s \to \infty} f(s) = \infty \qquad\text{and}\qquad \limsup\limits_{s \to \infty} \frac{f(s)}{\log(s)} < \infty.\N\]\NRequiring these assumptions, the authors prove that the above problem has a unique global classical solution which is uniformly bounded.\N\NAn important step in the proof is the derivation of estimates by using the inverse operator of \(I - \Delta\) and comparison arguments. Further steps involve e.g. several a priori estimates, parabolic regularity theory, and properties of the Neumann heat semigroup.