Local well-posedness for the nonlinear heat and Schr\"{o}dinger equations with nonlinear Wentzell boundary conditions on time-dependent compact Riemannian manifolds

TL;DR

This paper establishes local well-posedness for nonlinear heat and Schrödinger equations with nonlinear Wentzell boundary conditions on evolving compact Riemannian manifolds, advancing understanding of such PDEs in geometric and dynamic settings.

Contribution

It introduces a novel analysis for semilinear PDEs with nonlinear boundary conditions on time-dependent manifolds, extending existing theories to dynamic geometric contexts.

Findings

Proves local existence and uniqueness of solutions

Handles subcritical nonlinearities in complex geometric settings

Extends PDE theory to time-dependent manifolds

Abstract

We prove a local well-posedness result for the semilinear heat and Schr\"{o}dinger equations with subcritical nonlinearities posed on a time-dependent compact Riemannian manifold and supplied with a nonlinear dynamical boundary condition of Wentzell type.