Modica type gradient estimates for reaction-diffusion equations and a parabolic counterpart of a conjecture of De Giorgi
Abstract
We continue the study of Modica type gradient estimates for non-homogeneous parabolic equations initiated in 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 e0 below in × [0, T], where is an epigraph such that the mean curvature of ∂ 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 main3, Theorem main5 and Theorem T:ricci), this is is true for × (-∞, 0] and × (-∞, 0], where is an epigraph satisfying the geometric assumption mentioned above, and for M × (-∞,0], where M is a connected, compact Riemannian manifold with nonnegative Ricci tensor. As a consequence of the gradient estimate mo2, we establish a rigidity result (see Theorem main6 below) for solutions to e0 which is the analogue of Theorem 5.1 in CGS. Finally, motivated by Theorem main6, we close the paper by proposing a parabolic version of the famous conjecture of De Giorgi also known as the -version of the Bernstein theorem.