Global gradient estimates for a general type of nonlinear parabolic equations

Abstract

We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations, possibly in a Riemannian geometry setting.Our result is new in comparison with the existing ones in the literature, in light of the validity of the estimates in the global domain, and it detects several additional regularity effects due to special parabolic data. Moreover, our result comprises a large number of nonlinear sources treated by a unified approach, and it recovers many classical results as special cases.

Global gradient estimates for nonlinear parabolic operators

“…which coincides with the quantity in (1.4) of [8]; also, in (1.20), (1.22) and (1.23) one finds µ 2 = γ 2 = γ 3 = 0, therefore Theorem 1.5 here recovers, in the special setting of (1.29), the result given in Theorem 1.1 of [8].…”

“…To obtain our result, we will perform a number of rather involved and ad-hoc computations and exploit also the cut-off function method that was introduced in [8] to address global estimates. We also remark that, as far as we know, our results are new also in the case of nonlinear parabolic equations in the Euclidean space when k = 0.…”