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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 2

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

Pages: 299 – 322

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n2.a1

#### Authors

#### Abstract

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.

Received 31 January 2022

Received revised 8 May 2022

Accepted 8 May 2022

Published 1 February 2023