Mathematical Sciences: Semilinear Partial Differential Equations and Their Applications

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Research will be performed in three areas: (i) stability, finite time blow-up, decay rates, and asymptotics of life span of solutions to reaction-diffusion equations in the whole space; (ii) locations and rates of blow-up of solutions to spatially non-uniform nonlinear heat equations (model equations including the ones arising in the channel flow problem); (iii) concentration phenomena of solutions of semilinear elliptic equations (including Schrodinger equations and the equations arising in chemotaxis and binary mixture problems in biology and physics). The problems proposed are of mathematical, physical and biological importance. Some of the equations in the project have been well-known to mathematicians for years, yet new and challenging questions about these equations remain unanswered. The solutions for these problems will not only deepen our understanding of these concrete models, but also have impact on the understanding of and resolution to other related problems in the fields of mathematics, physics and biology.