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Jump processes and nonlinear fractional heat equations on metric measure spaces
Author(s) -
Hu Jiaxin,
Zähle Martina
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310352
Subject(s) - mathematics , heat kernel , dirichlet form , measure (data warehouse) , sobolev space , minkowski–bouligand dimension , mathematical analysis , hausdorff measure , heat equation , hausdorff dimension , nonlinear system , space (punctuation) , dimension (graph theory) , pure mathematics , dirichlet distribution , fractal dimension , fractal , boundary value problem , physics , linguistics , philosophy , quantum mechanics , database , computer science
Jump processes on metric‐measure spaces are investigated by using heat kernels. It is shown that the heat kernel corresponding to a σ ‐stable type process decays at a polynomial rate rather than at an exponential rate as a Brownian motion. The domain of the Dirichlet form associated with the jump process is a Sobolev–Slobodeckij space, and the embedding theorems for this space are derived by using the heat kernel technique. As an application, we investigate nonlinear fractional heat equations of the form$$ { {\partial u} \over {\partial t} }(t, x) = - (- {\rm \Delta})^{\sigma} u(t, x) + u(t, x)^{p} $$with non‐negative initial values on a metric‐measure space F , and show the non‐existence of non‐negative global solution if 1 < p ≤ 1 + $ { {\sigma \beta} \over {\alpha} } $ , where α is the Hausdorff dimension of F whilst β is the walk dimension of F . (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)