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On selfsimilar solutions to the surface diffusion flow equations with contact angle boundary conditions
Title:  On selfsimilar 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:  17Sep2013 
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) selfsimilar solution when the angle is sufficiently small.We further prove the stability of this selfsimilar 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 selfsimilar solution we solves the initialboundary 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 semidivergence structure to construct a solution. 2010 Mathematics Subject Classification: Primary 35C06; Secondary 35G31, 35K59, 74N20. Keywords: Selfsimilar 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

Submitter: 数学紀要登録作業用
