Non-uniqueness of mild solutions for 2d-heat equations with singular initial data

In a recent article by the authors [15] it was shown that wide classes of semilinear elliptic equations with exponential type nonlinearities admit singular radial solutions $U$ on the punctured disc in ${\mathbb{R}}^2$ which are also distributional solutions on the whole disc. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: $u_s(t,x)\equiv U(x)$ is a stationary solution, and there exists also a solution $u_r(t,x)$ departing from $U$ which is bounded for $t>0$. While such non-uniqueness results have been known in higher dimensions by Ni–Sacks [33], Terraneo [40] and Galaktionov–Vazquez [16], only two very specific results have recently been obtained in two dimensions by Ioku–Ruf–Terraneo [22] and Ibrahim–Kikuchi–Nakanishi–Wei [21].