Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations

2019/08/21 by Huang, Jizheng, Li, Pengtao, Liu, Yu +1
#31C15 #31E05 #35K05 #47D03 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1908.07895

Abstract

Let (\mathbb M, d,μ) be a metric measure space with upper and lower densities: \begincases |||μ|||_β:=sup_{(x,r)∈ \mathbb M×(0,∞)} μ(B(x,r))r^-β0, \endcases where β, β^⋆ are two positive constants which are less than or equal to the Hausdorff dimension of \mathbb M. Assume that p_t(⋅,⋅) is a heat kernel on \mathbb M satisfying Gaussian upper estimates and \mathcal L is the generator of the semigroup associated with p_t(⋅,⋅). In this paper, via a method independent of Fourier transform, we establish the decay estimates for the kernels of the fractional heat semigroup \e^{-t Lα}\_t>0 and the operators \L^θ/2 e^{-t Lα}\_t>0, respectively. By these estimates, we obtain the regularity for the Cauchy problem of the fractional dissipative equation associated with \mathcal L on (\mathbb M, d,μ). Moreover, based on the geometric-measure-theoretic analysis of a new L^p-type capacity defined in \mathbbM×(0,∞), we also characterize a nonnegative Randon measure ν on \mathbb M×(0,∞) such that R_αL^p(\mathbb M)⊆ L^q(\mathbb M×(0,∞),ν) under (α,p,q)∈ (0,1)×(1,∞)×(1,∞), where u=R_αf is the weak solution of the fractional diffusion equation (∂_t+ L^α)u(t,x)=0 in \mathbb M×(0,∞) subject to u(0,x)=f(x) in \mathbb M.

Related

• Regularity of fractional heat semigroup associated with Schrödinger operators
• Heat kernel estimates for time fractional equations
• A unified framework for pointwise convergence to the initial data of heat equations in metric measure spaces
• Fractional heat equations with subcritical absorption having a measure as initial data
• Compactness and Density Estimates for Weighted Fractional Heat Semigroups