Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation

TL;DR

This paper proves the existence and uniqueness of weak solutions for a fractional single-phase-lag heat equation using variational methods and Rothe's time discretization, extending properties of Mittag-Leffler functions.

Contribution

It introduces a rigorous framework for solving the fractional heat equation with Caputo derivatives, including a new representation formula and bounds for solutions.

Findings

Existence of a unique weak solution under low regularity assumptions

Development of a variational and Rothe's method-based approach

Derivation of explicit solution bounds using Mittag-Leffler functions

Abstract

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $\cal{D}_t^\alpha(u_t)$ and $\cal{D}_t^\alpha u $ (with $\alpha \in(0,1)$), where $\cal{D}_t^\alpha$ denotes the Caputo fractional derivative in time of constant order $\alpha\in(0,1)$. We consider homogeneous Dirichlet boundary data for the temperature. We rigorously show the existence of a unique weak solution under low regularity assumptions on the data. Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe's method. We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution. The variational approach is employed to show the uniqueness of this weak solution to the problem. We also consider the one-dimensional problem and derive a representation formula for the solution. We establish bounds on this explicit solution and its time derivative by extending properties of the multinomial Mittag-Leffler function.