Title: A universal method for evolution partial differential equations posed in domains with a boundary
Abstract: Evolution partial differential equations are omnipresent in applications from a broad spectrum of disciplines. A wide class of associated problems that arise spontaneously within these applications are known as initial-boundary value problems (IBVPs). These problems differ from the classical initial value problems in that they model phenomena in settings that involve a boundary. The presence of a boundary is natural not only when having in mind purely physical settings (e.g. waves in a semi-infinite channel) but also other applications such as the development of new numerical schemes. In this talk, we will discuss various IBVPs for evolution equations via a novel method for the rigorous analysis of these problems developed with A. Himonas and other collaborators. Existence and uniqueness of solution to IBVPs for nonlinear equations will be our primary focus; however, perhaps surprisingly, new results will be presented even at the level of linear equations. The results discussed in this talk are the outcome of joint works with A. Himonas, A. Fokas, F. Yan, T. Gartman and C.M. Johnston.