Blow-Up and Extinction of Solutions

Abstract

For large nonlinearities, semilinear parabolic equations can undergo dramatic effects: blow-up or extinction. This means that solutions do not exist for all times or simply vanish in finite time, two scenarios that are the first signs of visible nonlinear effects. The mechanism is the same that for ordinary differential equations and the question is to understand how diffusion can prevent or not the blow-up of solutions. We explain several methods allowing to answer this question: the method of the eigenfunction, the energy method, the method of moments. We illustrate the question of non-extinction with a counter-intuitive example modeling a fertilization strategy used by various benthic invertebrates.