## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 54, Number 4 (2017), 689-706.

### Longtime convergence for epitaxial growth model under Dirichlet conditions

Somayyeh Azizi, Gianluca Mola, and Atsushi Yagi

#### Abstract

This paper continues our study on the initial-boundary value problem for a semilinear parabolic equation of fourth order which has been presented by Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr [12] to describe the large-scale features of a growing crystal surface under molecular beam epitaxy. In the preceding paper [1], we already constructed a dynamical system generated by the problem and verified that the dynamical system has a finite-dimensional attractor (especially, every trajectory has nonempty $\omega$-limit set) and admits a Lyapunov function (of the form (3.1)). This paper is then devoted to showing longtime convergence of trajectory. We shall prove that every trajectory converges to some stationary solution as $t \to \infty$. As a matter of fact, we have obtained in [10] the similar result for the equation but under the Neumann like boundary conditions $\frac{\partial u}{\partial n}=\frac\partial{\partial n}\varDelta u=0$ on the unknown function $u$. In this paper, we want as in [1] to handle the Dirichlet boundary conditions $u=\frac{\partial u}{\partial n}=0$, maybe physically more natural conditions than before.

#### Article information

**Source**

Osaka J. Math. Volume 54, Number 4 (2017), 689-706.

**Dates**

First available in Project Euclid: 20 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1508486574

**Subjects**

Primary: 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations 74E15: Crystalline structure

#### Citation

Azizi, Somayyeh; Mola, Gianluca; Yagi, Atsushi. Longtime convergence for epitaxial growth model under Dirichlet conditions. Osaka J. Math. 54 (2017), no. 4, 689--706.https://projecteuclid.org/euclid.ojm/1508486574