Nonuniqueness for the heat flow of harmonic maps on the disk (Q1347510)

The authors deal with the equation \[ {\partial u\over\partial t}= \Delta u+|\nabla u|^2u,\;x\in\Omega,\;t>0,\tag{1} \] where \(u(x,t)\) denotes a unit vector in \(\mathbb{R}^3\), \(u:\Omega \times\mathbb{R}^+\to S^2\). The main result of this paper is that if \(\Omega\) is the unit disk in \(\mathbb{R}^2\), then there exists \(u_0\in C^\infty (\overline\Omega)\) such that the Cauchy problem for (1) with initial data \(u_0\) has more than one solution satisfying \(\int_\Omega |\nabla u(t)|^2 dx\leq\int_\Omega|\nabla u_0|^2dx\) for \(t>0\).