UC Riverside

This thesis has been on the pointwise time analyticity of several evolutionary partial differential equations, including the heat equation, the biharmonic heat equation, the heat equation with potentials, some nonlinear heat equations and nonlocal parabolic equations.

For the first there equations, we prove if $u$ satisfies some growth conditions in $(x,t)\in \mathrm{M}\times [0,1]$, then $u$ is analytic in time $(0,1]$. Here $\mathrm{M}$ is $R^d$or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that $u(x,t)$ is analytic in time at $t=0$. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general. An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable $x$, implying that the analyticity of space and time can be independent. Actually, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.

For the nonlinear heat equation with power nonlinearity of order $p$, we prove that a solution is analytic in time $t\in (0,1]$ if it is bounded in $\mathrm{M}\times[0,1]$ and $p$ is a positive integer. In addition, we investigate the case when $p$ is a rational number with a stronger assumption $0

We also investigate pointwise time analyticity of solutions to nonlocal parabolic equations in the settings of $\mathrm{R}^d$ and a complete Riemannian manifold $\mathrm{M}$. On one hand, in $\mathrm{R}^d$, 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 $(t,x)\in (0,1]\times \mathrm{R}^d$ and $\epsilon\in(0,\alpha)$. We also obtain pointwise estimates for $\p_t^kp_\alpha(t,x;y)$, 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 $\mathrm{R}^d$ 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 nonlocal parabolic equations and time analyticity of some nonlinear nonlocal parabolic equations with power nonlinearity of order $p$. These results are related to those in \cite{DZ20} and \cite{Zeng} which deal with local equations.