On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation

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DOI: 10.4236/ijmnta.2017.61002    366 Downloads   479 Views  

ABSTRACT

In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy.

Cite this paper

Lin, G. , Gao, Y. and Sun, Y. (2017) On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation. International Journal of Modern Nonlinear Theory and Application, 6, 11-25. doi: 10.4236/ijmnta.2017.61002.

References

[1] Kirchhoff, G. (1883) Vorlesungen über Mechanik. Teubner, Leipzig.
[2] Ball, J.M. (1997) Remarks on Blow-Up and Nonexistence Theorems for Nonlinear Evolution Equations. The Quarterly Journal of Mathematics, Oxford Series, 28, 473-486.
[3] Kopácková, M. (1989) Remarks on Bounded Solutions of a Semilinear Dissipative Hyperbolic Equation. Commentationes Mathematicae Universitatis Carolinae, 30, 713-719.
[4] Haraux, A. and Zuazua, E. (1988) Decay Estimates for Some Semilinear Damped Hyperbolic Problems. Archive for Rational Mechanics and Analysis, 100, 191-206.
https://doi.org/10.1007/BF00282203
[5] Yang, Z.F. and Qiu, D.H. (2009) Energy Decaying and Blow-Up of Solution for a Kirchhoff Equation with Strong Damping. Journal of Mathematical Research & Exposition, 29, 707-715.
[6] Kosuke, O. (1997) On Global Existence, Asymptotic Stability and Blowing up of Solutions for Some Degenerate Non-Linear Wave Equations of Kirchhoff Type with a Strong Dissipation. Mathematical Methods in the Applied Sciences, 20, 151-177.
https://doi.org/10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0
[7] Lin, X.L. and Li, F.S. (2013) Global Existence and Decay Estimates for Nonlinear Kirchhoff-Type Equation with Boundary Dissipation. Differential Equations & Applications, 5, 297-317.
https://doi.org/10.7153/dea-05-18
[8] Ye, Y.J. (2013) Global Existence and Energy Decay Estimate of Solutions for a Higher-Order Kirchhoff Type Equation with Damping and Source Term. Nonlinear Analysis: Real World Applications, 14, 2059-2067.
https://doi.org/10.1016/j.nonrwa.2013.03.001
[9] Li, M.R. and Tsai, L.Y. (2003) Existence and Nonexistence of Global Solutions of Some System of Semilinear Wave Equations. Nonlinear Analysis, 54, 1397-1415.
https://doi.org/10.1016/S0362-546X(03)00192-5
[10] Li, F.C. (2004) Global Existence and Blow-Up of Solutions for a Higher-Order Kirchhoff-Type Equation with Nonlinear Dissipation. Applied Mathematics Letters, 17, 1409-1414.
https://doi.org/10.1016/j.am1.2003.07.014
[11] Wu, S. and Tsai, L. (2006) Blow-Up of Solutions for Some Nonlinear Wave Equations of Kirchhoff Type with Some Dissipation. Nonlinear Analysis: Theory, Methods & Applications, 65, 243-264.
https://doi.org/10.1016/j.na.2004.11.023
[12] Gazzola, F. and Squassina, M. (2006) Global Solutions and Finite Time Blow up for Semilinear Wave Equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 23, 185-207.
[13] Lin, G.G. (2011) Nonlinear Evolution Equation. Yunnan University Press, Kunming.

  
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