Nonuniqueness and failure of positivity preservation for doubly critical semilinear heat equations

At the Fujita exponent $p=1+2/N$ we construct a strictly positive mild solution of $u_t-Δu=|u|^{p-1}u$ in $\mathbb R^N$ whose initial trace in $L^1$ is zero. In similarity variables the solution approaches a Gaussian profile, and $\|U(t)\|_1\sim C_N(\log(1/t))^{-N/2},\qquad \|U(t)\|_\infty\sim c_Nt^{-N/2}(\log(1/t))^{-N/2},$ with explicit constants $C_N$ and $c_N$. In particular, uniqueness fails at the borderline logarithmic rate, and the solution lies outside every stronger logarithmic uniqueness class; this answers two questions raised by Miyamoto [J. Evol. Equ. 21 (2021), 151–166, Remark~1.6]. We also prove that, up to time translation, this is the only ancient solution of fixed sign of the rescaled equation that decays in a Gaussian weighted supremum norm. On a bounded smooth domain the whole-space solution can be localized at finitely many interior points, with independently chosen signs, on top of an arbitrary bounded initial datum. The resulting solutions are classical for $t>0$ and continuous at $t=0$ in $L^1$. Consequently, uniqueness fails for every bounded datum, and positivity is lost for every bounded nonnegative datum, $u_0\equiv1$ included. These results provide affirmative answers to Open Problems~3 and~4 posed by Brezis and Cazenave in [J. Anal. Math. 68 (1996), 277–304].