# Time-fractional equations with reaction terms: fundamental solutions and asymptotics

## TL;DR

This paper investigates the fundamental solutions of time-fractional equations with reaction terms, establishing existence, uniqueness, and asymptotic invasion speeds, especially in one-dimensional cases with specific fractional exponents.

## Contribution

It provides new results on the existence, uniqueness, and asymptotic behavior of solutions to time-fractional reaction equations, focusing on invasion speed estimates in one dimension.

## Key findings

- Existence and uniqueness of fundamental solutions.
- Invasion speed is at least almost of square root type.
- Invasion speed exceeds any multiple of t^β for β in (0, 1/2).

## Abstract

We analyze the fundamental solution of a time-fractional problem, establishing existence and uniqueness in an appropriate functional space. We also focus on the one-dimensional spatial setting in the case in which the time-fractional exponent is equal to, or larger than, $\frac12$. In this situation, we prove that the speed of invasion of the fundamental solution is at least `almost of square root type', namely it is larger than~$ct^\beta$ for any given~$c>0$ and~$\beta\in\left(0,\frac12\right)$.