Representation of solutions and largetime behavior for fully nonlocal diffusion equations 

Author:  Kemppainen, Jukka^{1}; Siljander, Juhana^{2}; Zacher, Rico^{3} 
Organizations: 
^{1}Applied and Computational Analysis, University of Oulu, P.O. Box 4500, 90014, Finland ^{2}Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland ^{3}Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany 
Format:  article 
Version:  accepted version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.5 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife201903057141 
Language:  English 
Published: 
Elsevier,
2017

Publish Date:  20190307 
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 L^{2}decay of mild solutions in all dimensions, (iv) L^{2}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 L^{2}norm is actually a subsolution to a purely timefractional problem which allows us to use the known theory to obtain the result. see all

Series: 
Journal of differential equations 
ISSN:  00220396 
ISSNE:  10902732 
ISSNL:  00220396 
Volume:  263 
Issue:  1 
Pages:  149  201 
DOI:  10.1016/j.jde.2017.02.030 
OADOI:  https://oadoi.org/10.1016/j.jde.2017.02.030 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
Copyright information: 
© 2017. This manuscript version is made available under the CCBYNCND 4.0 license http://creativecommons.org/licenses/byncnd/4.0/ 
https://creativecommons.org/licenses/byncnd/4.0/ 