Qualitative Studies On Three Kinds Of Chemotaxis Models

Chemotaxis refers to the property that allows cells or organisms to make directional movements in response to the chemical gradients.Hence cells or organisms are able to find food and stay away from toxins.Chemotaxis phenomenon exists widely in nature,and has been applied in biological decontamination,microbial enhanced oil recovery,biofilm formation and wound healing.In 1970s,E.F.Keller and L.A.Segel introduced the original mathematical model describing chemotaxis,which is later called Keller-Segel chemotaxis model.The mathe-matical studies on Keller-Segel chemotaxis model received widespread attention.In this paper,we apply the energy estimate,semigroup theorem,Lyapunov functional and other methods to study the asymptotic behavior of the traveling waves,global existence and large time behavior of classical solutions to chemotaxis models.To be more precise,we have the following results.?.In the second chapter,we study the asymptotic stability of the composite wave consisting two traveling waves to the Keller-Segel chemotaxis model with logarithm sensitivity in one-dimensional space.Applying the energy estimate method,we show that the composite wave is asymptotically stable if the initial perturbation is small in H1-norm.?.In the third chapter,we consider the initial-boundary problem of the attraction-repulsion chemotaxis model with logistic source.Precisely,in a bounded smooth domain ?(?)Rn,we establish the global existence and boundedness of classical so-lutions to the model in higher dimensional space through a coupled energy estimate and Moser type iteration method.Moreover,we study the large time behavior of the solutions by establishing a Lyapunov functional.Furthermore,we find that when attraction cancels repulsion,the logistic source plays an important role in the behavior of solutions to the attraction-repulsion chemotaxis model with logistic source.?.In the last chapter,we are concerned with the initial-boundary problem of an indi-rect chemotaxis model arising from tumor invasion.Firstly,under some conditions,we establish the uniform-in-time bound of the global classical solution to the model by a coupled energy estimate,semigroup estimate and Moser type iteration method.Secondly,using the semigroup estimate method and the parabolic regularity theory,we show that the classical solution converges to the constant steady state exponen-tially.Lastly,we study the differences of the critical index between the direct chemotaxis model and the indirect chemotaxis model.