Transient behavior of solutions to a class of nonlinear boundary value problems
Authors:
Kurt Bryan and Michael S. Vogelius
Journal:
Quart. Appl. Math. 69 (2011), 261290
MSC (2000):
Primary 35B05, 35B40
Published electronically:
March 3, 2011
MathSciNet review:
2814527
Fulltext PDF
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Abstract: In this paper we consider the asymptotic behavior in time of solutions to the heat equation with nonlinear Neumann boundary conditions of the form , where is a function that grows superlinearly. Solutions frequently exist for only a finite time before ``blowing up.'' In particular, it is well known that solutions with initial data of one sign must blow up in finite time, but the situation for signchanging initial data is less well understood. We examine in detail conditions under which solutions with signchanging initial data (and certain symmetries) must blow up, and also conditions under which solutions actually decay to zero. We carry out this analysis in one space dimension for a rather general , while in two space dimensions we confine our analysis to the unit disk and of a special form.
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 J.M. Arrieta and A. RodriguezBernal, Localization on the boundary of blowup for reactiondiffusion equations with Nonlinear Boundary Conditions, Comm. in Partial Diff. Equations, 29, 2004, pp. 11271148. MR 2097578 (2005f:35159)
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 K. Bryan and M.S. Vogelius, Singular solutions to a nonlinear elliptic boundary value problem originating from corrosion modeling , Quarterly of Applied Math, 60, 2002, pp. 675694. MR 1939006 (2003i:35108)
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 K. Bryan and M.S. Vogelius, Precise bounds for finite time blowup of solutions to very general onespacedimensional nonlinear Neumann problems, to appear in the Quarterly of Applied Math.
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 M. Fila and J. Guo, Complete blowup and incomplete quenching for the heat equation with a nonlinear boundary condition, Nonlinear Analysis, 48, 2002, pp. 9951002. MR 1880259
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 M. Fila and P. Quittner, The blowup rate for the heat equation with a nonlinear boundary condition, Mathematical Methods in the Applied Sciences, 14, 1991, pp. 197205. MR 1099325 (92a:35023)
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Additional Information
Kurt Bryan
Affiliation:
Department of Mathematics, RoseHulman Institute of Technology, Terre Haute, Indiana 47803
Michael S. Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
DOI:
http://dx.doi.org/10.1090/S0033569X2011012045
PII:
S 0033569X(2011)012045
Keywords:
Blowup,
heat equation,
nonlinear Neumann boundary condition
Received by editor(s):
June 2, 2009
Published electronically:
March 3, 2011
Article copyright:
© Copyright 2011 Brown University
