Existence and nonexistence of a global solution to the Kuramoto-Sivashinsky equation

A study was conducted to discuss the existence and nonexistence of a global solution to the Kuramoto-Sivashinsky equation. The study considered initial boundary-value problems (IBV) for the Kuramoto-Sivashinsky equation in one dimension. It proved the existence of a global solution both with Dirichlet and with 'Navier' boundary conditions, and also proved a blow-up of solutions of IBV problem with boundary conditions of other kind. The Kuramoto-Sivashinsky equation arises as a model in hydrodynamics, in the combustion theory, phase turbulence, and plasmas, as one model for spatiotemporal chaos and in many other physical phenomena. The study also used a classical approach on the Galerkin method for the proof of existence, while for the proof of blow-up, the 'nonlinear capacity' method, based on special characteristic test functions (eigenfunctions), was used.