Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities

Two relaxation features of the migration-consumption chemotaxis system involving signal-dependent motilities,\begin{align*}\begin{cases}u_t = \Delta (u \phi (v)) \; , \\v_t = \Delta v - uv \; ,\end{cases}&&&& (\star)\end{align*}are studied in smoothly bounded domains $\Omega \subset \mathbb{R}^n , n\geq 1$: It is shown that if $\phi \in C^0 ([0,\infty))$ is positive on $[0,\infty)$, then for any initial data $(u_0,v_0)$ belonging to the space $(C^0 (\overline{\Omega}))^\star \times L^\infty (\Omega)$ an associated no-flux type initial-boundary value problem admits a global very weak solution. Beyond this initial relaxation property, it is seen that under the additional hypotheses that $\phi \in C^1 ([0,\infty))$ and $n \leq 3$, each of these solutions stabilizes toward a semi-trivial spatially homogeneous steady state in the large time limit.

By thus applying to irregular and partially even measure-type initial data of arbitrary size, this firstly extends previous results on global solvability in $(\star)$ which have been restricted to initial data not only considerably more regular but also suitably small. Secondly, this reveals a significant difference between the large time behavior in $(\star)$ and that in related degenerate counterparts involving functions $\phi$ with $\phi (0) = 0$, about which, namely, it is known that some solutions may asymptotically approach nonhomogeneous states.

Keywords

chemotaxis, instantaneous regularization, large time behavior

2010 Mathematics Subject Classification

Primary 35B40. Secondary 35D30, 35K55, 35Q92, 92C17.

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Received 31 January 2022

Received revised 8 May 2022

Accepted 8 May 2022

Published 1 February 2023