# American Institute of Mathematical Sciences

March  2020, 40(3): 1847-1878. doi: 10.3934/dcds.2020096

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

 1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China 2 School of Mathematical Sciences, Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Handan Road 220, Shanghai 200433, China

* Corresponding author: Hao Wu

Received  July 2019 Revised  October 2019 Published  December 2019

Fund Project: K. F. Lam is supported by a Direct Grant of CUHK (project 4053288), H. Wu is supported by NNSFC grant No. 11631011 and the Shanghai Center for Mathematical Sciences at Fudan University

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 Łojasiewicz–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.

Citation: Kei Fong Lam, Hao Wu. Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1847-1878. doi: 10.3934/dcds.2020096
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