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

<p style='text-indent:20px;'>In this paper, we investigate the pointwise time analyticity of three differential equations. They are biharmonic heat equation, equation with potentials and some nonlinear equations power nonlinearity order <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>. The include all nonnegative ones. For first two equations, prove if id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> satisfies growth conditions in id="M3">\begin{document}$ (x,t)\in \mathrm{M}\times [0,1] $\end{document}</tex-math></inline-formula>, then id="M4">\begin{document}$ is analytic id="M5">\begin{document}$ (0,1] Here id="M6">\begin{document}$ \mathrm{M} id="M7">\begin{document}$ R^d or a complete noncompact manifold Ricci curvature bounded from below by constant. Then obtain necessary sufficient condition such that id="M8">\begin{document}$ u(x,t) at id="M9">\begin{document}$ t = 0 Applying method, also for solvability backward which ill-posed general.</p><p style='text-indent:20px;'>For id="M10">\begin{document}$ solution id="M11">\begin{document}$ t\in it id="M12">\begin{document}$ \mathrm{M}\times[0,1] id="M13">\begin{document}$ positive integer. In addition, case when id="M14">\begin{document}$ rational number stronger assumption id="M15">\begin{document}$ 0<C_3 \leq |u(x,t)| C_4 It shown may not be allowed to id="M16">\begin{document}$ As lemma, an estimate id="M17">\begin{document}$ \partial_t^k \Gamma(x,t;y) where id="M18">\begin{document}$ kernel on manifold, explicit estimation coefficients.</p><p style='text-indent:20px;'>An interesting point even smooth space variable id="M19">\begin{document}$ x implying can independent. Besides, general manifolds, hold since requires certain bounds its derivatives.</p>