Periodic Solutions And Dirichlet-Neumann Mapping Of The Initial-boundary Value Problem To Keller-Segel Equations

The thesis concerns with the properties of solutions for Keller-Segel equations arising in biology. First of all, we consider the periodic solutions to Keller-Segel equations. By utilizing the method of characteristics, we are able to prove that the smooth periodic solutions of the Keller-Segel equations will blow up in finite time.Moreover, the lifespan for the solutions is also given. Then, we use the Fourier and Laplace transform and its inverse transformation to study the initial boundary value problem for Keller-Segel equations. The corresponding Dirichlet-Neumann mapping is established, and the explicit solution formula is determined by constructing the Green's function.