Jump processes and nonlinear fractional heat equations on metric measure spaces

• Published in: Mathematische Nachrichten 2006-01, Vol.279 (1-2), p.150-163
• Main Authors: Hu, Jiaxin, Zähle, Martina
• Format: Article
• Language: English
• Subjects: Dirichlet form; fractional heat equation; Hausdorff dimension; heat kernel; Jump process; Sobolev-Slobodeckij space; walk dimension
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.200310352

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)