Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion

By Johannes Lankeit. 2016

122 citations recorded by the source corpus.

We show the existence of locally bounded global solutions to the chemotaxis system \[ u_t = \nabla\cdot(D(u)\nabla u) - \nabla\cdot(\frac{u}{v} \nabla v) \] \[ v_t = \Delta v - uv \] with homogeneous Neumann boundary conditions and suitably regular positive initial data in smooth bounded domains $\Omega \subset \mathbb{R}^N$, $N\geq2$, for $D(u)\geq \delta u^{m-1}$ with some $\delta>0$, provided that $m>1+\frac N4$.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/857fca4022049ae068c44f259dbec24486235154

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Modeling and Simulation

In Public Health, Environmental and Occupational Health

Citations in the corpus

A quantitative strong parabolic maximum principle and application to a taxis-type migration–consumption model involving signal-dependent degenerate diffusion (2022, background): “The statements on existence, positivity and extensibility can be verified by following standard approaches in local existence theories of taxis-type parabolic systems ([2], [20]).”
Analysis of a chemotaxis model with indirect signal absorption (2019, background): “For instance, coupled chemotaxis-fluid systems [14, 26, 29, 28], systems with nonlinear diffusion and/or nonlinear chemotactic sensitivity [3, 9, 13, 31] or systems with zeroth order terms accounting for logistic growth [10] or competition between species [23] have been analyzed.”
Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement (2017, unknown): “By means of standard arguments from the local existence theories of taxis-type cross diffusive parabolic systems and the Stokes evolution equation ([1], [30], [25], [43]), it follows that there exist Tmax,ε ∈ (0,∞] and at least one classical solution (nε, cε, uε, Pε) ∈ ( C0(Ω × [0, Tmax,ε);R(5)) ∩ C”
Depleting the signal: Analysis of chemotaxis‐consumption models—A survey (2023, background): “with D being suitably smooth and satisfying D(s) ≥ Csm−1 for all s ≥ 0 with some C > 0, then for such locally bounded global weak solutions to exist in n-dimansional domains, to date the literature requires the assumption that m > 1+ n4 ([63]) which already in the case n = 2 is considerably stronger”