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Anomaly of the fractional heat propagator in abstract settings

Joel E. Restrepo

TL;DR

The paper analyzes time-fractional heat equations with a Caputo derivative on Banach spaces, focusing on the propagator $E_{\alpha}(-t^{\alpha}\mathscr{L})$ tied to the Mittag-Leffler function. It compares two representations of the propagator—the functional-calculus/Hankel form and a Wright-type integral against the $C_0$-semigroup—and derives $L^{p}$–$L^{q}$ decay estimates on locally compact unimodular groups under spectral-trace growth conditions. A key finding is that the Wright-type integral can be suboptimal at endpoint decay, while the direct bound $|E_{\alpha}(-t^{\alpha}x)|\le C/(1+x)$ yields sharper, often optimal, decay rates; the results are illustrated through examples involving classical and subelliptic operators. The work clarifies how the choice of propagator representation affects the sharpness of decay estimates in abstract fractional diffusion settings and guides the selection of representation for optimal $L^{p}$–$L^{q}$ bounds.

Abstract

We study the following time-fractional heat equation: \begin{equation*} ^{C}\partial_{t}^αu(t)+\mathscr{L}u(t)=0,\quad u(0)=u_0\in X, \quad t\in[0,T],\quad T>0,\quad 0<α<1, \end{equation*} where $^{C}\partial_{t}^α$ 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 $E_α(-t^α\mathscr{L})$ for any $t\geqslant0$, closely related with the Mittag-Leffler function $E_α(-x)$ 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: \[ E_α(-t^α\mathscr{L})=\int_0^{+\infty}M_α(s)e^{-st^α\mathscr{L}}{\rm d}s,\quad t\geqslant0, \] where $M_α(s)$ 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 $|E_α(-x)|\leqslant \frac{C}{1+x},$ $x\geqslant0.$

Anomaly of the fractional heat propagator in abstract settings

TL;DR

The paper analyzes time-fractional heat equations with a Caputo derivative on Banach spaces, focusing on the propagator tied to the Mittag-Leffler function. It compares two representations of the propagator—the functional-calculus/Hankel form and a Wright-type integral against the -semigroup—and derives – decay estimates on locally compact unimodular groups under spectral-trace growth conditions. A key finding is that the Wright-type integral can be suboptimal at endpoint decay, while the direct bound yields sharper, often optimal, decay rates; the results are illustrated through examples involving classical and subelliptic operators. The work clarifies how the choice of propagator representation affects the sharpness of decay estimates in abstract fractional diffusion settings and guides the selection of representation for optimal – bounds.

Abstract

We study the following time-fractional heat equation: \begin{equation*} ^{C}\partial_{t}^αu(t)+\mathscr{L}u(t)=0,\quad u(0)=u_0\in X, \quad t\in[0,T],\quad T>0,\quad 0<α<1, \end{equation*} where is the Djrbashian-Caputo fractional derivative, is a complex Banach space and is a closed linear operator. The solution operator of the equation above is given by the strongly continuous operator for any , closely related with the Mittag-Leffler function for There are different ways to present explicitly this operator and one of the most popular is given in terms of the -semigroup generated by as follows: where 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

Paper Structure

This paper contains 6 sections, 2 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Fractional heat equations and their time-decay estimates
  3. Heat equation and its time-decay rate
  4. Time-fractional heat equation and its time-decay rate
  5. Alternative approach to study the time-fractional heat equation
  6. Conclusion

Key Result

Theorem 2.1

Let $G$ be a locally compact separable unimodular group and let $1<p\leqslant 2\leqslant q<+\infty$. Let $\mathscr{L}$ be any positive left invariant operator on $G$ (unbounded) such that If $0<\alpha<1$ and $w_0\in L^p(G)$ then the solution for the time-fractional $\mathscr{L}$-heat equation (heat-f) is given by and also it is in $L^q(G).$ In particular, if the condition (trace) holds, we obtai

Theorems & Definitions (4)

  • Theorem 2.1
  • Remark 1
  • Theorem 2.2
  • proof