Nonlinear Evolution Equations

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文件名称：Nonlinear Evolution Equations
文件大小：1.48MB
文件格式：PDF
更新时间：2013-05-29 00:25:06
半群，吸引子 This book is designed to introduce some important methods for nonlinear evolution equations.、 Nonlinear evolution equations, i.e., partial differential equations with time t as one of the independent variables, arise not only from many fields of mathematics, but also from other branches of science such as physics, mechanics and material science. For example, Navier-Stokes and Euler equations from fluid mechanics, nonlinear reaction-diffusion equations from heat transfers and biological sciences, nonlinear Klein- Gorden equations and nonlinear Schr¨odinger equations from quantum mechanics and Cahn-Hilliard equations from material science, to name just a few, are special examples of nonlinear evolution equations. Complexity of nonlinear evolution equations and challenges in their theoretical study have attracted a lot of interest from many mathematicians and scientists in nonlinear sciences. The first question to ask in the theoretical study is whether for a nonlinear evolution equation with given initial data there is a solution at least locally in time, and whether it is unique in the considered class. Generally speaking, this problem has been solved for a wide class of nonlinear evolution equations by two powerful methods in nonlinear analysis, i.e., the contraction mapping theorem and the Leray-Schauder fixed-point theorem. Roughly speaking, since the 1960s, much more attention has been paid to the global existence and uniqueness of a solution, i.e., when a local solution can be extended to become a global one in time. Furthermore, if there is a global solution for a given nonlinear evolution equation, one also wants to know about the asymptotic behavior of the solution as time goes to infinity.