Modica type gradient estimates for reaction-diffusion equations and a parabolic counterpart of a conjecture of De Giorgi

We continue the study of Modica type gradient estimates for non-homogeneous parabolic equations initiated in \cite{BG}. First, we show that for the parabolic minimal surface equation with a semilinear force term if a certain gradient estimate is satisfied at $t=0$, then it holds for all later times $t>0$. We then establish analogous results for reaction-diffusion equations such as \eqref{e0} below in $\Om \times [0, T]$, where $\Om$ is an epigraph such that the mean curvature of $\partial \Om$ is nonnegative. We then turn our attention to settings where such gradient estimates are valid without any a priori information on whether the estimate holds at some earlier time. Quite remarkably (see Theorem \ref{main3}, Theorem \ref{main5} and Theorem \ref{T:ricci}), this is is true for $\Rn \times (-\infty, 0]$ and $\Om \times (-\infty, 0]$, where $\Om$ is an epigraph satisfying the geometric assumption mentioned above, and for $M \times (-\infty,0]$, where $M$ is a connected, compact Riemannian manifold with nonnegative Ricci tensor. As a consequence of the gradient estimate \eqref{mo2}, we establish a rigidity result (see Theorem \ref{main6} below) for solutions to \eqref{e0} which is the analogue of Theorem 5.1 in \cite{CGS}. Finally, motivated by Theorem \ref{main6}, we close the paper by proposing a parabolic version of the famous conjecture of De Giorgi also known as the $\ve$-version of the Bernstein theorem.