Abstract
Existence of global solutions to chemotaxis fluid system with logistic source
Highlights
Chemotaxis is the oriented movement of biological cells or microscopic organisms toward or away from the concentration gradient of certain chemicals in their environment
We may use cells to denote the biological objects whose movement we are interested in and chemo attractants or repellents to denote chemicals which attract or repell the cells. This type of movement exists in many biological phenomena, such as the movement of bacteria toward certain chemicals [1], or the movement of endothelial cells toward the higher concentration of chemoattractant that cancer cells produce [4]
We have studied a chemotaxis model where a compressible fluid model for cells and a diffusive Lotka–Volterra model for chemoattractants and repellents are used
Summary
Chemotaxis is the oriented movement of biological cells or microscopic organisms toward or away from the concentration gradient of certain chemicals in their environment. There exists a positive number 0 which is small enough such that if [ρ0, u0, c1,0, c2,0] HN ≤ 0, the Cauchy problem (2.2)–(2.3) has a unique solution (ρ, u, c1, c2)(t) globally in time which satisfies (ρ, u, c1, c2)(t) ∈ X(0, ∞) and there are constants C0 > 0, λ1 > 0 and λ1 > 0 such that t [ρ, u, c1, c2]. Let U(t) = [ρ, u, c1, c2] be the solution to the Cauchy problem (2.2)–(2.3) obtained in Proposition 2.1, which satisfies the following Lq-time decay estimates for any t ≥ 0: ρ. The goal of this section is to prove the global existence of solutions to the Cauchy problem (2.2) when initial data is a small, smooth perturbation near the steady state (n∞, 0, 0, 0). The proof is based on some uniform a priori estimates combined with the local existence, which will be shown in Subsections 3.1 and 3.2
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