Anomaly of the fractional heat propagator in abstract settings

TL;DR

This paper investigates the peculiar behavior of the fractional heat propagator in abstract mathematical settings, revealing an anomaly in its properties and limitations of common integral representations.

Contribution

It identifies the limitations of the integral representation of the solution operator for fractional heat equations and proposes direct use of the Mittag-Leffler function for better estimates.

Findings

Integral representation may lose endpoint restrictions.

Direct use of Mittag-Leffler function provides better norm estimates.

Highlights anomalies in fractional heat propagator behavior.

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 $|E_{\alpha}(-x)|\leqslant \frac{C}{1+x},$ $x\geqslant0.$