Abstract
In this paper we study an attraction–repulsion chemotaxis system with a free boundary in one space dimension. First, under some conditions, we investigate existence, uniqueness and uniform estimates of the global solution. Next, we prove a spreading–vanishing dichotomy for this model. In the vanishing case, the species fail to establish and die out in the long run. In the spreading case, we provide some sufficient conditions to prove that the species successfully spread to infinity as trightarrowinfty and stabilize at a constant equilibrium state. The criteria for spreading and vanishing are also obtained.
Highlights
1 Introduction This paper is devoted to studying the dynamics of solutions to the following attraction– repulsion chemotaxis system with a free boundary:
We prove a spreading–vanishing dichotomy for this model, that is, the species either fails to establish and vanishes eventually or the species successfully spreads to infinity as t → ∞ and stabilizes at a constant equilibrium state under some sufficient conditions
We provide a sufficient condition (1.3) to prove that system (1.1) has a unique positive equilibrium (a/b, μ1a/λ1b, μ2a/λ2b)
Summary
Λ2)+], the following is true: (i) There exists a T > 0 such that problem (1.1) admits a unique solution (u, v1, v2, h) ∈ Wp1,2(DT ) ∩ C(1+θ)/2,1+θ (DT ) × C(1+θ)/2,1+θ DTv 2 × C1+θ/2 [0, T ] , (2.6) (2.7) For the given w ∈ WT1 , we have the following initial-boundary value problem For such a defined function w, the initial value problem (2.12) has a unique solution, denoted by h(t) = h(t; h). Similar to the proofs of Theorem 2.1, Lemma 2.2 and Theorem 2.3 in [7], we can obtain the following global existence result.
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