Failure of the Strong Maximum Principle in Nonlinear Diffusion. Existence of Needles

The Strong Maximum Principle is a basic tool in the theory of elliptic and parabolic equations. Here we examine the family of nonlinear heat equations ut = ∇ · (um−1∇u), for different values of m ∈ R, with the purpose of finding out when and how the Strong Maximum Principle fails for these degenerate parabolic equations. We classify the situations into three groups: (1) finite speed, (2) infinite propagation with extinction, (3) a phenomenon of partial quenching at isolated times that we call “Space Needles”. The latter appear in the range −1 < m ≤ 0 and seem to be a new item in the parabolic literature. The existence of needles admits an interpretation in terms of control. After a change of variables, it can also be seen as a blow-up problem, where it displays a phenomenon of momentary regional blow-up at a time T with smooth continuation for t > T .