Asymptotic Properties of Solutions of the Fractional Diffusion-Wave Equation

TL;DR

This paper investigates the long-term behavior of solutions to the fractional diffusion-wave equation with a Caputo derivative, establishing analogs of classical principles like limiting amplitude and stabilization.

Contribution

It introduces new asymptotic properties for solutions of the fractional diffusion-wave equation, extending classical principles to fractional hyperbolic-parabolic equations.

Findings

Proves an analog of the limiting amplitude principle for the fractional diffusion-wave equation.

Establishes a pointwise stabilization property of solutions.

Extends classical wave and heat equation principles to fractional derivatives.

Abstract

For the fractional diffusion-wave equation with the Caputo-Dzhrbashyan fractional derivative of order $\alpha \in (1,2)$ with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).