Is the heat equation unique under general boundary conditions?

Question

Energy methods can be used to show that the heat equation has a unique solution, but this requires specific boundary conditions (from what I know, that $u=0$ on the boundary, though I assume one can generalize further). But what if the boundary conditions are absolutely general functions? Is the solution unique in this case?

Answer 1

For the most general case, the solution to the heat equation is not unique. This result is first due to Tychonoff for functions with smooth intial data that do not necessarily decay as $|x|\to\infty$. The constructed counterexample is compactly supported in time but grows rapidly in $x$ when it is nonzero. This implies that the solution to the heat equation is nonunique as initial data that is identically zero corresponds both to the zero solution and this counterexample. This is an exercise in one of Terence Tao's note sets on the incompressible Navier-Stokes equations link.

Tychonoff, A., Théoremes d’unicité pour l’équation de la chaleur, Rec. Math. Moscou 42, 199-215 (1935). ZBL0012.35501.