Precise bounds for finite time blowup of solutions to very general onespacedimensional nonlinear Neumann problems
Authors:
Kurt Bryan and Michael S. Vogelius
Journal:
Quart. Appl. Math. 69 (2011), 5778
MSC (2000):
Primary 35B05, 35B40, 45D05, 45G10, 45M05
Published electronically:
January 20, 2011
MathSciNet review:
2807977
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Abstract: In this paper we analyze the asymptotic finite time blowup of solutions to the heat equation with a nonlinear Neumann boundary flux in one space dimension. We perform a detailed examination of the nature of the blowup, which can occur only at the boundary, and we provide tight upper and lower bounds for the blowup rate for ``arbitrary'' nonlinear functions , subject to very mild restrictions.
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 K. Deng, The blowup behavior of the heat equation with Neumann boundary conditions, J. Math. Anal. Appl., 188, 1994, pp. 641650. MR 1305473 (95i:35120)
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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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 S. Fu, J. Guo and J. Tsai Blowup behavior for a semilinear heat equation with a nonlinear boundary condition, Tohoku Math. J. (2), 55, 2003, pp. 565581. MR 2017226 (2004h:35112)
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 S. Kichenassamy, Recent Progress on Boundary Blowup, pp. 329341 in ``Elliptic and Parabolic Problems'', Volume 63 of the Book Series ``Progress in Nonlinear Differential Equations and Their Applications'', Birkhäuser, Basel, 2005. MR 2176725 (2006e:35022)
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 H. Levine and L. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations, 16, 1974, pp. 319334. MR 0470481 (57:10235)
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 Z. Lin and M. Wang, The blowup properties of solutions to semilinear heat equations with nonlinear boundary conditions, Z. Angew. Math. Phys., 50, 1999, pp. 361374. MR 1697712 (2000c:35132)
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 C. Roberts, Recent results on blowup and quenching for nonlinear Volterra equations, J. Comput. Applied Math., 205, 2007, pp. 736743. MR 2329649 (2008c:45006)
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 W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. Math. Anal., 6, 1975, pp. 8590. MR 0364868 (51:1122)
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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/S0033569X2011012032
PII:
S 0033569X(2011)012032
Keywords:
Blowup,
heat equation,
nonlinear Neumann boundary condition
Received by editor(s):
June 2, 2009
Published electronically:
January 20, 2011
Article copyright:
© Copyright 2011 Brown University
