On the Exponential Decay of Solutions for Some Kirchhoff-Type Modelling Equations with Strong Dissipation

ABSTRACT

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equations with strong dissipative and source terms in a bounded domain, where and are constants. We obtain the global existence of solutions by constructing a stable set in and show the energy exponential decay estimate by applying a lemma of V. Komornik.

This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equations with strong dissipative and source terms in a bounded domain, where and are constants. We obtain the global existence of solutions by constructing a stable set in and show the energy exponential decay estimate by applying a lemma of V. Komornik.

KEYWORDS

Kirchhoff-type Equation; Initial Boundary Value Problem; Stable Set; Exponential Decay Estimate

Kirchhoff-type Equation; Initial Boundary Value Problem; Stable Set; Exponential Decay Estimate

Cite this paper

nullY. Ye, "On the Exponential Decay of Solutions for Some Kirchhoff-Type Modelling Equations with Strong Dissipation,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 529-533. doi: 10.4236/am.2010.16070.

nullY. Ye, "On the Exponential Decay of Solutions for Some Kirchhoff-Type Modelling Equations with Strong Dissipation,"

References

[1] K. Narasimha, “Nonlinear Vibration of an Elastic String,” Journal of Sound and Vibration, Vol. 8, No. 1, 1968, pp. 134-146.

[2] K. Nishihara and Y. Yamada, “On Global Solutions of Some Degenerate Quasilinear Hyperbolic Equations with Dissipative Terms,” Funkcialaj Ekvacioj, Vol. 33, No. 1, 1990, pp. 151-159.

[3] M. Aassila and A. Benaissa, “Existence Globale et Com- portement Asymptotique des Solutions des Equations de Kirchhoff Moyennement Degenerees avce un Terme Nonlinear Dissipatif,” Funkcialaj Ekvacioj, Vol. 43, No. 2, 2000, pp. 309-333.

[4] K. Ono and K. Nishihara, “On a Nonlinear Degenerate Integro-Differential Equation of Hyperbolic Type with a Strong Dissipation,” Advances in Mathematics Seciences and Applications, Vol. 5, No. 2, 1995, pp. 457-476.

[5] K. Ono, “Global Existence, Decay and Blowup of Solutions for Some Mildly Degenerate Nonlinear Kirchhoff Strings,” Journal of Differential Equations, Vol. 137, No. 1, 1997, pp. 273-301.

[6] P. D. Ancona and S. Spagnolo, “Nonlinear Perturbations of the Kirchhoff Equation,” Commnicathins on Pure and Applied Mathematics, Vol. 47, No. 7, 1994, pp. 1005-1029.

[7] M. Ghisi and M. Gobbino, “Global Existence for a Mildly Degenerate Dissipativehyperbolic Equation of Kirchhoff Type,” Preprint, Dipartimento di Matematica Universita di Pisa, Pisa, 1997.

[8] M. Hosoya and Y. Yamada, “On Some Nonlinear Wave Equations II: Global Existence and Energy Decay of Solutions,” Journal of the Faculty of Science, The University of Tokyo, Section IA, Mathematics, Vol. 38, No. 1, 1991, pp. 239-250.

[9] L. E. Payne and D. H. Sattinger, “Saddle Points and Instability of Nonlinear Hyperbolic Equations,” Israel Jounal of Mathematics, Vol. 22, No. 3-4, 1975, pp. 273-303.

[10] D. H. Sattinger, “On Global Solutions of Nonlinear Hyperbolic Equations,” Archive for Rational Mechanics Analysis, Vol. 30, No. 2, 1968, pp. 148-172.

[11] V. Komornik, “Exact Controllability and Stabilization, The Multiplier Method, RAM: Research in Applied Mathematics,” Masson-John, Wiley, Paris, 1994.

[1] K. Narasimha, “Nonlinear Vibration of an Elastic String,” Journal of Sound and Vibration, Vol. 8, No. 1, 1968, pp. 134-146.

[2] K. Nishihara and Y. Yamada, “On Global Solutions of Some Degenerate Quasilinear Hyperbolic Equations with Dissipative Terms,” Funkcialaj Ekvacioj, Vol. 33, No. 1, 1990, pp. 151-159.

[3] M. Aassila and A. Benaissa, “Existence Globale et Com- portement Asymptotique des Solutions des Equations de Kirchhoff Moyennement Degenerees avce un Terme Nonlinear Dissipatif,” Funkcialaj Ekvacioj, Vol. 43, No. 2, 2000, pp. 309-333.

[4] K. Ono and K. Nishihara, “On a Nonlinear Degenerate Integro-Differential Equation of Hyperbolic Type with a Strong Dissipation,” Advances in Mathematics Seciences and Applications, Vol. 5, No. 2, 1995, pp. 457-476.

[5] K. Ono, “Global Existence, Decay and Blowup of Solutions for Some Mildly Degenerate Nonlinear Kirchhoff Strings,” Journal of Differential Equations, Vol. 137, No. 1, 1997, pp. 273-301.

[6] P. D. Ancona and S. Spagnolo, “Nonlinear Perturbations of the Kirchhoff Equation,” Commnicathins on Pure and Applied Mathematics, Vol. 47, No. 7, 1994, pp. 1005-1029.

[7] M. Ghisi and M. Gobbino, “Global Existence for a Mildly Degenerate Dissipativehyperbolic Equation of Kirchhoff Type,” Preprint, Dipartimento di Matematica Universita di Pisa, Pisa, 1997.

[8] M. Hosoya and Y. Yamada, “On Some Nonlinear Wave Equations II: Global Existence and Energy Decay of Solutions,” Journal of the Faculty of Science, The University of Tokyo, Section IA, Mathematics, Vol. 38, No. 1, 1991, pp. 239-250.

[9] L. E. Payne and D. H. Sattinger, “Saddle Points and Instability of Nonlinear Hyperbolic Equations,” Israel Jounal of Mathematics, Vol. 22, No. 3-4, 1975, pp. 273-303.

[10] D. H. Sattinger, “On Global Solutions of Nonlinear Hyperbolic Equations,” Archive for Rational Mechanics Analysis, Vol. 30, No. 2, 1968, pp. 148-172.

[11] V. Komornik, “Exact Controllability and Stabilization, The Multiplier Method, RAM: Research in Applied Mathematics,” Masson-John, Wiley, Paris, 1994.