Decay estimates for evolution equations with classical and fractional time-derivatives

TL;DR

This paper establishes decay estimates for a broad class of evolution equations with classical and fractional time-derivatives, using energy methods to analyze polynomial and exponential decay behaviors in various nonlocal and nonlinear settings.

Contribution

It provides a unified framework for decay estimates in evolution equations involving standard and Caputo derivatives, including complex, fractional, and nonlocal operators.

Findings

Power-law and exponential decay estimates are proved for diverse evolution equations.

The decay behavior depends critically on the structure of the time-derivative.

Results encompass equations with both local and nonlocal, linear and nonlinear features.

Abstract

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators. Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviors, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation.