Representation of solutions and large-time behavior for fully nonlocal diffusion equations
By Jukka Kemppainen, Juhana Siljander, Rico Zacher. 2015
110 citations recorded by the source corpus.
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 $L^2$-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 time-fractional problem which allows us to use the known theory to obtain the result.
Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation
View Paper: https://www.semanticscholar.org/paper/38ef1764735b11c7af31328a1f7501d4a7fe049f
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In Applied Mathematics
In Mathematical Physics
In Modeling and Simulation
Citations in the corpus
- Fundamental solutions and decay of fully non-local problems (2018, background): “2) takes the form of the equation studied in [20] and [21].”