On solvability of a time-fractional doubly critical semilinear equation, and its quantitative approach to the non-existence result on the classical counterpart

We study a time-fractional semilinear heat equation $${\mathrm{\partial }}_t^{\alpha }u-\mathrm{\Delta }u=u^p,in(0,T)\times {\mathbb{R}}^N,u(0)=u_0\ge 0$$ with $u_0\in L^1({\mathbb{R}}^N)$ and $p=1+2\mathrm{/}N$. Here ${\mathrm{\partial }}_t^{\alpha }$ denotes the Caputo derivative of order $\alpha \in (0,1)$. Since the space $L^1({\mathbb{R}}^N)$ is scale critical with $p=1+2\mathrm{/}N$, this type of equation is known as a doubly critical problem. It is known that the usual doubly critical equation ${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=u^p$ does not have nonnegative global-in-time solutions, while the time-fractional problem does. Moreover, there exists a singular initial data which admits no local-in-time solution, while the time-fractional equation is solvable for any $L^1({\mathbb{R}}^N)$ initial data. In this paper, we deduce a necessary condition imposed on $u_0$ for the existence of a nonnegative solution. Furthermore, we obtain corollaries that describe the collapse of the local and global solvability for the time-fractional equation as $\alpha \to 1$.