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On self-similar solutions to the surface diffusion flow equations with contact angle boundary conditions
| Title: | On self-similar solutions to the surface diffusion flow equations with contact angle boundary conditions |
| Authors: | Asai, Tomoro Browse this author | | GIGA, YOSHIKAZU Browse this author |
| Issue Date: | 17-Sep-2013 |
| Publisher: | Department of Mathematics, Hokkaido University |
| Journal Title: | Hokkaido University Preprint Series in Mathematics |
| Volume: | 1039 |
| Start Page: | 1 |
| End Page: | 25 |
| Abstract: | We consider the surface diffusion flow equation when the curve is given as the graph of a function v(x; t) defined in a half line R+ = {x > 0} under the boundary conditions vx = tan > 0 and vxxx = 0 at x = 0. We construct a unique (spatially bounded) self-similar solution when the angle is sufficiently small.We further prove the stability of this self-similar solution. The problem stems from an equation proposed by W. W. Mullins (1957) to model formation of surface grooves on the grain boundaries, where the second boundary condition vxxx = 0 is replaced by zero slope condition on the curvature of the graph. For construction of a self-similar solution we solves the initial-boundary problem with homogeneous initial data. However, since the problem is quasilinear and initial data is not compatible with the boundary condition a simple application of an abstract theory for quasilinear parabolic equation is not enough for our purpose. We use a semi-divergence structure to construct a solution. 2010 Mathematics Subject Classification: Primary 35C06; Secondary 35G31, 35K59, 74N20. Keywords: Self-similar solution; Surface diffusion flow; Stability; Analytic semigroup; Mild solution. |
| Type: | bulletin (article) |
| URI: | http://hdl.handle.net/2115/69843 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics
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Submitter: 数学紀要登録作業用
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