Anomaly of the fractional heat propagator in abstract settings

Abstract

We study the following time-fractional heat equation: \begin{equation*} ^{C}\partial_{t}^{\alpha}u(t)+\mathscr{L}u(t)=0,\quad u(0)=u_0\in X, \quad t\in[0,T],\quad T>0,\quad 0<\alpha<1, \end{equation*} where $^{C}\partial_{t}^{\alpha}$ is the Djrbashian-Caputo fractional derivative, $X$ is a complex Banach space and $\mathscr{L}:\mathcal{D}(\mathscr{L})\subset X\to X$ is a closed linear operator. The solution operator of the equation above is given by the strongly continuous operator $E_\alpha(-t^{\alpha}\mathscr{L})$ for any $t\geqslant0$, closely related with the Mittag-Leffler function $E_\alpha(-x)$ for $x\geqslant0.$ There are different ways to present explicitly this operator and one of the most popular is given in terms of the $C_0$-semigroup generated by $-\mathscr{L}$ $\big(\{e^{-t\mathscr{L}}\}_{t\geqslant0}\big)$ as follows: \[ E_\alpha(-t^{\alpha}\mathscr{L})=\int_0^{+\infty}M_{\alpha}(s)e^{-st^{\alpha}\mathscr{L}}{\rm d}s,\quad t\geqslant0, \] where $M_{\alpha}(s)$ is a Wright-type function. We will see that the latter expression is not always optimal (regarding restrictions: endpoint lost) to estimate different norms. An additional restriction appears while bounding the above integral, which can be avoided by using directly the function itself and its well-known uniform bound $|E_{\alpha}(-x)|\leqslant \frac{C}{1+x},$ $x\geqslant0.$