# On a non-local problem for a multi-term fractional diffusion-wave equation

## TL;DR

This paper investigates a generalized multi-term time-fractional diffusion-wave equation with nonlocal conditions, providing existence, uniqueness, and explicit solutions for operators with discrete spectrum and hypoelliptic operators on Lie groups.

## Contribution

It extends the analysis of fractional diffusion-wave equations to multi-term and nonlocal settings, including hypoelliptic operators on Lie groups, with explicit solution formulas.

## Key findings

- Existence and uniqueness of solutions established.
- Explicit representation formulas derived.
- Applicable to operators with discrete spectrum and hypoelliptic operators on Lie groups.

## Abstract

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions.