Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition

Abstract

<p style='text-indent:20px;'>In this paper, we establish global <inline-formula><tex-math id="M1">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> a priori estimates for solutions to the uniformly parabolic equations with Neumann boundary condition on the smooth bounded domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula> by a blow-up argument. As a corollary, we obtain that the solutions converge to ones which move by translation. This generalizes the viscosity results derived before by Da Lio.</p>