Representation formula, regularity, and decay of solutions for sub-diffusion equations
Sandro Coriasco, Giovanni Girardi, Stevan Pilipović
TL;DR
The paper analyzes regularity and decay for time-fractional Cauchy problems $\partial_t^r u + \mathrm{Op}(a)u = f$ with Caputo derivative $0<r<1$, in spaces with polynomially bounded coefficients. It develops a representation formula via a parameter-dependent parametrix and Mittag-Leffler kernels, first for constant symbols $a(\xi)$ and then for variable-symbol operators in SG- and Hörmander-calculus, enabling precise propagation and smoothing results. The main contributions are explicit kernel expansions $K_0(t)$ and $K_1(t)$ expressed through $E_{r,1}$ and $E_{r,r}$, SG-type symbol estimates, and a framework to control singularities of homogeneous solutions through global wavefront sets. These results provide sharp regularity and decay information for sub-diffusive evolution in both homogeneous and inhomogeneous media, with implications for memory effects and diffusion processes in heterogeneous environments.
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.