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arXiv 1908.07895math.AP
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有限密度度量测度空间上的分数阶热半群及其在分数阶耗散方程中的应用

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

Jizheng Huang, Pengtao Li, Yu Liu, Shaoguang Shi

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48 pages
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摘要：

设$(\mathbb M, d,μ)$是具有上密度和下密度的度量测度空间：$$ \begin{cases} |||μ|||_β:=\sup_{(x,r)\in \mathbb M\times(0,\infty)} μ(B(x,r))r^{-β}<\infty;\\ |||μ|||_{β^{\star}}:=\inf_{(x,r)\in \mathbb M\times(0,\infty)} μ(B(x,r))r^{-β^{\star}}>0, \end{cases} $$其中$β, β^{\star}$是两个小于或等于$\mathbb M$的Hausdorff维数的正常数.假设$p_t(\cdot,\cdot)$是$\mathbb M$上的热核，满足高斯上估计，$\mathcal L$是与$p_t(\cdot,\cdot)$相关联的半群的生成元。本文通过一种不依赖于Fourier变换的方法，分别建立了分数热半群$\{e^{-t \mathcal{L}^α}\}_{t>0}$和算子$\{\mathcal{L}^{θ/2} e^{-t \mathcal{L}^α}\}_{t>0}$的核的衰减估计.利用这些估计，我们得到了与$\mathcal L$相关的分数阶耗散方程在$(\mathbb M, d,μ)$上的Cauchy问题的正则性.此外，基于对$\mathbb{M}\times(0,\infty)$中定义的一个新的$L^p$型容量的几何测度理论分析，我们还刻画了$\mathbb M\times(0,\infty)$上的一个非负Randon测度$ν$，使得$R_αL^p(\mathbb M)\subseteq L^q(\mathbb M\times(0,\infty),ν)$在$(α,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$下，其中$u=R_αf$是分数阶扩散方程$(\partial_t+ \mathcal{L}^α)u(t,x)=0$在$\mathbb M\times(0,\infty)$中服从$u(0,x)=f(x)$在$\mathbb M$中的弱解.

英文摘要：

Let $(\mathbb M, d,μ)$ be a metric measure space with upper and lower densities: $$ \begin{cases} |||μ|||_β:=\sup_{(x,r)\in \mathbb M\times(0,\infty)} μ(B(x,r))r^{-β}<\infty;\\ |||μ|||_{β^{\star}}:=\inf_{(x,r)\in \mathbb M\times(0,\infty)} μ(B(x,r))r^{-β^{\star}}>0, \end{cases} $$ where $β, β^{\star}$ 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 $\{e^{-t \mathcal{L}^α}\}_{t>0}$ and the operators $\{\mathcal{L}^{θ/2} e^{-t \mathcal{L}^α}\}_{t>0}$, 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 $\mathbb{M}\times(0,\infty)$, we also characterize a nonnegative Randon measure $ν$ on $\mathbb M\times(0,\infty)$ such that $R_αL^p(\mathbb M)\subseteq L^q(\mathbb M\times(0,\infty),ν)$ under $(α,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$, where $u=R_αf$ is the weak solution of the fractional diffusion equation $(\partial_t+ \mathcal{L}^α)u(t,x)=0$ in $\mathbb M\times(0,\infty)$ subject to $u(0,x)=f(x)$ in $\mathbb M$.

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