Blow-up Solutions of Quasilinear Hyperbolic Equations With Critical Sobolev Exponent

S. Ibrahim; A. Lyaghfouri

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 2, page 66-76
  • ISSN: 0973-5348

Abstract

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

How to cite

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Ibrahim, S., and Lyaghfouri, A.. "Blow-up Solutions of Quasilinear Hyperbolic Equations With Critical Sobolev Exponent." Mathematical Modelling of Natural Phenomena 7.2 (2012): 66-76. <http://eudml.org/doc/222260>.

@article{Ibrahim2012,
abstract = {In this paper, we show finite time blow-up of solutions of the p−wave equation in ℝN, with critical Sobolev exponent. Our work extends a result by Galaktionov and Pohozaev [4]},
author = {Ibrahim, S., Lyaghfouri, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {p−wave equation; Blow-up; critical Sobolev exponent; -wave equation},
language = {eng},
month = {2},
number = {2},
pages = {66-76},
publisher = {EDP Sciences},
title = {Blow-up Solutions of Quasilinear Hyperbolic Equations With Critical Sobolev Exponent},
url = {http://eudml.org/doc/222260},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Ibrahim, S.
AU - Lyaghfouri, A.
TI - Blow-up Solutions of Quasilinear Hyperbolic Equations With Critical Sobolev Exponent
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/2//
PB - EDP Sciences
VL - 7
IS - 2
SP - 66
EP - 76
AB - In this paper, we show finite time blow-up of solutions of the p−wave equation in ℝN, with critical Sobolev exponent. Our work extends a result by Galaktionov and Pohozaev [4]
LA - eng
KW - p−wave equation; Blow-up; critical Sobolev exponent; -wave equation
UR - http://eudml.org/doc/222260
ER -

References

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  1. 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).  
  2. T. Aubin. Problème isopérimétrique et espaces de Sobolev, J. Differential Geometry. 11, pp. 573–598 (1976).  Zbl0371.46011
  3. 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.  Zbl1211.35199
  4. V.A. Galaktionov, S.I. Pohozaev. Blow-up and critical exponents for nonlinear hyperbolic equations. Nonlinear Analysis 53, pp. 453–466 (2003).  Zbl1012.35058
  5. G. Hongjun, Z. Hui. Global nonexistence of the solutions for a nonlinear wave equation with theq-Laplacian operator. J. Partial Diff. Eqs. 20 pp. 71–79(2007) .  Zbl1142.35493
  6. 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.  Zbl1270.35132
  7. 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).  Zbl1115.35125
  8. 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).  Zbl1183.35202
  9. J. Shatah. Unstable ground state of nonlinear Klein-Gordon equations. Trans. Amer. Math. Soc. 290, No. 2, pp. 701–710 (1985).  Zbl0617.35072
  10. G. Talenti. Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110, pp. 353–372 (1976).  Zbl0353.46018
  11. Z. Wilstein. Global Well-Posedness for a Nonlinear Wave Equation withp-Laplacian Damping. Ph.D. thesis, University of Nebraska.  Zbl1262.35151URIhttp://digitalcommons.unl.edu/mathstudent/24

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