Mathematical Sciences: Semilinear Partial Differential Equations and Their Applications

研究人员研究了一些缺乏紧性或李亚普诺夫泛函的反应扩散方程解的稳定性。 这里的具体问题包括解的全局存在的初始值的最佳条件、全局解的限制集以及平衡或行波稳定的拓扑。 他还研究了无界域中椭圆方程解的点集中现象。 集中的速度和地点是主要问题之一。 所有研究的方程都是生物学、化学或数学物理学的模型方程。 更好地理解这些重要的方程可以深入了解潜在的科学问题。 但由于缺乏处理非线性方程的通用工具，对方程进行彻底的研究很困难。因此，问题和解决这些问题的技术也具有内在的数学兴趣。

The investigator studies stability of solutions of some reaction-diffusion equations that lck compactness or Liapunov functionals. Here specific issues include optimal conditions on the initial value for global existence of a solution, limiting sets of global solutions, and topologies in which equilibria or traveling waves are stable. He also studies point-concentration phenomena of solutions of elliptic equations in unbounded domains. The rates and locations of concentration are among the main questions. All the equations studied are model equations from biology, chemistry, or mathematical physics. Better understanding of these important equations provides insight into the underlying scientific questions. But a thorough study of the equations is difficult because of a lack of general tools to deal with nonlinear equations. Hence the problems and techniques for solving them are of intrinsic mathematical interest also.