标题： 非局部抛物方程的时间解析性 显示英文标题

标题： Time analyticity for nonlocal parabolic equations

摘要： 在本文中，我们研究分数热方程在 $\mathbb{R}^d$ 和完备黎曼流形 $\mathrm{M}$ 设置下的解的逐点时间解析性。 一方面，在$\mathbb{R}^d$中，我们证明了任何解$u=u(t,x)$对于$u_t(t,x)-\mathrm{L}_\alpha^{\kappa} u(t,x)=0$，其中$\mathrm{L}_\alpha^{\kappa}$是一个阶数为$\alpha$的非局部算子，在$(0,1]$中如果$u$满足增长条件$|u(t,x)|\leq C(1+|x|)^{\alpha-\epsilon}$对于任何$(t,x)\in (0,1]\times \mathbb{R}^d$和$\epsilon\in(0,\alpha)$。 我们还得到了关于$\partial_t^kp_\alpha(t,x;y)$的逐点估计，其中$p_\alpha(t,x;y)$是分数热核。 此外，在相同的增长条件下，我们证明了温和解是唯一的解。 另一方面，在流形$\mathrm{M}$上，我们在相同的增长条件以及分数热核的时间解析性下，证明了温和解的时间解析性，当$\mathrm{M}$满足 Poincaré 不等式和体积加倍条件时。 此外，我们还使用傅里叶变换和围道积分的方法研究了$\mathbb{R}^d$中分数热核的时间和空间导数。 我们发现当$\alpha\in (0,1]$时，分数热核在$t=0$处是时间解析的，当$x\neq 0$时，这与标准热核不同。 作为推论，我们得到了向后分数热方程的精确可解条件以及一些具有幂非线性次序$p$的非线性分数热方程的时间解析性。 这些结果与[8]和[11]中的结果相关，这些文献处理的是局部方程。 显示英文摘要

摘要： In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $\mathbb{R}^d$ and a complete Riemannian manifold $\mathrm{M}$. On one hand, in $\mathbb{R}^d$, we prove that any solution $u=u(t,x)$ to $u_t(t,x)-\mathrm{L}_\alpha^{\kappa} u(t,x)=0$, where $\mathrm{L}_\alpha^{\kappa}$ is a nonlocal operator of order $\alpha$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|\leq C(1+|x|)^{\alpha-\epsilon}$ for any $(t,x)\in (0,1]\times \mathbb{R}^d$ and $\epsilon\in(0,\alpha)$. We also obtain pointwise estimates for $\partial_t^kp_\alpha(t,x;y)$, where $p_\alpha(t,x;y)$ is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold $\mathrm{M}$, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel, when $\mathrm{M}$ satisfies the Poincar\'e inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in $\mathbb{R}^d$ using the method of Fourier transform and contour integrals. We find that when $\alpha\in (0,1]$, the fractional heat kernel is time analytic at $t=0$ when $x\neq 0$, which differs from the standard heat kernel. As corollaries, we obtain sharp solvability condition for the backward fractional heat equation and time analyticity of some nonlinear fractional heat equations with power nonlinearity of order $p$. These results are related to those in [8] and [11] which deal with local equations.