Global existence and slow grow-up in a quasilinear Keller–Segel system with exponentially decaying diffusivity

The Neumann initial-boundary value problem for the chemotaxis system {ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u,(⋆) is considered in a bounded domain $\Omega \subset R^n$, $n⩾1$, with smooth boundary. In compliance with refined modeling approaches, the diffusivity function D therein is allowed to decay considerably fast at large densities, where a particular focus will be on the mathematically delicate case when D(s) decays exponentially as $s\to \infty$. In such situations, namely, straightforward Moser-type recursive arguments for the derivation of $L^{\infty }$ estimates for u from corresponding L ^p bounds seem to fail. Accordingly, results on global existence, and especially on quantitative upper bounds for solutions, so far mainly concentrate on cases when D decays at most algebraically, and hence are unavailable in the present context.

This work develops an alternative approach, at its core based on a Moser-type iteration for the quantity ${\text{e}}^u$, to establish global existence of classical solutions for all reasonably regular initial data, as well as a logarithmic upper estimate for the possible growth of $\parallel u(\cdot ,t){\parallel }_{L^{\infty }\left(\Omega \right)}$ as $t\to \infty$, under the assumptions that with some K ₁  >  0, K ₂  >  0, ${\beta }^{-}>0$ and ${\beta }^{+}\in (-\infty ,{\beta }^{-}]$ we have $K_1{\text{e}}^{-{\beta }^{-}s}⩽D\left(s\right)⩽K_2{\text{e}}^{-{\beta }^{+}s}$ for all $s⩾0$, and that the size of S relative to D can be estimated according to $\frac{S\left(s\right)}{D\left(s\right)}⩽K_3{\text{e}}^{\gamma s}$ for all $s⩾0$ with some K ₃  >  0 and $\gamma \in [\frac{{\beta }^{+}-{\beta }^{-}}{2},\frac{{\beta }^{+}}{2})$.

Making use of the fact that this allows for certain superalgebraic growth of $\frac{S}{D}$, as a particular consequence of this and known results on nonexistence of global bounded solutions we shall see that in the prototypical case when $D\left(s\right)={\text{e}}^{-\beta s}$ and $S\left(s\right)=s{\text{e}}^{-\alpha s}$ for all $s⩾0$ and some positive α and β, the assumptions that $n⩾2$ and that β>0and{α∈(β2,β)if n=2,α∈(β2,β]if n⩾3, warrant the existence of classical solutions which are global but unbounded, and for which this infinite-time blow-up is slow in the sense that the corresponding grow-up rate is at most logarithmic.

To the best of our knowledge, this inter alia seems to constitute the first quantitative information on a blow-up rate in a parabolic Keller–Segel system of type ($\star$) for widely arbitrary initial data, hence independent of a particular construction of possibly non-generic exploding solutions. (paper)