On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart

We are concerned with the following time-fractional semilinear heat equation in the $N$-dimensional whole space ${\mathbf{R}}^N$ with $N\ge 1$. $${(\mathrm{P})}_{\alpha }{\mathrm{\partial }}_t^{\alpha }u-\mathrm{\Delta }u=u^p,t>0,x\in {\mathbf{R}}^N,u(0)=\mu in{\mathbf{R}}^N,$$ where ${\mathrm{\partial }}_t^{\alpha }$ denotes the Caputo derivative of order $\alpha \in (0,1)$, $p>1$, and $\mu$ is a nonnegative Radon measure on ${\mathbf{R}}^N$. The case $\alpha =1$ formally gives the Fujita-type equation (P)${}_1$ \ ${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=u^p$. In particular, we mainly focus on the Fujita critical case where $p=p_F:=1+2\mathrm{/}N$. It is well known that the Fujita exponent $p_F$ separates the ranges of $p$ for the global-in-time solvability of (P)${}_1$. In particular, (P)${}_1$ with $p=p_F$ possesses no global-in-time solutions, and does not locally-in-time solvable in its scale critical space $L^1({\mathbf{R}}^N)$. It is also known that the exponent $p_F$ plays the same role for the global-in-time solvability for (P)${}_{\alpha }$. However, the problem (P)${}_{\alpha }$ with $p=p_F$ is globally-in-time solvable, and exhibites local-in-time solvability in its scale critical space $L^1({\mathbf{R}}^N)$. The purpose of this paper is to clarify the collapse of the global and local-in-time solvability of (P)${}_{\alpha }$ as $\alpha$ approaches $1-0$.