# Asymptotics of fundamental solutions for time fractional equations with convolution kernels

## TL;DR

This paper analyzes the long-time behavior of fundamental solutions to time fractional evolution equations with convolution kernels, revealing their asymptotic properties across all space-time regions.

## Contribution

It provides a detailed description of the point-wise asymptotic behavior of fundamental solutions for fractional equations with super-exponentially decaying kernels, extending existing knowledge.

## Key findings

- Asymptotic behavior characterized for all space-time regions.
- Fundamental solutions exhibit specific decay rates at large times.
- Results apply to symmetric, integrable kernels with super-exponential decay.

## Abstract

The paper deals with the large time asymptotic of the fundamental solution for a time fractional evolution equation for a convolution type operator. In this equation we use a Caputo time derivative of order $\alpha$ with $\alpha\in(0,1)$, and assume that the convolution kernel of the spatial operator is symmetric, integrable and shows a super-exponential decay at infinity. Under these assumptions we describe the point-wise asymptotic behavior of the fundamental solution in all space-time regions.