On The Asymptotic Behavior Of Solutions To Two Evolutionary Partial Differential Equations

The study of nonlinear dynamics is a fascinating field which is at the very heart of the understanding of many important problems of the natural sciences. In mathematics, many new ideas and new tools have been developed to solve the infinite dimensional dynamical system problems. In the terms of partial differential equation, the important part is to obtain uniform prior estimates on the solution of the initial-boundary problem for time t. It can help us to analyze whether the system has some asymptotic properties (e.g. the invariant property and attractivity). This paper is devoted to two problems, one of which is about a reaction-diffusion equation: where v >0, λ( x ) ≥λ₀>0, |▽λ( x)| ≤C₀, f (u )u≥0, f ( 0)=0, f ' (u)≥-C. We extend the result of Bixiang Wang in [26] to the case that the parameter λdepends on space variables and prove the existence of the global attractor in R n. The other problem is the following nonlinear Airy equation: In this paper, we succeed to prove the global existence and uniqueness of an L2 solution and the continuous dependence of the solution on the initial value u 0. If we assume the data of the equation in some asymmetric spaces, we can obtain some asymptotic properties of the solution (refer to Theorems 3.2 and 3.3).