The aim of the project is to address fundamental questions of uniqueness for solutions of semilinear heat equations with unbounded initial data. Such questions remain at the heart of modern well-posedness theories of nonlinear partial differential equations and have wider applicability to other branches of Analysis (functional analysis, harmonic analysis, probability theory, dynamical systems, numerical analysis) and as models in the natural sciences. For many cases of interest the question of uniqueness has remained wide open since partial results were first obtained in the 1980s and 90s but recent advances in local existence theories, and in particular their connection with comparison principles and uniqueness, have suggested new methods of attack. The objective is to exploit these recent developments to shed some light on the uniqueness problems and establish new, stronger results. We anticipate that this project will be of significant interest to mathematicians locally, nationally and internationally.​