The Well-posedness Of Solutions To The Nonlinear Chemotaxis Model And Shallow Water Wave Model

The well-posedness of solutions has always been the research frontier and hot spot in the partial differential equations field.By means of studying this issue to equations with singularity or degeneration,we can explain and predict some special phenomenon in subjects like Physics,Mechanics and biology etc.In the present study,we mainly consider two types of nonlinear evolution equations: The biological chemotaxis model equation and the shallow water wave model equation.For the biological chemotaxis model,we mainly study the initial boundary value problem of the parabolic-elliptic coupled chemotaxis-competition model,we obtain the global existence,uniform boundedness and large time behavior of the smooth solutions.For the shallow water wave model,we primarily consider the Camassa-Holm equation with the coriolis effect(R-CH equation),under the energy space 1H(),we obtain the global existence,uniqueness and the generic regularity result to the weak solutions,furthermore,we construct the Lipschitz metric of weak solutions to the R-CH equation,under which the solutions have Lipschitz continuous dependence on the initial data.This thesis is divided into five chapters as follows:In Chapter 1,the introduction part.We mainly introduce the background of the chemotaxis models and the shallow water wave models,also,we present the main content of this thesis.In Chapter 2,we study the initial boundary value problem for the parabolic-elliptic coupled chemotaxis-competition model.Under the homogeneous Neumann boundary condition,suppose that the initial data satisfy suitable regularity condition,first,we prove the global existence and uniform boundedness for the solution when the ratio of the parameters in the equations are sufficiently small.Next,when some parameters in the equations are sufficiently large,we obtain that the solution decays exponentially to the steady-state solution,moreover,we calculate the convergence rates accurately.(The main results of this chapter are published in Discrete Contin.Dyn.Syst.A,2018(38): 3617-3636.)In Chapter 3,we consider an important and special shallow water wave equation—Rotation-Camassa-Holm equation(R-CH equation).The research content is mainly divided into two parts: For the first part,we study the global existence for the weak solution to the R-CH equation.First,by denoting new energy variables,we transfer the original equation to a semi-linear ordinary differential equation(ODE)system.Next,in light of the standard ODE theorem,we prove the global existence and uniqueness of the solution for the semi-linear ODE system.Finally,from the inverse transformation to solutions of the ODE system,we obtain that the weak solution of the primary equation exists globally in the energy space1H().For the second part,we consider the uniqueness of the weak solution.First,by introducing new variables,we obtain a new semi-linear ODE system.Next,by proving the Lipschitz continuity for the right hand side of the equations,we obtain the uniqueness of the solution to the new ODE system.Then making use of the proof by contradiction,we prove the uniqueness of the weak solution to the original equation.(The main results of this aspect are published in J.Differential Equations,2019(266): 4864-4900.).In Chapter 4,we construct the Lipschitz metric for the R-CH equation.Consider that the wave-breaking phenomenon will happen in finite time on this R-CH equation even for smooth initial data,thus the solution flow obtained in Chapter 3 is not Lipschitz continuous under usual Sobolev metric.To address this problem,we first establish the Lipschitz metric for the smooth solution.Next,making use of the Transversality lemma,we prove the generic regularity result of the solution.Then,based on the generic regularity result,we generalize the Lipschitz metric from the smooth solution to the general weak solution.Finally,we compare this Lipschitz metric with other metrics(Sobolev metric,1L metric and Kantorovich-Rubinstein metric).In Chapter 5,we summarize the research works in this thesis and give the prospect of future research problems.