The author studies classical solutions to the fully parabolic chemotaxis system \[ \begin{cases} u_t = \nabla \cdot ( D(u) \nabla u ) - \nabla \cdot (S(u) \nabla v), \quad & (x,t) \in \Omega \times (0,\infty), \\ v_t = \Delta v -v+u, \quad & (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0, v_0 \in W^{1,\infty} (\Omega)\), where \(\Omega \subset \mathbb{R}^n\), \(n \in \mathbb{N}\), is a bounded domain with smooth boundary. Moreover, \(D,S \in C^{1+\delta} ([0,\infty))\) with \(\delta >0\) are assumed to satisfy \[ S(s) \geq 0, \quad S(0) =0, \quad K_1 e^{-\beta^- s} \leq D(s) \leq K_2 e^{-\beta^+ s}, \quad \frac{S(s)}{D(s)} \leq K_3 e^{\gamma s} \] for all \(s \geq 0\) with some \(\beta^- >0\), \(\beta^+ \in (-\infty, \beta^-]\), \(\gamma \in [\frac{\beta^+ - \beta^-}{2}, \frac{\beta^+}{2})\), and positive constants \(K_1,K_2,K_3\).
The author shows the existence of a unique global classical solution \((u,v)\). Moreover, for any \(\varepsilon >0\) the existence of a constant \(C(\varepsilon) >0\) is established such that \[ \|u(\cdot,t) \|_{L^\infty (\Omega)} \leq \left( \frac{1}{\beta^+ - 2\gamma} + \varepsilon \right) \ln (1+t) + C(\varepsilon) \] for all \(t>0\) is satisfied. This extends known global existence results, where the diffusivity \(D\) is assumed to decay at most algebraically, to exponentially decaying \(D\).
In the particular case \(D(s) = e^{-\beta s}\) and \(S(s) = se^{-\alpha s}\) with \(0 < \frac{\beta}{2} < \alpha \leq \beta\) for \(n \geq 2\) (where the additional restriction \(\alpha <\beta\) is needed for \(n=2\)), the author proves the existence of radially symmetric global solutions \((u,v)\) which blow up in infinite time with blow-up rates which are not faster than logarithmic. This seems to be the first time that such slow rates of infinite time blow-up are observed for a Keller-Segel system.
The main idea in the proof of the global existence result is a Moser-type iteration for \(e^u\) which finally results in an estimate for \(\|e^{u(\cdot,t)} \|_{L^\infty (\Omega)}\) and the above logarithmic estimate for \(u\).