A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data

By Johannes Lankeit, Michael Winkler. 2017

134 citations recorded by the source corpus.

The chemotaxis system $$\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta u - \chi \nabla \cdot \left( \frac{u}{v}\nabla v\right) , \\ v_t=\Delta v - v+u, \end{array} \right. \end{aligned}$$ut=Δu-χ∇·uv∇v,vt=Δv-v+u,is considered in a bounded domain $$\Omega \subset \mathbb {R}^n$$Ω⊂Rn with smooth boundary, where $$\chi >0$$χ>0. An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter $$\chi $$χ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis that $$\begin{aligned} \chi <\left\{ \begin{array}{ll} \infty \qquad &{} \text{ if } n=2, \\ \sqrt{8} \qquad &{} \text{ if } n=3, \\ \frac{n}{n-2} \qquad &{} \text{ if } n\ge 4, \end{array} \right. \end{aligned}$$χ<∞ifn=2,8ifn=3,nn-2ifn≥4,for all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that $$\begin{aligned} u\in L^1_{loc}(\overline{\Omega }\times [0,\infty )). \end{aligned}$$u∈Lloc1(Ω¯×[0,∞)).

Subfields: Cancer Research, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/3012c9123b5cf45cc62bf826af5d65346ea89c8e

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Citations in the corpus

Analysis of a chemotaxis model with indirect signal absorption (2019, background): “4) have also already been studied with p ∈ (0, 1) in [11, 18], in both cases with φ(s) = s, s ≥ 0, for some q > 0.”
Local energy estimates and global solvability in a three-dimensional chemotaxis-fluid system with prescribed signal on the boundary (2021, background): “Approximation and basic estimates Let us first specify the solution concept which we pursue throughout the sequel, and which by referring to products of functions simultaneously involving the first two solution components partially parallels a similar approach designed in [34] for a fluid-free chemo”
Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions (2020, background): “We accordingly may refrain from detailing a discussion about how far this concept is consistent with that of classical solvability here, and rather refer the reader to ([47] and the related precedent in [20] instead.”