When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions?

Abstract

We construct global generalized solutions to the chemotaxis system \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + λ(x) u - μ(x) u^κ,\\ v_t = Δv - v + u \end{cases} \end{align*} in smooth, bounded domains $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, for certain choices of $\lambda ,\mu$ and $\kappa$. Here, inter alia, the selections $\mu (x)=|x{|}^{\alpha }$ with $\alpha <2$ and $\kappa =2$as well as $\mu \equiv {\mu }_1>0$ and $\kappa >\min \{\frac{2n-2}{n},\frac{2n+4}{n+4}\}$ are admissible (in both cases for any sufficiently smooth $\lambda$). While the former case appears to be novel in general, in the two- and three-dimensional setting, the latter improves on a recent result by Winkler (Adv. Nonlinear Anal. 9 (2019), no. 1, 526-566), where the condition $\kappa >\frac{2n+4}{n+4}$ has been imposed. In particular, for $n=2$, our result shows that taking any $\kappa >1$ suffices to exclude the possibility of collapse into a persistent Dirac distribution.

arXiv comment16 pages

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Cited by in corpus (4)

• Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source
• Blow-up phenomena in a parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with superlinear logistic degradation
• Global existence of solutions to Keller-Segel chemotaxis system with heterogeneous logistic source and nonlinear secretion
• Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source