Representation of solutions and large-time behavior for fully nonlocal diffusion equations
| Autor: | Juhana Siljander, Jukka Kemppainen, Rico Zacher |
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| Rok vydání: | 2015 |
| Předmět: |
Riemann-Liouville derivative
Riemann–Liouville derivative nonlocal diffusion 01 natural sciences decay of solutions symbols.namesake Mathematics - Analysis of PDEs Fundamental solution FOS: Mathematics Initial value problem Applied mathematics 0101 mathematics Mathematics fundamental solution Spacetime Applied Mathematics 010102 general mathematics ta111 energy inequality Random walk 010101 applied mathematics Primary 35R11 Secondary 45K05 35C15 47G20 Fourier analysis Norm (mathematics) Bounded function symbols Heat equation fractional Laplacian Analysis Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1505.02803 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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