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Blow-up Solutions of Quasilinear Hyperbolic Equations WithCritical Sobolev Exponent

Published online by Cambridge University Press:  29 February 2012

S. Ibrahim  and
A. Lyaghfouri
S. Ibrahim*
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada
A. Lyaghfouri
Affiliation:
Fields Institute, Toronto, ON, Canada
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

In this paper, we show finite time blow-up of solutions of the p−waveequation in ℝN, with critical Sobolev exponent. Our workextends a result by Galaktionov and Pohozaev [4]

Keywords

p−wave equationBlow-upcritical Sobolev exponent
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 66 - 76
Copyright
© EDP Sciences, 2012

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References

M. Agueh. A new ODE approach to sharp Sobolev inequalities. Nonlinear Analysis Research Trends. Nova Science Publishers, Inc. Editor : Inès N. Roux, pp. 1–13 (2008).
T. Aubin. Problème isopérimétrique et espaces de Sobolev, J. Differential Geometry. 11, pp. 573–598 (1976).
C. Chen, H. Yao, L. Shao. Global Existence, Uniqueness, and Asymptotic Behavior of Solution for p-Laplacian Type Wave Equation. Journal of Inequalities and Applications Volume 2010, Article ID 216760, 15 pages.
V.A. Galaktionov, S.I. Pohozaev. Blow-up and critical exponents for nonlinear hyperbolic equations. Nonlinear Analysis 53, pp. 453–466 (2003).
G. Hongjun, Z. Hui. Global nonexistence of the solutions for a nonlinear wave equation with the q -Laplacian operator. J. Partial Diff. Eqs. 20 pp. 71–79(2007) .
S. Ibrahim, N. Masmoudi, K. Nakanishi. Scattering threshold for the focusing nonlinear Klein-Gordon equation. Analysis and PDE 4, No. 3, pp. 405–460, 2011.
C. E. Kenig, F. Merle. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166, No. 3, pp. 645–675 (2006).
C. E. Kenig, F. Merle. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201, No. 2, pp. 147–212 (2008).
J. Shatah. Unstable ground state of nonlinear Klein-Gordon equations. Trans. Amer. Math. Soc. 290, No. 2, pp. 701–710 (1985).
G. Talenti. Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110, pp. 353–372 (1976).
Z. Wilstein. Global Well-Posedness for a Nonlinear Wave Equation with p -Laplacian Damping. Ph.D. thesis, University of Nebraska. http://digitalcommons.unl.edu/mathstudent/24