An evolutionary partial differential equation (PDE) is a differential equation which describes the time evolution of a phenomenon distributed in space. Some examples include: 1) the wave equation modeling the displacement of a guitar string, 2) the heat equation modeling the temperature distribution on a surface, 3) the plate equation modeling the oscillations of a thin panel on an aircraft fuselage, and 4) the Schrodinger equation modeling the wave function of a particle in a potential well.
In this general audience talk we examine an evolutionary PDE from its derivation, through various notions of solutions, as well as some qualitative properties of solutions. We will use the Kirchoff plate equation in two dimensions as our model as we explore the relationship between abstract theorems at the level of operators, Hilbert spaces, and dynamical systems and ``hard" analysis done at the level of the equation itself.
As we move through the various stages of analysis we will be careful to motivate and define basic notions in differential equations; we will also discuss fundamental concepts in the modern study of evolution equations, including semigroups, stability, and global attractors. The goal for the talk is two-fold: (1) for non PDE specialists the hope is to use a specific evolution equation to highlight the principal questions of interest and some key techniques in the modern study of evolution PDEs; (2) for those experienced in the analysis of differential equations, recent results on the asymptotic properties (quasistability theory) of solutions to some nonlinear plate equations (Berger, von Karman) will be discussed, time permitting of course.