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Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model

Michael Winkler

Michael Winkler

Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

E-mail Address: michael.winkler@math.uni-paderborn.de

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https://doi.org/10.1142/S1664360722500126Cited by:27 (Source: Crossref)

Abstract

A no-flux initial-boundary value problem for the cross-diffusion system

{u t = Δ (u ϕ (v)), v t = Δ v - u v <math display="block" altimg="eq-00001.gif"><mrow><mfenced separators="" open="{" close=""><mrow><mtable equalrows="false" columnlines="" equalcolumns="false" class="array"><mtr><mtd class="array" columnalign="left"><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mi>ϕ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>v</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo></mtd></mtr><mtr><mtd class="array" columnalign="left"><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>v</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>u</mi><mi>v</mi></mtd></mtr></mtable></mrow></mfenced></mrow></math>

is considered in smoothly bounded domains

$Ω \subset ℝ^{n}$ with

$n \leq 2$ . It is shown that whenever

$ϕ \in C^{0} ([0, \infty))$ is positive on

$(0, \infty)$ and such that

lim infξ↘0ϕ(ξ)ξα>0(⋆)<math display="block" altimg="eq-00006.gif"><mrow><munder class="msub"><mrow><mo class="qopname">lim inf</mo></mrow><mrow><mi>ξ</mi><mo class="MathClass-rel">↘</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>ϕ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>ξ</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>ξ</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac><mo class="MathClass-rel">&gt;</mo><mn>0</mn><mspace width="1em" class="quad"></mspace><mo class="MathClass-open" stretchy="false">(</mo><mo stretchy="false" class="MathClass-bin">⋆</mo><mo class="MathClass-close" stretchy="false">)</mo></mrow></math>

for some

$α > 0$ , for all suitably regular positive initial data a global very weak solution, particularly preserving mass in its first component, can be constructed. This extends previous results which either concentrate on non-degenerate analogs, or are restricted to the special case

$α = 1$ .

To appropriately cope with the considerably stronger cross-degeneracies thus allowed through $(⋆)$ when $α$ is large, in its core part the analysis relies on the use of the Moser–Trudinger inequality in controlling the respective diffusion rates $ϕ (v)$ from below.

Communicated by Neil Trudinger

Keywords:

AMSC: 35K65, 35K59, 35Q92, 92C17, 35K57

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Vol. 13, No. 02

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History

Received 15 June 2022

Accepted 28 October 2022

Published: 6 December 2022

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model

Abstract

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