Show simple item record

dc.contributor.authorKemppainen, Jukka
dc.contributor.authorSiljander, Juhana
dc.contributor.authorZacher, Rico
dc.date.accessioned2017-04-25T06:35:06Z
dc.date.available2019-03-07T22:35:16Z
dc.date.issued2017
dc.identifier.citationKemppainen, J., Siljander, J., & Zacher, R. (2017). Representation of solutions and large-time behavior for fully nonlocal diffusion equations. <em>Journal of Differential Equations</em>, 263 (1), 149-201. <a href="https://doi.org/10.1016/j.jde.2017.02.030">doi:10.1016/j.jde.2017.02.030</a>
dc.identifier.otherTUTKAID_73186
dc.identifier.urihttps://jyx.jyu.fi/handle/123456789/53671
dc.description.abstractWe 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.
dc.language.isoeng
dc.publisherElsevier
dc.relation.ispartofseriesJournal of Differential Equations
dc.subject.othernonlocal diffusion
dc.subject.otherRiemann-Liouville derivative
dc.subject.otherfractional Laplacian
dc.subject.otherdecay of solutions
dc.subject.otherenergy inequality
dc.subject.otherfundamental solution
dc.titleRepresentation of solutions and large-time behavior for fully nonlocal diffusion equations
dc.typearticle
dc.identifier.urnURN:NBN:fi:jyu-201704242047
dc.contributor.laitosMatematiikan ja tilastotieteen laitosfi
dc.contributor.laitosDepartment of Mathematics and Statisticsen
dc.contributor.oppiaineMatematiikka
dc.type.urihttp://purl.org/eprint/type/JournalArticle
dc.date.updated2017-04-24T06:15:04Z
dc.type.coarjournal article
dc.description.reviewstatuspeerReviewed
dc.format.pagerange149-201
dc.relation.issn0022-0396
dc.relation.volume263
dc.type.versionacceptedVersion
dc.rights.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.
dc.rights.accesslevelopenAccessfi
dc.relation.doi10.1016/j.jde.2017.02.030


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record