Study On The Spatial Model Of Solutions Of Chemotaxis Models With Strong Interactions

The formation and diffusion of tumors are related to the diffusion and accumulation of cells closely,and the migration of invasive cells is directed either by a mechanism termed chemotaxis or by a mechanism termed haptotaxis.The spatial behavior of solutions to the corresponding systems has always been a hot issue in ODE and PDE,that is whether coexistence of all species occurs or the internal growth law leads to extinction.In recent years,great interest has been shown in chemotaxis-competition models.Compared with the single-cell chemotaxis models,the multicellular chemotaxis-competition models have richer kinetic behavior.In particular,under strong competition,the solutions of system are asymptotic separation,and thus a free boundary problem arises.Phase separation phenomenon caused by strong interaction and the free boundary problem have became frontier problems in the partial differential equations.It has attracted many famous scholars,such as the winner of Wolf Prize and Abel Prize,American mathematician L.A.Caffarelli,the famous Chinese mathematician Lin Fanghua,the academician of the Australian Academy of Sciences E.N.Dancer,and the famous Italian mathematician S.Terracini.On the one hand,the researches on these problems have promoted the development of the theory of partial differential equation.On the other hand,their research results have made a scientific explanation of the corresponding biological phenomena and play a certain guiding role.In this paper,we mainly concern the global existence and asymptotic behavior of the solutions of chemotactaxis models with logistic motion mechanism.The specific content consists of the following four chapters:In this paper,we mainly concern the global existence and asymptotic behavior of the solutions of chemotactaxis models with logistic motion mechanism.The specific content consists of the following four chapters:In Chapter 1,we introduce the background and development status of our research problem.In Chapter 2,we concern a parabolic-elliptic model with fractional diffusion and logistic terms,and prove the global existence and uniqueness of the classical solutions of the model.In Chapter 3,we study a parabolic-parabolic-elliptic model with logistic terms,and prove the asymptotic separation of the support sets of solutions under strong competition.We extend Terracini’s blow up method to the chemotaxis model,and prove the uniform bound estimate in solutions with respect to competition parameters.Finally,overall results are summarized in Chapter 4,together with some problems for future research.