Non-uniqueness of mild solutions for 2d-heat equations with singular initial data
Yohei Fujishima, Norisuke Ioku, Bernhard Ruf, Elide Terraneo
TL;DR
The paper studies the Cauchy problem for the two-dimensional semilinear heat equation with exponential-type nonlinearities and singular elliptic data. It constructs a regular mild solution with initial data $U$, a singular radial solution of $-\Delta U=f(U)$ that blows up at the origin and is distributional on the full ball, by building a supersolution via a Cole–Hopf-type transformation and applying a Perron monotone method, complemented by a Lorentz-space fixed-point argument. It simultaneously shows that the stationary mild solution $u_s(t,x)=U(x)$ is also a mild solution, yielding non-uniqueness of mild solutions for the same initial data. The work also highlights a threshold phenomenon: with scaled initial data $u_0=\lambda U$, the problem is well-posed for $0<\lambda<1$, but admits no nonnegative local solution for $\lambda>1$. These results extend 2D non-uniqueness phenomena to broad exponential-type nonlinearities using a blend of PDE tools including heat-kernel methods, Perron’s principle, and singular elliptic data.
Abstract
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].