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 R2 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: us(t,x)≡U(x) is a stationary solution, and there exists also a solution ur(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].