For the fractional heat equation \begin{equation} \partial_t u + (-\Delta^s)u=0 \text{ in } \mathbb{R}^d \times (0,\infty), \end{equation} where $s \in (0,1)$ where the fractional laplacian is the Fourier multiplier by $|\xi|^{2s}$ i.e. \begin{equation} \mathcal{F}( (-\Delta)^s \phi) = |\xi|^{2s} \hat{\phi} \end{equation} Green's function has the following asymptotic formula \begin{equation} P(t,x;s) \sim \frac{t}{(t^{1/s}+|x|^2)^{(n+2s)/2}} \end{equation} where the $\sim$ means bounded above and below by constants (that depend on $s$).
I was wondering if there's some asymptotic formula for the implied constants in the Green's function formula. I was also wondering since $(-\Delta^s) \rightarrow -\Delta$ as $s \rightarrow 1^-$, if $P(t,x;s)$ actually converges to Green's function of the heat equation i.e. \begin{equation} P(t,x;1):= \frac{1}{(4 \pi t)^{d/2}} e^{-|x|^2/(2t)}. \end{equation} My guess is the answer is no since the fractional heat equation kernel is polynomial decay with the heat equation has exponential tails but I am worried about how the implied constants blow up as $s \rightarrow 1^-$. Also I was wondering if there were any good references to learn fractional parabolic differential equations.
Thank you so much!
