Estimates for a class of Hessian type fully nonlinear parabolic equations on Riemannian manifolds

TL;DR

This paper establishes nearly optimal a priori estimates for solutions to Hessian type fully nonlinear parabolic equations on Riemannian manifolds, leading to existence results for smooth solutions over infinite time without boundary restrictions.

Contribution

It provides new a priori estimates for a broad class of nonlinear parabolic equations on manifolds, with minimal geometric boundary conditions, and proves existence of solutions over infinite time.

Findings

Derived nearly optimal gradient and second derivative estimates.

Proved existence of smooth solutions for infinite time.

No geometric boundary restrictions required.

Abstract

In this paper, we derive a priori estimates for the gradient and second order derivatives of solutions to a class of Hessian type fully nonlinear parabolic equations with the first initial-boundary value problem on Riemannian manifolds. These a priori estimates are derived under conditions which are nearly optimal. Especially, there are no geometric restrictions on the boundary of the Riemannian manifolds. And as an application, the existence of smooth solutions to the first initial-boundary value problem even for infinity time is obtained.