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Longtime convergence for epitaxial growth model under Dirichlet conditions. (English) Zbl 1516.35068

Summary: This paper continues our study on the initial-boundary value problem for a semilinear parabolic equation of fourth order which has been presented by M. D. Johnson et al. [Phys. Rev. Lett. 72, No. 1, 116–119 (1994; doi:10.1103/PhysRevLett.72.116)] to describe the large-scale features of a growing crystal surface under molecular beam epitaxy. In the preceding paper [Sci. Math. Jpn. 80, No. 2, 109–122 (2017; Zbl 1382.35129)], 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 [M. Grasselli et al., Osaka J. Math. 48, No. 4, 987–1004 (2011; Zbl 1233.35036)] 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 [Zbl 1382.35129] to handle the Dirichlet boundary conditions \(u=\frac{\partial u}{\partial n}=0\), maybe physically more natural conditions than before.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations
35K58 Semilinear parabolic equations
74E15 Crystalline structure
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Full Text: Euclid