Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations

Affiliation(s)

Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan.

Nakhchivan State University, Nakhchivan, Azerbaijan.

Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan.

Nakhchivan State University, Nakhchivan, Azerbaijan.

ABSTRACT

In this paper, we investigate the effect of weight function in the nonlinear part on global solvability of the Cauchy problem for a class of semi-linear hyperbolic equations with damping.

KEYWORDS

Cauchy Problem; Wave Equation; Global Solvability; Weight Function; Semi-Linear Hyperbolic Equation

Cauchy Problem; Wave Equation; Global Solvability; Weight Function; Semi-Linear Hyperbolic Equation

Cite this paper

A. Aliev and A. Kazimov, "Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 1, 2013, pp. 102-106. doi: 10.4236/ijmnta.2013.21A013.

A. Aliev and A. Kazimov, "Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations,"

References

[1] A. B. Aliev and A. A. Kazymov, “Global Weak Solutions of the Cauchy Problem for Semi-Linear Pseudo-Hyperbolic Equations,” Differential Equations, Vol. 45, No. 2, 2009, pp. 1-11.

[2] A. B. Aliev and B. H. Lichaei, “Existence and Non-Existence of Global Solutions of the Cauchy Problem for Higher Semi-Linear Pseudo-Hyperbolic Equations,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 72, No. 7-8, 2010, pp. 3275-3288.

[3] B. R. Ikehata, Y. Miataka and Y. Nakatake, “Decay Estimates of Solutions of Dissipative Wave Equations in R_{n} with Lower Power Nonlinearities,” Journal of the Mathematical Society of Japan, Vol. 56, No. 2, 2004, pp. 365- 373.

[4] T. Li and Y. Zhou, “Breakdown of Solutions u+u_{2}=|u|^{1+α},” Discrete and Continuous Dynamical Systems, Vol. 1, No. 4, 1995, pp. 503-520. doi:10.3934/dcds.1995.1.503

[5] I. E. Segal, “Dispersion for Non-Linear Realistic Equations II,” Journal of the American Mathematical Society, Vol. 4, No. 16, 1968, pp. 459-497.

[6] Q. S. Zhag, “A Blow-Up Result for a Nonlinear Wave Equation with Damping,” The Critical Case, dans Comptesrendus de l’Académie des Sciences de Paris, Serie I, Vol. 333, 2001, pp. 109-114.

[7] G. Todorova and B. Yordanov, “Critical Exponent for a Nonlinear Wave Equation with Damping,” dans Comptes rendus de l’Académie des Sciences de Paris, Serie I, Vol. 330, 2000, pp. 557-562.

[8] E. Mitidieri and S. I. Pokhozhaev, “A Priori Estimates and the Absence of Solutions of Nonlinear Partial Differential Equations and Inequalities,” Steklov Mathematical Institute Seminar, Vol. 234, 2001, pp. 1-384.

[9] A. B. Aliev, “Solvability in the Large of the Cauchy Problem for Quasilinear Equations of Hyperbolic Type,” Doklady Akademii Nauk SSSR, Vol. 240, No. 2, 1978, pp. 249-252.

[10] O. V. Besov, V. P. Ilin and S. M. Nikolski, “Integral Representation of Functions and Embedding Theorem,” V.H. Wilson and Sons, Washington DC, 1978.

[1] A. B. Aliev and A. A. Kazymov, “Global Weak Solutions of the Cauchy Problem for Semi-Linear Pseudo-Hyperbolic Equations,” Differential Equations, Vol. 45, No. 2, 2009, pp. 1-11.

[2] A. B. Aliev and B. H. Lichaei, “Existence and Non-Existence of Global Solutions of the Cauchy Problem for Higher Semi-Linear Pseudo-Hyperbolic Equations,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 72, No. 7-8, 2010, pp. 3275-3288.

[3] B. R. Ikehata, Y. Miataka and Y. Nakatake, “Decay Estimates of Solutions of Dissipative Wave Equations in R

[4] T. Li and Y. Zhou, “Breakdown of Solutions u+u

[5] I. E. Segal, “Dispersion for Non-Linear Realistic Equations II,” Journal of the American Mathematical Society, Vol. 4, No. 16, 1968, pp. 459-497.

[6] Q. S. Zhag, “A Blow-Up Result for a Nonlinear Wave Equation with Damping,” The Critical Case, dans Comptesrendus de l’Académie des Sciences de Paris, Serie I, Vol. 333, 2001, pp. 109-114.

[7] G. Todorova and B. Yordanov, “Critical Exponent for a Nonlinear Wave Equation with Damping,” dans Comptes rendus de l’Académie des Sciences de Paris, Serie I, Vol. 330, 2000, pp. 557-562.

[8] E. Mitidieri and S. I. Pokhozhaev, “A Priori Estimates and the Absence of Solutions of Nonlinear Partial Differential Equations and Inequalities,” Steklov Mathematical Institute Seminar, Vol. 234, 2001, pp. 1-384.

[9] A. B. Aliev, “Solvability in the Large of the Cauchy Problem for Quasilinear Equations of Hyperbolic Type,” Doklady Akademii Nauk SSSR, Vol. 240, No. 2, 1978, pp. 249-252.

[10] O. V. Besov, V. P. Ilin and S. M. Nikolski, “Integral Representation of Functions and Embedding Theorem,” V.H. Wilson and Sons, Washington DC, 1978.