Failure of the Strong Maximum Principle in Nonlinear Diffusion. Existence of Needles
Juan Luis Vázquez · Communications in Partial Differential Equations · 2005
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 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 T.