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We study a time-fractional semilinear heat equation $$\partial^{\alpha}_t u-\Delta u = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^{\alpha}$denotes the Caputo derivative of order $\alpha \in (0,1)$. Since the space$L^1(\mathbb{R}^N)$ is scale critical with $p=1+2/N$, this type of equation isknown as a doubly critical problem. It is known that the usual doubly criticalequation $\partial_t u-\Delta u=u^p$ does not have nonnegative global-in-timesolutions, while the time-fractional problem does. Moreover, there exists asingular initial data which admits no local-in-time solution, while thetime-fractional equation is solvable for any $L^{1}(\mathbb{R}^N)$ initialdata. In this paper, we deduce a necessary condition imposed on $u_0$ for theexistence of a nonnegative solution. Furthermore, we obtain corollaries thatdescribe the collapse of the local and global solvability for thetime-fractional equation as $\alpha \rightarrow 1$.

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