# On the Lojasiewicz-Simon gradient inequality on submanifolds

## TL;DR

This paper establishes conditions under which the Lojasiewicz-Simon gradient inequality applies to submanifolds in Banach spaces, aiding the analysis of long-term behavior of constrained quasilinear parabolic equations.

## Contribution

It offers new sufficient conditions for the Lojasiewicz-Simon inequality on submanifolds, including discussions on the optimality of these assumptions.

## Key findings

- Provides a framework for applying the inequality to constrained PDEs
- Identifies optimality conditions for the assumptions
- Facilitates analysis of asymptotic behavior in quasilinear parabolic equations

## Abstract

We provide sufficient conditions for the Lojasiewicz-Simon gradient inequality to hold on a submanifold of a Banach space and discuss the optimality of our assumptions. Our result provides a tool to study asymptotic properties of quasilinear parabolic equations with (nonlinear) constraints.