Representation formula, regularity, and decay of solutions for sub-diffusion equations

TL;DR

This paper investigates the regularity and decay behavior of solutions to time-fractional PDEs with tempered initial data, providing a representation formula that helps analyze solution singularities and their relation to initial data properties.

Contribution

It introduces a new representation formula for solutions of sub-diffusion equations, enabling detailed analysis of regularity, decay, and singularity propagation in terms of initial data.

Findings

Solutions exhibit controlled decay and smoothness properties.

Singularities are characterized by wavefront sets of initial data.

Representation formula facilitates analysis of solution regularity.

Abstract

We study regularity and decay properties for the solutions of the Cauchy problem for time-fractional partial differential equations, with tempered initial data, belonging to suitable (weighted) Sobolev spaces, associated with a differential operator on space variables with polynomially bounded coefficients. We obtain a representation formula for the solution, modulo time-regular functions, smooth and rapidly decreasing with respect to the space variables. By means of the representation formula, the (decay and smoothness) singularities of the solution of the homogeneous Cauchy problem can be controlled, in terms of (global) wavefront sets of the initial data.