University of Jyväskylä | JYX Digital Repository

  • English  | Give feedback |
    • suomi
    • English
 
  • Login
JavaScript is disabled for your browser. Some features of this site may not work without it.
View Item 
  • JYX
  • Artikkelit
  • Matemaattis-luonnontieteellinen tiedekunta
  • View Item
JYX > Artikkelit > Matemaattis-luonnontieteellinen tiedekunta > View Item

Representation of solutions and large-time behavior for fully nonlocal diffusion equations

ThumbnailFinal Draft
View/Open
466.3Kb

Downloads:  
Show download detailsHide download details  
Kemppainen, J., Siljander, J., & Zacher, R. (2017). Representation of solutions and large-time behavior for fully nonlocal diffusion equations. Journal of Differential Equations, 263 (1), 149-201. doi:10.1016/j.jde.2017.02.030
Published in
Journal of Differential Equations
Authors
Kemppainen, Jukka |
Siljander, Juhana |
Zacher, Rico
Date
2017
Discipline
Matematiikka
Copyright
© 2017 Elsevier Inc. This is a final draft version of an article whose final and definitive form has been published by Elsevier. Published in this repository with the kind permission of the publisher.

 
We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal L2 -decay of mild solutions in all dimensions, (iv) L2 -decay of weak solutions via energy methods. The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of (ii). Next we use Fourier analysis techniques to obtain the optimal decay rate for mild solutions. Here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions. Consequently, the decay rate does not anymore improve when the dimension increases. The theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis. Finally, we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation. Our main idea is to show that the L2–norm is actually a subsolution to a purely time-fractional problem which allows us to use the known theory to obtain the result. ...
Publisher
Elsevier
ISSN Search the Publication Forum
0022-0396
Keywords
nonlocal diffusion Riemann-Liouville derivative fractional Laplacian decay of solutions energy inequality fundamental solution
DOI
10.1016/j.jde.2017.02.030
URI

http://urn.fi/URN:NBN:fi:jyu-201704242047

Metadata
Show full item record
Collections
  • Matemaattis-luonnontieteellinen tiedekunta [3697]
  • Browse materials
  • Browse materials
  • Articles
  • Conferences and seminars
  • Electronic books
  • Historical maps
  • Journals
  • Tunes and musical notes
  • Photographs
  • Presentations and posters
  • Publication series
  • Research reports
  • Research data
  • Study materials
  • Theses

Browse

All of JYXCollection listBy Issue DateAuthorsSubjectsPublished inDepartmentDiscipline

My Account

Login

Statistics

View Usage Statistics
  • How to publish in JYX?
  • Self-archiving
  • Publish Your Thesis Online
  • Publishing Your Dissertation
  • Publication services

Open Science at the JYU
 
Data Protection Description

Accessibility Statement
Open Science Centre