The goal of this project is to investigate the evolution dynamics of a class of nonlinear diffusion equations that appear as mathematical models in life and social science: (1) Analysis of existence and long time behavior of solutions of a free boundary problem in one and two dimensional space; (2) Study of the regularity of solutions to a Hamilton-Jacobi-Bellman equation coupled to a Fokker-Planck diffusion equation; (3) Analysis of long-time behavior of parabolic equations with nonlinear singular sources. These equations present several difficulties due to high differential order as well as the underlying nonlinear structure, and new analytical tools need to be developed: some key techniques involve combination of classical partial differential equations technique with methods from statistical mechanics, optimal control and kinetic theory. Of particular interest is the study of stability and finite time blow-up phenomena in the model: in case of stability, behavior of solution for large time will be investigated (possible bifurcations or convergence towards equilibrium state). In case of blow-up, relation between formation of singularities and blow-up of the related quantities will be analyzed.
Complex, real-life systems in sociology, economics, and life sciences often contain a large number of individuals that interact and can develop a collective behavior; these are exactly the areas of sciences where this project draws the topics from and where mathematics could offer great insights and contributions. The understanding of diffusion processes and collective behavior is essential in many areas of science. The project focuses on investigating a class of mean field and kinetic models that can describe very well interactions among agents, collective behavior and averaging processes. Applications of such a study range from game theory (decisional strategies, behavior of investors), socio-economic (opinion formation, population dynamics), to biology (tumor growth, flocking) and neuroscience. The project will also provide education and training through research to undergraduate and graduate students.