Almost Automorphic Solutions for the Heat Equation Involving the Discrete Fractional Laplacian in Continuous and Discrete Time

ABSTRACT In this work, we study the almost automorphic property of solutions to the one‐dimensional fractional heat equation involving the discrete fractional Laplacian with in the Lebesgue space , considering both continuous and discrete time cases. The lattice system is formulated as an initial value problem, whose solutions are expressed through a subordination formula involving the continuous and discrete Lévy functions, as well as the semi‐discrete heat kernel defined in terms of the modified Bessel functions. We establish sufficient conditions to guarantee the existence and uniqueness of almost automorphic solutions under suitable Lipschitz‐type assumptions, relying on fixed point theorems. To achieve this, we prove invariance under convolution and the superposition principle for the almost automorphic property.