Jukka Kemppainen, Juhana Siljander, Rico Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, Journal of Differential Equations, Volume 263, Issue 1, 2017, Pages 149-201, ISSN 0022-0396, https://doi.org/10.1016/j.jde.2017.02.030

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

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Author: Kemppainen, Jukka1; Siljander, Juhana2; Zacher, Rico3
Organizations: 1Applied and Computational Analysis, University of Oulu, P.O. Box 4500, 90014, Finland
2Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland
3Institute 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)
Language: English
Published: Elsevier, 2017
Publish Date: 2019-03-07
Description:

# Abstract

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.

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Series: Journal of differential equations
ISSN: 0022-0396
ISSN-E: 1090-2732
ISSN-L: 0022-0396
Volume: 263
Issue: 1
Pages: 149 - 201
DOI: 10.1016/j.jde.2017.02.030