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Register a new userFractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations
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Let $(mathbb M, d,mu)$ be a metric measure space with upper and lower densities: $$ begin{cases} |||mu|||_{beta}:=sup_{(x,r)in mathbb Mtimes(0,infty)} mu(B(x,r))r^{-beta}<infty; |||mu|||_{beta^{star}}:=inf_{(x,r)in mathbb Mtimes(0,infty)} mu(B(x,r))r^{-beta^{star}}>0, end{cases} $$ where $beta, beta^{star}$ are two positive constants which are less than or equal to the Hausdorff dimension of $mathbb M$. Assume that $p_t(cdot,cdot)$ is a heat kernel on $mathbb M$ satisfying Gaussian upper estimates and $mathcal L$ is the generator of the semigroup associated with $p_t(cdot,cdot)$. In this paper, via a method independent of Fourier transform, we establish the decay estimates for the kernels of the fractional heat semigroup ${e^{-t mathcal{L}^{alpha}}}_{t>0}$ and the operators ${{mathcal{L}}^{theta/2} e^{-t mathcal{L}^{alpha}}}_{t>0}$, respectively. By these estimates, we obtain the regularity for the Cauchy problem of the fractional dissipative equation associated with $mathcal L$ on $(mathbb M, d,mu)$. Moreover, based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in $mathbb{M}times(0,infty)$, we also characterize a nonnegative Randon measure $ u$ on $mathbb Mtimes(0,infty)$ such that $R_alpha L^p(mathbb M)subseteq L^q(mathbb Mtimes(0,infty), u)$ under $(alpha,p,q)in (0,1)times(1,infty)times(1,infty)$, where $u=R_alpha f$ is the weak solution of the fractional diffusion equation $(partial_t+ mathcal{L}^alpha)u(t,x)=0$ in $mathbb Mtimes(0,infty)$ subject to $u(0,x)=f(x)$ in $mathbb M$.
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