Global Existence And Asymptotic Behavior For A Chemotaxis System With Attraction And Repulsion

None kind of species in the nature is isolated, and changes in the external environment will have more or less impacts on them. For instance, microorganisms or cells can capture the external stimulus and move towards the chemicals which are good for them and away from the chemicals which are bad for them. This phenomena is called chemotaxis in biology. Keller-Segel system is a famous model describing the chemotaxis, which has practical significance for revealing the essence of some biological phenomena.In this thesis, we consider the Cauchy problem for the attraction-repulsion chemotaxis Keller-Segel system where β,δ≥0, χ,ξ,α,γ are positive real number. Applying the Lp-Lq estimates of heat semigroup and the fixed point theorem, we study the existence and uniqueness of the global solution for small initial data (u0, u0, ω0). Moreover, we obtain the asymptotic behavior of the solution in the large time.In Chapter 1, we give the research background.In Chapter 2, we introduce some main concepts about PDE.In Chapter 3, we study the existence of the global solution to the attraction-repulsion chemotaxis system. First, under appropriate assumptions for the initial data, applying the fixed point theorem and Lp-Lq space estimates of heat semigroup, we proved the existence and uniqueness of the global solution. Next, we improve the regularity of solution by the bootstrap argument, so that the integrability can be raised from p∈(4/3,2) to p∈ (4/3,∞]. Finally, we obtain the continuous dependence of solutions upon the small initial data.In Chapter 4, we study the asymptotic behavior of solution for the attraction-repulsion chemotaxis system with β=δ=0. Using the self-similar solutions, we describe the asymp-totic behavior of solution in the large time. The result reads as follows:if β=δ=0, then for the initial data (uo,vo,wo) such that (|x|2+1)u0∈L1(R2),(?)u0,(?)w0∈L1(R2), the solution satisfies for any q>4/3, where u is the self-similar solution.In Chapter 5, we study the asymptotic behavior of solution for the attraction-repulsion chemotaxis system with β> 0,δ> 0. Using the properties of heat kernel, we obtain the asymptotic behavior for the global solution. The result reads as follows:if β> 0,δ> 0, then for the initial data (u0,u0,u0) such that (u0,u0,ω0) ∈ (L1 (R2))3, the solution satisfies for any q>4/3, where Mu:=∫R2 u0dy, and Gt{·) denotes the Gaussian heat kernel.