Global Existence And Long Time Behavior Of Solutions In Chemotaxis Systems

The present work deals with global existence and long time behavior of solutions in chemo-taxis systems. There are two kinds of topics:Global existence and global boundedness of solu-tions; Long time behavior, including long time convergence and convergence rate, to the global solutions.The thesis consists of four parts.In Chapter1, we give an overview to the chemotaxis models involved by this thesis, includ-ing the background of chemotaxis, as well as the history and the current progress of the related studies, with useful references of the field. We also state the main results of the thesis.Chapter2is devoted to a quasilinear parabolic Keller-Segel system with logistic source. We find global existence solutions under different conditions on the parameters.In chapter3, we study the semilinear fully parabolic Keller-Segel in bounded domain. We derive a smallness condition on the initial data in optimal Lebesgue spaces. If N≥2, one can find an upper bound for u0in LN/2(Ω) and▽v0in LN(Ω), such that the solution is global, globally bounded, and converges to the trivial steady state. In addition, we study a chemotaxis system with consumption. In contrast to the classical Keller-Segel model, we obtain much weaker smallness conditions for this model to ensure the global boundedness and the long time convergence.Chapter4deals with the1D fully parabolic Keller-Segel system with logistic source in bounded domain under Neumann boundary condition. It is shown that if the Logistic source is much weaker than the aggregation term, then any nontrivial solution goes to the spatially homogenous steady state. Moreover, the convergence is exponential.Chapter5summarizes the main results of the thesis, and proposes some interesting prob-lems to be done.