Very singular diffusion equations. (English) Zbl 1002.35074

Maruyama, Masaki (ed.) et al., Taniguchi conference on mathematics Nara ’98. Papers from the conference, Nara, Japan, December 15-20, 1998. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 31, 93-125 (2001).
This paper deals with equations of the type \[ \dfrac{\partial u}{\partial t}=\dfrac{1}{b} \text{div}(a\dfrac{\nabla u}{|\nabla u|}), \] where \(a, b\) are positive given functions. The main scope is to analyze the initial-boundary problem in the 1-D case, describing the very peculiar behaviour of the solutions. However, some results are reviewed in the general framework, starting from the formulation \[ \dfrac{du}{dt} \in -\partial \phi (u),\;u/_{t=0}=u_0 \] in a Hilbert space, for which existence and uniqueness is known. For the one-dimensional problem the emphasis is on the behaviour of solutions developing from piecewise constant initial data. Nontrivial conditions are found on \(a, b\) for the solutions to develop “plateaus”. A detailed numerical study is performed.
For the entire collection see [Zbl 0980.00032].


35K65 Degenerate parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations