Mathematical behavior of the solutions of a class of hyperbolic-type equation

Öz

In this paper, we consider hyperbolic-type equations with initial and Dirichlet boundary conditions in a bounded domain. Under some suitable assumptions on the initial data and source term, we obtain nonexistence of global solutions for arbitrary initial energy.

Anahtar Kelimeler

• Hyperbolic equation,nonexistence,damping term

Hiperbolik tipten bir denklemin çözümlerinin matematiksel davranışı

Öz

Bu makalede sınırlı bir bölgede hiperbolik tipten başlangıç ve Dirichlet sınır koşullu problem ele alınmıştır. Başlangıç ve kaynak terim üzerine bırakılan bazı uygun koşullar altında çözümlerin global yokluğu keyfi başlangıç enerjisi için çalışılmıştır.

Anahtar Kelimeler

• Hiperbolik denklem,yokluk,damping terim

Kaynakça

1. Georgiev, V., Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source term, Journal of Differential Equations, 109, 295-308, (1994).
2. Levine, H.A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = -Au + F(u), Transactions of the American Mathematical Society,, 192, 1-21, (1974).
3. Levine, H.A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM Journal on Applied Mathematics, 5, 138-146, (1974).
4. Messaoudi, S.A., Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231, 105-111, (2001).
5. Vitillaro, E., Global existence theorems for a class of evolution equations with dissipation, Archive for Rational Mechanics and Analysis, 149, 155-182 (1999).
6. Messaoudi, S. A., Global existence and nonexistence in a system of Petrovsky, Journal of Mathematical Analysis and Applications, 265(2), 296-308, (2002).
7. Wu, S.T., Tsai, L.Y., On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese Journal of Mathematics, 13(2A), 545-558 (2009).
8. Chen, W., Zhou, Y., Global nonexistence for a semilinear Petrovsky equation, Nonlinear Analysis, 70, 3203-3208, (2009).

9. Alshin, A.B., Korpusov, M.O., Sveshnikov, A.G., Blow up in nonlinear Sobolev type equations, Walter De Gruyter, 2011.
10. Hu, B., Blow up theories for semilinear parabolic equations, Springer, 2011.
11. Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, 1995.
12. Adams, R.A., Fournier, J.J.F., Sobolev Spaces, Academic Press, 2003.
13. Pişkin, E., Sobolev Spaces, Seçkin Publishing, 2017. (in Turkish).
14. Pişkin, E., Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Mathematics, 13, 408-420, (2015).
15. Li, M.R., Tsai, L.Y., Existence and nonexistence of global solutions of some system of semi-linear wave equations, Nonlinear Analysis, 54(8), 1397-1415, (2003).
16. Zhou, Y., Global existence and nonexistence for a nonlinear wave equation with damping and source terms, Mathematische Nachrichten, 278(11), 1341-1358, (2005).