Representation of solutions and large-time behavior for fully nonlocal diffusion equations
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. https://doi.org/10.1016/j.jde.2017.02.030
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Journal of Differential EquationsDate
2017Copyright
© 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.
...


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