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Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties

Tobias Black

Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

E-mail Address: tblack@math.uni-paderborn.de

https://doi.org/10.1142/S0218202520400072Cited by:26 (Source: Crossref)

This article is part of the issue:

Special Issue on "Chemotaxis Systems in Complex Frameworks:Pattern Formation, Qualitative Analysis and Blowup Prevention"
Guest Editors: N. Bellomo, Y. Tao and M. Winkler

Abstract

In this paper, we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the forager group. Specifically, we will investigate an initial boundary-value problem for a prototypical forager–exploiter model of the form

in a smoothly bounded domain

$Ω \subset ℝ^{2}$ , where

$μ \geq 0$ ,

$r \in C^{1} (\bar{Ω} \times [0, \infty)) \cap L^{\infty} (Ω \times (0, \infty))$ is nonnegative and the functions

$f, g \in C^{1} ([0, \infty))$ are assumed to satisfy

$f (0) \geq 0$ ,

$g (0) \geq 0$ as well as

- k f s α - l f \leq f (s) \leq - K f s α + L f and - k g s β - l g \leq g (s) \leq - K g s β + L g for s \geq 0, <math display="block" altimg="eq-00008.gif"><mrow><mtext> </mtext><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>f</mi></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>f</mi></mrow></msub><mo class="MathClass-rel">\leq</mo><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>s</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\leq</mo><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>f</mi></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mi>α</mi></mrow></msup><mo stretchy="false" class="MathClass-bin">+</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub><mtext> </mtext><mstyle><mtext class="textrm" mathvariant="normal">and</mtext></mstyle><mtext> </mtext><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>g</mi></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mi>β</mi></mrow></msup><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>g</mi></mrow></msub><mo class="MathClass-rel">\leq</mo><mi>g</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>s</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\leq</mo><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>g</mi></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mi>β</mi></mrow></msup><mo stretchy="false" class="MathClass-bin">+</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>g</mi></mrow></msub><mtext> </mtext><mstyle><mtext class="textrm" mathvariant="normal">for</mtext></mstyle><mi>s</mi><mo class="MathClass-rel">\geq</mo><mn>0</mn><mo class="MathClass-punc">,</mo></mrow></math>

respectively, with constants

$α, β > 1$ ,

$k_{f}, K_{f}, k_{g}, K_{g} > 0$ and

$l_{f}, L_{f}, l_{g}, L_{g} \geq 0$ and

$α, β > 1$ .

Assuming that $α > 1 + \sqrt{2}$ , $min {α, β} > \frac{α + 1}{α - 1}$ and that $r$ satisfies certain structural conditions, we establish the global solvability of this system with respect to a suitable generalized solution concept and then, for the more restrictive case of $α, β > 1 + \sqrt{2}$ and $μ > 0$ , investigate an eventual regularity effect driven by the decay of the nutrient density $w$ .

Communicated by N. Bellomo, Y. Tao and M. Winkler

Keywords:

AMSC: 35D99, 35B65 (primary), 35B40, 35K55, 35Q91, 35Q92, 92C17

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