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

Abstract

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