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
Dtα​(ut​) and Dtα​u (with α∈(0,1)),
where Dtα​ denotes the Caputo fractional derivative in time of
constant order α∈(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.Comment: 27 page