Some Partial Differential Equations Arising From Physics And Biology

This dissertation is devoted to some partial differential equations arising from Physics and Biology. The three problems, as Mathematical problems, are researched in this dissertation from different point of views. As practical problems, our results reflect some characteristic properties of the Physical or Biological problems them-selves.Chapter one is about the mathematical model from MEMS problem, i.e. the regularity of extremal solution u* of-Δu+c(x)·▽u=λf(u) in Rn with Dirichlet boundary condition. Here,f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). The regularity of extremal solution in low dimension has been shown in this part.The problem from Allen-Cahn equation and thin film problem has been demon-strated in Chapter two, i.e. for someδ0> 0,σ> 0 and f(0)=f'(0)=0. We obtain the asymptotic behavior of the oscillatory solution and apply to Allen-Cahn equation and thin film problem.The last chapter is devoted to Gierer-Meinhardt system which comes from the regeneration phenomena of hydra. This system has drawn many mathematicians' attention in last decade. Some beautiful results have been achieved, including the concentration phenomena of solution to stationary Gierer-Meinhardt system and the dynamic behavior of the kinetic part. In this chapter, we consider the dynamic behavior of kinetic Gierer-Meinhardt system with positive self-productive rate, in-cluding the global stability, the conditions of existence/non-existence of periodic solutions and the asymptotical behavior of finite time blow-up solution.