The Controllability Of A Nonlinear Keller-Segel Equation

The Keller-Segel equation,as a model to describe the chemotaxis progress of slime mold morphogenesis,is widely used in biology and medical science,etc.It has unique mathematical structure which causes difficulties in research.Currently,the research on the theory and application are active and the results are abundant.Especially in the field of partial differential equations,a wide range of topics have been involved,such as the existence of the solution,the stability of the solution,etc.However,the discussion in controllability is relatively rare.In this thesis,we study the controllability for a nonlinear Keller-Segel equation coupled by an elliptic partial differential equation and a parabolic one,in which nonlinearity lies both on its drift-diffusion term and population growth.Compared with others parabolic systems of equations with diffusion operator,the nonlinear drift-diffusion term ▽(u▽Φ(v))brings some difficulties in the research,such as the regularity of the equation,the observability estimate of the linear equation,etc.Our nonlinear Keller-Segel model contains some classical chemotaxis types.The results are richer,though the problem is more difficult.Firstly,we establish both the approximate controllability and exact null controllability for a linear heat equation with Neumann boundary condition.To do that,we adopt the Carleman inequality with explicit expression of time T for the conjugate equation,thereby get the unique continuation and observability estimate.Hence,the approximate controllability and exact null controllability of the linear heat equation follows respectively.Secondly,we establish the controllability property for the nonlinear Keller-Segel equation by utilizing the nonlocal structure of the elliptic-parabolic system of equations.Since it can be treated as a single nonlinear parabolic equation,the controllability of the nonlinear Keller-Segel equation was solved by analogy.We obtain that the nonlinear Keller-Segel equation is local null controllable by using the controllability of the linear heat equation and some fixed point theorem.The method in this thesis can be applied to other models having similar structure,such as the semiconductor equation.Finally,the controllability problem of Keller-Segel equation with one-control and other similar types of parabolic equations is not yet fully covered.There are many scientific problems worthy of discussion.The research of this paper may supplement the research in this research field.