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LOWER BOUNDS FOR BLOW-UP TIME IN SOME NON-LINEAR PARABOLIC PROBLEMS UNDER NEUMANN BOUNDARY CONDITIONS

Published online by Cambridge University Press:  10 March 2011

CRISTIAN ENACHE
CRISTIAN ENACHE*
Affiliation:
Department of Mathematics and Informatics, Ovidius University, Constanta, 900597, Romania e-mail: cenache@univ-ovidius.ro
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Abstract

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This paper deals with some non-linear initial-boundary value problems under homogeneous Neumann boundary conditions, in which the solutions may blow up in finite time. Using a first-order differential inequality technique, lower bounds for blow-up time are determined.

Keywords

35K2035B4535B30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 569 - 575
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Ball, J. M., Remarks on blow-up and non-existence theorems for nonlinear evolution equations, Q. J. Math. 28 (1977), 473–486.CrossRefGoogle Scholar
2.Galaktionov, V. A. and Vazquez, J. L., The problem of blow-up in nonlinear parabolic equations, Discrete Cont. Dyn. Syst. 8 (2) (2002), 399–433.CrossRefGoogle Scholar
3.Kielhöfer, H., Existenz und Regularität von Lösungen semilinearer parabolischer Anfangs-Randwertprobleme, Math. Z. 142 (1975), 131–160.CrossRefGoogle Scholar
4.Levine, H. A., Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics. The method of unbounded Fourier coefficients, Math. Ann. 214 (1975), 205–220.CrossRefGoogle Scholar
5.Payne, L. E., Philippin, G. A. and Schaefer, P. W., Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal. 69 (2008), 3495–3502.CrossRefGoogle Scholar
6.Payne, L. E. and Schaefer, P. W., Lower bounds for blow-up time in parabolic problems under Neumann conditions, Appl. Anal. 85 (2006), 1301–1311.CrossRefGoogle Scholar
7.Payne, L. E. and Schaefer, P. W., Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl. 328 (2007), 1196–1205.CrossRefGoogle Scholar
8.Payne, L. E. and Schaefer, P. W., Bounds for the blow-up time for the heat equation under nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A. 139 (6) (2009), 1289–1296.CrossRefGoogle Scholar
9.Payne, L. E. and Song, J. C., Lower bounds for blow-up time in a nonlinear parabolic problem, J. Math. Anal. Appl. 354 (2009), 294–396.CrossRefGoogle Scholar
10.Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations (Springer-Verlag, Berlin, 1975).Google Scholar
11.Quittner, R. and Souplet, P., Superlinear parabolic problems. Blow-up, global existence and steady states (Birkhauser, Basel, 2007).Google Scholar
12.Straughan, B., Explosive instabilities in mechanics (Springer-Verlag, Berlin, 1998).CrossRefGoogle Scholar