Long time behavior and stability of special solutions of nonlinear partial differential equations.

Abstract

In this dissertation, in the first part, I study the long-time behavior of the solutions of the Kuramoto-Sivashinsky equation and the Burgers-Sivashinsky equation. First, I work on a two-dimensional modified KS equation and prove the existence of a global attractor on a bounded domain. Next, I study the long-time behavior of the solutions of the one-dimensional BS equation for general initial data as opposed to the usually considered odd initial data. Third, I study the long-time behavior of radially symmetric solutions of the KS equation in a shell domain in three-dimensions. In the second part, we deal with the conditional stability of radial steady state solutions for the one-dimensional Klein-Gordon equation. I consider the one-dimensional case and construct the infinite-dimensional invariant manifolds explicitly. The result is a precise center-stable manifold theorem, which includes the co-dimension of the manifolds and the decay rates.