Blow up for the wave equation with a fractional damping

• Autorzy: Alaimia M. R. , Tatar N.-E.
• Języki publikacji: EN

Abstrakty

EN

We consider the wave equation with a fractional damping of order between o and 1 and a polynomial source. Introducing a new functional and using an argument due to Georgiev and Todorova [1] together with some appropriate estimates, it is proved that some solutions blow up in finite time.

Słowa kluczowe

EN

blow up; Caputo's fractional derivative; integro-differential problem; modified energy functional; singular kernel

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2005
• Tom: Vol. 11, nr 1
• Strony: 133--144
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Alaimia M. R.

• King Fahd University of Petroleum and Minerals, Department of Mathematical Sciences, Dhahran, 31261, Saudi Arabia

autor

Tatar N.-E.

• King Fahd University of Petroleum and Minerals, Department of Mathematical Sciences, Dhahran, 31261, Saudi Arabia

Bibliografia

• [1] Georgiev, V., Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994), 295-308.
• [2] Kirane, M., Tatar, N.-E., Exponential growth for a fractionally damped utai>e equation, Z. Anal. Anwendungen 22(1) (2003), 167-177.
• [3] Matignon, D., Audounet, J., Montseny, G., Energy decay for wave equations with damping of fractional order, Proc. of the 4th International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena (Golden, Colorado, June 1998), 638-640, INRIA-SIAM.
• [4] Messaoudi, S., A blow up result in a multidimensional semilinear thermoelastic system, Electron. J. Differential Equations 2001(30) (2001), 9 pp., (electronic).
• [5] Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, New York-London, 1974.
• [6] Podlubny, I., Fractional Differential Equations, Math. Sci. Engrg. 198, Academic Press, San Diego, CA, 1999.
• [7] Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals a,nd Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. (Translated from the 1987 Russian original).
• [8] Tatar, N.-E., A wave equation with fractional damping, Z. Anal. Anwendungen 22(3) (2003), 609-617.
• [9] Tatar, N.-E., A blow up result for a fractionally damped wave equation, NoDEA Nonlinear Differential Equations Appl., (to appear).