Jump processes and nonlinear fractional heat equations on fractals (2003)
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BibTeX
@MISC{Hu03jumpprocesses,
author = {Jiaxin Hu and Martina Zähle},
title = {Jump processes and nonlinear fractional heat equations on fractals},
year = {2003}
}
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Abstract
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 on a metric-measure space 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 ∂u ∂t (t, x) = −(−∆)σu(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 + σβ, where α is the Hausdorff dimension of α F whilst β is the walk dimension of F.
