A Chemotaxis Staggered Existence Of Diffusion Equations

We investigate a biological model for chemotaxis is as follows:We study the local existence of the solution in different space and the global existence of the solution in different cases for the above system.(1) Local existence of the solution:We establish the local existence of the solution by employing Banach's fixed point theorem in L^∞(Ω)×W^1,q(Ω). And using Galerkin method to obtain the local existence of the solution in Hilbert space.(2) Global existence of the solution:The global existence of the solution is always the difficult problem of the chemotaxis model. For low dimension cases especially for n=1, there are a lot of articals which give the global existence of the solution. The study of high dimension cases is the important problem of the chemotaxis model in recent years.In case x(v)= v,F(u,v)= 0,G(u,v) = -v + u, Dirk Horstmann and Michael Winkier proved the global existence and boundedness of the solution under the condition f(u) ≤ cu^α, u≥ 1, 0 < α < 2/n in 2004.Their condition of f(u)≤ cu^α, u≥ 1, 0 < α < 2/n avoided the case of the existence of singularity in f(u).This paper mainly use semigroup theory and some iteration to obtain the global existence of the solution under the condition f(u) ≤ cu^α, u≥ 0, 0 < α < 1 and the assumption sup ,with . But the boundedness of the solution is realated to T.At the same time we consider the singularity in x(v). In this case, we just need the initial value 0≤ u₀, 0 < v₀. Then,we can obtain the boundedness of v(x, t) which has a sub-bound realated to time (t). This proof is following the method of Yagai, Mimura. Therefore, employing semigroup theory and some kinds of interpolation inequality to get the...