Ian Zimmermann - Reaction vs. Diffusion: Blow-up of solutions of an abstract semilinear parabolic equation

Consider a container filled with some type of fuel and an oxidant so that an exothermic chemical reaction takes place. Let us assume that the reaction rate depends on the temperature in a monotonically increasing fashion and on nothing else. Then the reaction will speed itself up due to the heat it produces, leading in turn to an increased production of heat. The only thing standing in the way of this self-accelerating process is the diffusion of heat from hotter to colder spots. We will examine a mathematical model describing this situation and investigate the competition between reaction and diffusion.