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

ABSTRACT 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 < m ≤ 0 and seem to be new 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 blowup problem, where it displays the phenomenon of momentary regional blowup at a time T with smooth continuation for t > T.