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On the global existence of solutions to chemotaxis system for two populations in dimension two

Part of: Physiological, cellular and medical topics Partial differential equations

Published online by Cambridge University Press:  09 January 2023

Ke Lin
Ke Lin*
Affiliation:
School of Mathematics, Southwestern University of Economics and Finance, Chengdu, 611130 Sichuan, China (linke@swufe.edu.cn)
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Abstract

We consider the global existence for the following fully parabolic chemotaxis system with two populations

\[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, & x\in\Omega,\ t>0, \\ v_t=\Delta v-v+u_1+u_2, & x\in\Omega,\ t>0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), & x\in\Omega, \end{array} \right. \]
where $\Omega =\mathbb {R}^2$ or $\Omega =B_R(0)\subset \mathbb {R}^2$ supplemented with homogeneous Neumann boundary conditions, $\kappa _i,\chi _i>0,$ $i=1,2$. The global existence remains open for the fully parabolic case as far as the author knows, while the existence of global solution was known for the parabolic-elliptic reduction with the second equation replaced by $0=\Delta v-v+u_1+u_2$ or $0=\Delta v+u_1+u_2$. In this paper, we prove that there exists a global solution if the initial masses satisfy the certain sub-criticality condition. The proof is based on a version of the Moser–Trudinger type inequality for system in two dimensions.

Keywords

Chemotaxisglobal solutionthe Moser–Trudinger inequality for system

MSC classification

Primary: 35A01: Existence problems
Secondary: 92C17: Cell movement (chemotaxis, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 2106 - 2128
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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