Whitney folds in phase of some semilinear Sobolev-type equations (Q1893637)

The authors study the degenerate parabolic equation \[ 0 = d_1 \Delta u_1 + f_1 (u_1, u_2),\quad{\partial u_2 \over \partial t} = d_2 \Delta u_2 + f_2 (u_1, u_2) \text{ on } \Omega \] \[ u_i (x,0) = u_{i0} (x) \text{ on } \Omega,\quad {\partial u_i \over \partial \nu} = 0 \text{ on } \partial \Omega \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^n\). They interpret this equation as defining a flow on the phase space \[ \mathbb{B} = \left\{ (u_1, u_2) \;d_1 \Delta u_1 + f_1 (u_1, u_2) = 0, \quad {\partial u_1 \over \partial \nu} = 0 \right\}. \] This paper describes the morphology of \(\mathbb{B}\) in some special situations when the equations defining this phase space do not satisfy the assumptions of the implicit function theorem which would guarantee smoothness. When singularities do occur they are equivalent to a Whitney \(k\)-fold.