趋化模型解的存在性与渐近性研究

发布时间：2018-07-08 20:01

本文选题：趋化模型 + 古典解　；参考：《东南大学》2015年博士论文

【摘要】：我们考虑了生物数学中描述趋化现象的偏微分方程模型.该模型刻画了细胞在自身扩散和化学物质刺激作用的影响下种群密度的时空发展分布.本文致力于几类趋化方程组的可解性以及解的渐近性质的研究.本文主要内容安排如下：在第二章中,我们讨论了拟线性方程组的Neumann初边值问题,其中Ω(?)RN(N≥1)为有界光滑区域,假设函数D(u)和S(u)适当光滑且D(u)≥Mu-α和S(u)≤Muβ对所有u≥1成立,这里M0,α∈R,β∈R,并且假设logistic项f(u)满足f(0)≥0和f(u)≤α-μuγ对所有u≥0都成立,这里a≥0,μ0及γ≥1.我们证明了如果则对于充分光滑的初值,该问题的古典解存在唯一且一致有界.第三章主要考虑两种群趋化模型的Cauchy问题,其中参数γ≥0,x1,x2,α1,α2都为实数.我们得到了如果初值的‖u0‖1,‖v0‖1和‖%絯0‖2适当小,则该问题的解整体存在,且当t→∞时·如果γ=0,该问题的解渐近趋向于其自相似解；·如果γ0,该问题的解渐近趋向于热核的某一个倍数.在第四章中,我们研究了不可压缩趋化-Navier-Stokes方程组的初边值问题,其中Ω(?)R2为有界区域.已有结果表明如果χ0,κ∈R和Φ∈C2(Ω),对于充分光滑的初值(n0,c0,u0),该问题的古典解整体存在并且满足当t→∞时,(n,c,u)→(n0,0,0)对于x∈Ω一致成立,其中no:1/|Ω|(?)n(x,0)dx.这里我们得到了解收敛于平衡点(n0,0,0)的速率是指数衰减的.第五章致力于处理处理带有多孔介质型扩散的不可压缩趋化-Navier-Stokes方程组的初边值问题,其中Ω(?)R3为有界凸区域,κ∈R,Φ∈W1,∞(Ω),0χ∈C2([0,∞)),且0≤f∈C1([0,∞))满足f(0)=0.我们证明了如果参函数f和χ满足一些结构上的假设,对足够光滑的初值(n0, c0, n0),当m≥2/3时,该问题的弱解整体存在.
[Abstract]:We consider the partial differential equation model which describes the chemotaxis in biological mathematics. This model depicts the spatio-temporal distribution of the cell density under the influence of self diffusion and chemical substance stimulation. This paper is devoted to the solvability of several chemotaxis equations and the asymptotic properties of solutions. The main contents of this paper are as follows In the second chapter, we discuss the Neumann initial boundary value problem for a set of quasilinear equations, where omega (?) RN (N > 1) is a bounded smooth region, assuming that the function D (U) and S (U) are properly smooth and D (U) > Mu- alpha and S (U) < < < < < 1 >. More than 0 are established, where a > 0, Mu 0, and gamma > 1. we prove that the classical solution of the problem has unique and uniform bounds if it is for the full smooth initial value. The third chapter mainly considers the Cauchy problem of the two population chemotaxis model, in which the parameters are equal to 0, x1, X2, alpha 1, and alpha 2 are all real. In the fourth chapter, we study the initial boundary value problem of the incompressible chemotactic -Navier-Stokes equations in the fourth chapter. In the fourth chapter, we study the initial boundary value problem of the incompressible chemotactic -Navier-Stokes equations, of which omega (?) R2, in the fourth chapter. For the bounded region, the existing results show that if Chi 0, kappa R and C2 (omega), for the full smooth initial value (N0, C0, U0), the whole existence of the classical solution of the problem and satisfies when T - infinity is satisfied, (n, C, U) - (n0,0,0) is unanimous, where no:1/| omega (?) is here to understand the rate of convergence at the equilibrium point. Decaying. The fifth chapter is devoted to dealing with the initial boundary value problem of the incompressible chemotactic -Navier-Stokes equations with porous media diffusion, in which omega (?) R3 is bounded convex region, kappa R, W1, infinity, C2 ([0, infinity), and 0 < f C1 ([0, infinity)) satisfies f (0) =0.) and we prove that if the reference function f and Chi satisfy some structures The assumption is that for a sufficiently smooth initial value (N0, C0, N0), when m is equal to 2/3, the weak solution of the problem exists as a whole.
【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175.2

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