Pattern Formation Of A Chemotaxis-diffusion-growth Model Of A Bistable Type

In this paper, we consider a nonlinear dynamics near an unstable constant equilibrium in the Neumann boundary value problem for the chemotaxis-diffusion-growth model in a d-dimensional box Td=（0, π）d （d=1,2,3）. It is shown that given any general perturbation of magnitude δ, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of ln1/δ. Which nonlinear instability as δâ†'0. The instability occurs before the possible blow-up time. On the other hand, each initial perturbation certainly can behaves drastically differently from another, which gives rise to the richness of patterns. The paper is divided into seven section.In section1, the equilibrium solution for ODE model the positive constant steady state of A3=（h,fh/g）is unstable. A1=（0,0） and the positive equilibrium A2=（1,f/g）are locally asymptotically stableIn section2, the equilibrium solution for cross-diffusion model without chemo-taxis A1=（0,0） and the positive equilibrium A2=（1,f/g）are locally asymptotically stable, where Td=（0,π）d （d=1,2,3）.In section3, the equilibrium solution for reaction-diffusion model A1=（0,0） and the positive equilibrium A2=（1,f/g）are locally asymptotically stable, where Td=（0,π）d （d=1,2,3）. In section4, from Young and Grownwall inequality, the growing modes of （0.7）model is investigated.In section5, we introduce Bootstrap Lemma and give its proof. i.e., Apply PDEtheory, we state existence of Local-in-time solution for （4.1） and （4.2）. We provethe model （4.1） and （4.2） satisfies the homogeneous solution of Neumann boundaryconditions and periodic boundary conditions.In section6, we complete the proof of the main theorem. We show the instabilitycriterion and pattern formation. Let θ be a small fixed constant, and λ_maxbe thedominant eigenvalue which is the maximal growth rate. We also denote the gapbetween the largest growth rate λ_maxand the rest by ν>0. Then for δ>0arbitrarysmall. We define the escape time T^δ. The dynamics of a general perturbation ischaracterized by such linear dynamics over a long time period of εT^δ≤t≤T^δ, for anyε>0.In section7, use Energy estimate, Duhamel’s principle, Sobolev imbeddingto control theorem, Gagliardo-Nirenberg and Young inequality and so on, for thechemotaxis-difusion-growth model we prove that its nonlinear evolution is domi-nated by the corresponding linear dynamics along a fixed finite number of fastestgrowing modes, over a time period of ln1δ.