Time-fractional equations with reaction terms: Fundamental solutions and asymptotics

Serena Dipierro; Benedetta Pellacci; Enrico Valdinoci; Gianmaria Verzini

doi:10.3934/dcds.2020137

Article Contents

2021, Volume 41, Issue 1: 257-275. Doi: 10.3934/dcds.2020137

This issue Previous Article Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps Next Article Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions

Time-fractional equations with reaction terms: Fundamental solutions and asymptotics

1.
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia
2.
Dipartimento di Matematica e Fisica, Università della Campania "Luigi Vanvitelli", Viale Lincoln 5, 81100 Caserta, Italy
3.
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia
4.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

^* Corresponding author: Enrico Valdinoci
^* Corresponding author: Enrico Valdinoci

Received: September 2019

Early access: February 2020

Published: January 2021

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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) $.

Keywords:

Mathematics Subject Classification: 35R11, 35C15, 35B40, 35K57, 35K08, 26A33.

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