Precise bounds for finite time blowup of solutions to very general one-space-dimensional nonlinear Neumann problems
Authors:
Kurt Bryan and Michael S. Vogelius
Journal:
Quart. Appl. Math. 69 (2011), 57-78
MSC (2000):
Primary 35B05, 35B40, 45D05, 45G10, 45M05
DOI:
https://doi.org/10.1090/S0033-569X-2011-01203-2
Published electronically:
January 20, 2011
MathSciNet review:
2807977
Full-text PDF Free Access
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Additional Information
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 $F$, subject to very mild restrictions.
References
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- K. Bryan and M.S. Vogelius, Transient behavior of solutions to a class of nonlinear boundary value problems, to appear in the Quarterly of Applied Math.
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References
- C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, Journal of Computational and Applied Mathematics, 97, 1998, pp. 3-22. MR 1651764 (99g:35061)
- K. Bryan and M.S. Vogelius, Transient behavior of solutions to a class of nonlinear boundary value problems, to appear in the Quarterly of Applied Math.
- K. Deng, The blow-up behavior of the heat equation with Neumann boundary conditions, J. Math. Anal. Appl., 188, 1994, pp. 641-650. MR 1305473 (95i:35120)
- K. Deng and M. Xu, Remarks on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions, Proc. Amer. Math. Soc., 127, 1999, pp. 167-172. MR 1485467 (99b:35102)
- M. Fila and J. Filo, Blow-up on the boundary: A survey, pp. 67-78 in Singularities and Differential Equations, S. Janeczko et al. (eds), Banach Center Publ. 33, Polish Acad. Sciences, Warsaw, 1996. MR 1449147 (98c:35076)
- M. Fila and J. Guo, Complete blow-up and incomplete quenching for the heat equation with a nonlinear boundary condition, Nonlinear Analysis, 48, 2002, pp. 995-1002. MR 1880259
- S. Fu, J. Guo and J. Tsai Blow-up behavior for a semilinear heat equation with a nonlinear boundary condition, Tohoku Math. J. (2), 55, 2003, pp. 565-581. MR 2017226 (2004h:35112)
- S. Kichenassamy, Recent Progress on Boundary Blow-up, pp. 329-341 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)
- 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. 319-334. MR 0470481 (57:10235)
- Z. Lin and M. Wang, The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions, Z. Angew. Math. Phys., 50, 1999, pp. 361-374. MR 1697712 (2000c:35132)
- F. Quiros, J. D. Rossi and J. L. Vazquez, Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions, Comm. in PDE, 27, 2002, pp. 395-424. MR 1886965 (2002k:35041)
- C. Roberts, Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Applied Math., 205, 2007, pp. 736-743. MR 2329649 (2008c:45006)
- 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. 85-90. MR 0364868 (51:1122)
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Additional Information
Kurt Bryan
Affiliation:
Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803
Michael S. Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
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