Modica Type Gradient Estimates for Reaction-Diffusion Equations

We continue the study of Modica type gradient estimates for inhomogeneous parabolic equations initiated in Banerjee and Garofalo (Nonlinear Anal. Theory Appl., to appear). 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 (5) below in Ω × [0, T], where Ω is an epigraph such that the mean curvature of ∂ Ω 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 Theorems 4.1, 4.2 and 5.1), this is true for ℝn×(−∞,0] and Ω×(−∞,0], where Ω 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 (7), we establish a rigidity result (see Theorem 6.1 below) for solutions to (5) which is the analogue of Theorem 5.1 in Caffarelli et al. (Commun. Pure Appl. Math. 47, 1457–1473, 1994). Finally, motivated by Theorem 6.1, we close the paper by proposing a parabolic version of the famous conjecture of De Giorgi also known as the ε-version of the Bernstein theorem.