# Convergence to equilibrium for a bulk--surface Allen--Cahn system coupled through a Robin boundary condition

## TL;DR

This paper proves that solutions to a coupled bulk-surface Allen--Cahn system with Robin boundary conditions converge to a single equilibrium over time, providing convergence rates and addressing new coupling challenges.

## Contribution

It establishes convergence to equilibrium for a bulk-surface Allen--Cahn system with Robin boundary conditions, extending previous results to more complex coupling scenarios.

## Key findings

- Global strong solutions converge to equilibrium
- An extended Lojasiewicz--Simon inequality is used
- Convergence rate estimates are provided

## Abstract

We consider a coupled bulk--surface Allen--Cahn system affixed with a Robin-type boundary condition between the bulk and surface variables. This system can also be viewed as a relaxation to a bulk--surface Allen--Cahn system with non-trivial transmission conditions. Assuming that the nonlinearities are real analytic, we prove the convergence of every global strong solution to a single equilibrium as time tends to infinity. Furthermore, we obtain an estimate on the rate of convergence. The proof relies on an extended Lojasiewicz--Simon type inequality for the bulk--surface coupled system. Compared with previous works, new difficulties arise as in our system the surface variable is no longer the trace of the bulk variable, but now they are coupled through a nonlinear Robin boundary condition.