Rui Tang, Yuxiang Li, Zhiguang Zhang
Abstract:
We study the following two-species doubly degenerate nutrient-taxis system in a smoothly
bounded domain \(\Omega \subset \mathbb{R}^2\) under homogeneous Neumann boundary
conditions:
$$\displaylines{
(u_1)_t = \nabla \cdot(u_1 v \nabla u_{1}) -\chi_1 \nabla \cdot(u_1^2 v \nabla v)
+ \ell_1 u_1v, \quad x \in \Omega, \; t > 0, \cr
(u_2)_t = \nabla \cdot(u_2 v \nabla u_2) -\chi_2 \nabla \cdot(u_2^2 v \nabla v)
+ \ell_2 u_2 v, \quad x \in \Omega,\; t > 0, \cr
v_t = \Delta v - (u_1 + u_2) v, \quad x \in \Omega, \; t > 0,
}$$
with \(\chi_i,\ell_i > 0\) for \(i = 1, 2\). This system describes the spatio-temporal
dynamics of two interacting populations which consume nutrients.
We show that for all reasonably regular initial data, the system possesses a global
bounded weak solution. We also identify a criterion regarding the initial smallness
of the third component such that the solution stabilizes to a nonconstant steady state.
Submitted March 29, 2026. Published July 6, 2026.
Math Subject Classifications: 35A01, 35B45, 35K65, 35Q92, 92C17.
Key Words: Chemotaxis system; double degeneracy; two-species; weak solutions.
DOI: 10.58997/ejde.2026.49
Show me the PDF file (463 KB), TEX file for this article.
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Rui Tang School of Mathematics Southeast University Nanjing 211189, China email: 230248242@seu.edu.cn |
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Yuxiang Li School of Mathematics Southeast University Nanjing 211189, China email: lieyx@seu.edu.cn |
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Zhiguang Zhang School of Mathematics Southeast University Nanjing 211189, China email: guangz_z@163.com |
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