Jump processes and nonlinear fractional heat equations on metric measure spaces

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)