Qualitative Study of Advection-Diffusion Systems

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The purpose of this research is to study the qualitative properties of an advection-diffusion equationcalled a chemotaxis system. This system is a mathematical model describing chemotactic aggregation of cellular slime molds which move towards relatively higher concentration of a chemical. We studied the global existence of solutions and finite-time blow-up of solutions to the chemotaxis system and obtained the following results.1. Two types of sensitivity functions are considered to study the structure of the global existence and blow-up of solutions, and it was shown that the structure of radial solutions entirely depends on the sensitivity functions.2. It was shown that the finite-time blow-up of solutions to an parabolic-elliptic system, which is a simplified version of the chemotaxis system, necessarily leads to chemotactic collapse (the blow-up with 6-function singularities) at an isolated blow-up point in two-dimensional domains.3. The application of syrnmetrization techniques to the parabolic-elliptic system mentioned above is effective to obtain Dr-estimates of solutions in terms of L^p norm of initial functions.