Abstract

The global dynamics of the real dissipative quadratic heat equation



and its younger time-reversible, complex-valued ``Schrödinger'' sibling



seem to have little in common.
Consider , with Neumann boundary conditions. By its gradient-like structure, all real eternal non-equilibrium orbits of (*) are heteroclinic among equilibria . Nonhomogeneous real of Morse index are rescaled real-valued Weierstrass elliptic functions .

We show that real heteroclinic orbits are accompanied by finite-time blow-up, via analytic extension to imaginary time s. In particular, there exist such that the sibling in (**), of in (*), blows up at some finite real time .

Technically, however, we have to assume asymptotically stable target equilibria W. We also have to except a discrete set of , and are currently limited to unstable dimensions , or to fast unstable manifolds at of dimensions .