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), 261-290

MSC (2000):
Primary 35B05, 35B40

DOI:
https://doi.org/10.1090/S0033-569X-2011-01204-5

Published electronically:
March 3, 2011

MathSciNet review:
2814527

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Abstract | References | Similar Articles | Additional Information

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 sign-changing initial data is less well understood. We examine in detail conditions under which solutions with sign-changing 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.

**1.**J.M. Arrieta and A. Rodriguez-Bernal,*Localization on the boundary of blow-up for reaction-diffusion equations with Nonlinear Boundary Conditions*, Comm. in Partial Diff. Equations,**29**, 2004, pp. 1127-1148. MR**2097578 (2005f:35159)****2.**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)****3.**K. Bryan and M. Vogelius,*A uniqueness result concerning the identification of a collection of cracks from finitely many electrostatic boundary measurements*, SIAM J. Math. Anal.,**23**, 1992, pp. 950-958. MR**1166567 (93f:35238)****4.**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. 675-694. MR**1939006 (2003i:35108)****5.**K. Bryan and M.S. Vogelius,*Precise bounds for finite time blowup of solutions to very general one-space-dimensional nonlinear Neumann problems*, to appear in the Quarterly of Applied Math.**6.**Y.P. Chen and C.H. Xie,*Global existence and nonexistence for a strongly coupled parabolic system with nonlinear boundary conditions*, Acta Mathematica Sinica,**22**, 2006, pp. 1297-1304. MR**2251391 (2007c:35063)****7.**M. Chlebik and M. Fila,*On the blow-up rate for the heat equation with a nonlinear boundary condition*, Mathematical Methods in the Applied Sciences,**23**, 2000, pp. 1323-1330. MR**1784465 (2001k:35176)****8.**R. Courant and D. Hilbert,*Methods of Mathematical Physics, Vol. I*, Wiley, New York, 1953. MR**0065391 (16:426a)****9.**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)****10.**M. Fila, J. Filo and G.M. Lieberman,*Blow-up on the boundary for the heat equation*, Calculus of Variations and PDE,**10**, 2000, pp. 85-99. MR**1803975 (2001k:35177)****11.**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****12.**M. Fila and P. Quittner,*The blow-up rate for the heat equation with a nonlinear boundary condition*, Mathematical Methods in the Applied Sciences,**14**, 1991, pp. 197-205. MR**1099325 (92a:35023)****13.**Y. Giga and R. Kohn,*Asymptotically self-similar blow-up of semilinear heat equations*, Comm. Pure Appl. Math,**38**, 1985, pp. 297-319. MR**784476 (86k:35065)****14.**Y. Giga and R. Kohn,*Characterizing blowup using similarity variables*, Indiana Univ. Math. J.,**36**, 1987, pp. 425-447. MR**876989 (88c:35021)****15.**Y. Giga and R. Kohn,*Nondegeneracy of blowup for semilinear heat equations*, Comm. Pure Appl. Math**42**, 1989, pp. 845-884. MR**1003437 (90k:35034)****16.**J.S. Guo and B. Hu,*Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary*, Journal of Mathematical Analysis and Applications,**269**, 2002, pp. 28-49. MR**1907872 (2003c:35098)****17.**B. Hu and Z. Yin,*The profile near blowup time for solution of the heat equations with a nonlinear boundary condition*, Transactions Amer. Math. Soc.,**346**, 1994, pp. 117-135. MR**1270664 (95c:35040)****18.**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)****19.**H.A. Levine:*Some nonexistence and instability theorems for solutions of formally parabolic equations of the form*. Arch. Rational Mech. Anal.,**51**, 1973, pp. 371-386. MR**0348216 (50:714)****20.**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)****21.**Z. Lin and C. Xie,*The blow-up rate for a system of heat equations with nonlinear boundary conditions*, Nonlinear Analysis,**34**, 1998, pp. 767-778. MR**1634742 (2000a:35096)****22.**K. Medville,*Existence and blow up behavior of planar harmonic functions satisfying certain nonlinear Neumann boundary conditions*, Ph.D. dissertation, Department of Mathematics, Rutgers University (NJ), May 2005.**23.**M. Protter and H. Weinberger,*Maximum Principles in Differential Equations*, Springer, New York, 1999. MR**762825 (86f:35034)****24.**F. Quiros and J.D. Rossi,*Blow-up sets for linear diffusion equations in one dimension*, Z. Angew. Math. Phys.,**55**, 2004, pp. 357-362. MR**2047293 (2004m:35153)****25.**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)****26.**P. Quittner and A. Rodriguez-Bernal,*Complete and energy blow-up in parabolic problem with nonlinear boundary conditions*, Nonlinear Analysis,**62**, 2005, pp. 863-875. MR**2153217 (2006a:35156)****27.**P. Quittner and P. Souplet,*Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States*, Birkhaüser, Basel, 2007. MR**2346798 (2008f:35001)****28.**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)****29.**A. Rodriguez-Bernal and A. Tajdine,*Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up*, J. Diff. Eqns.,**169**, 2001, pp. 332-372. MR**1808470 (2002c:35162)****30.**J. Rossi,*The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition*, Acta Math. Univ. Comenianae,**2**, 1998, pp. 343-350. MR**1739446 (2000k:35133)****31.**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)****32.**M. Wang and Y. Wu,*Global existence and blow-up problems for quasilinear parabolic equations with nonlinear boundary conditions*, SIAM J. Math. Anal.,**24**, 1993, pp. 1515-1521. MR**1241155 (95c:35132)****33.**X.H. Yang, F.C. Li and C.H. Xie,*Global existence and blow up of solutions for parabolic system involving cross-diffusions and nonlinear boundary conditions*, Acta Mathematica Sinica,**21**, 2005, pp. 923-928. MR**2156972 (2006i:35141)****34.**L. Zhao and S. Zheng,*Blow-up estimates for system of heat equations coupled via nonlinear boundary flux*, Nonlinear Analysis,**54**, 2003, pp. 251-259. MR**1979732 (2004b:35151)****35.**S. Zheng, F. Li, and B. Liu,*Asymptotic behavior for a reaction-diffusion equation with inner absorbtion and boundary flux*, Applied Math. Letters,**19**, 2006, pp. 942-948. MR**2240489 (2007b:35202)**

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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

DOI:
https://doi.org/10.1090/S0033-569X-2011-01204-5

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