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- Title
Global solvability and asymptotic behavior in a two‐species chemotaxis system with two chemicals.
- Authors
Yang, Hongying; Tu, Xinyu; Yang, Li
- Abstract
This paper deals with a two‐species chemotaxis system with Lotka–Volterra competitive kinetics ut=Δu−χ1∇·(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇·(w∇z)+μ2w(1−w−a2u),x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0,$$ \left\{\begin{array}{ll}{u}_t=\Delta u-{\chi}_1\nabla \cdotp \left(u\nabla v\right)+{\mu}_1u\left(1-u-{a}_1w\right),& x\in \Omega, \kern0.30em t>0,\\ {}{v}_t=\Delta v-v+w,\kern0.30em & x\in \Omega, \kern0.30em t>0,\\ {}{w}_t=\Delta w-{\chi}_2\nabla \cdotp \left(w\nabla z\right)+{\mu}_2w\left(1-w-{a}_2u\right),\kern0.30em & x\in \Omega, \kern0.30em t>0,\\ {}{z}_t=\Delta z-z+u,\kern0.30em & x\in \Omega, \kern0.30em t>0,\end{array}\right. $$under homogeneous Neumann boundary condition in a smooth domain Ω⊂ℝn(n≥1)$$ \Omega \subset {\mathbb{R}}^n\left(n\ge 1\right) $$, where the parameters χi,μi$$ {\chi}_i,{\mu}_i $$, and ai$$ {a}_i $$ are positive constants for i=1,2$$ i=1,2 $$. Under appropriate regularity assumptions on the initial data, we obtain that the system possesses a globally bounded weak solution. Furthermore, we establish the asymptotic stabilization of this weak solution by constructing suitable energy functions in both coexistence and extinction cases.
- Subjects
CHEMICAL systems; CHEMOTAXIS; NEUMANN boundary conditions; ENERGY function
- Publication
Mathematical Methods in the Applied Sciences, 2022, Vol 45, Issue 12, p7663
- ISSN
0170-4214
- Publication type
Article
- DOI
10.1002/mma.8269