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Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations

In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order $p$. The potentials include all the nonnegative ones. For the first two 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. 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 $0<C_3 \leq |u(x,t)| \leq C_4$. It is also shown that a solution may not be analytic in time if it is allowed to be $0$. As a lemma, we obtain an estimate of $\partial_t^k Γ(x,t;y)$ where $Γ(x,t;y)$ is the heat kernel on a manifold, with an explicit estimation of the coefficients. 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. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.