Уважаемые авторы, просим обратить внимание!
Подача статьи осуществляется только через личный кабинет электронной редакции.
DOI: 10.23671/VNC.2015.4.5970
Несуществование решения затухающей системы нелинейных волновых уравнений типа Кирхгофа
Зеннир К. , Зитуни С.
Владикавказский математический журнал. 2015. Том 17. Выпуск 4.С.44-58.
Аннотация: Изучается влияние сильного источника на существование решений в пространстве с высоким порядком суммируемости в затухающей системе нелинейных волновых уравнений типа Кирхгофа.
Ключевые слова: взрыв, уравнение типа Кирхгофа, вырождающиеся затухающие системы, сильно нелинейный источник, положительная начальная энергия
Образец цитирования: Khaled Zennir, Salah Zitouni On the absence of solutions to damped
system of nonlinear wave equations of Kirchhoff-type // Владикавк.
мат. журн. 2015. Том 17. Выпуск 4. С.44-58.
DOI 10.23671/VNC.2015.4.5970
1. Abdelli M. and Benaissa A. Energy decay of solutions of degenerate Kirchhoff equation with a weak nonlinear dissipation // Nonlinear Analysis.---2008.---Vol. 69.---P. 1999--2008.
2. Agre K. and Rammaha M. A. Systems of nonlinear wave
equations with damping and source terms // Diff. Inte. Equ.---2007.---Vol. 19.---P. 1235--1270.
3. Benaissa A., Ouchenane D. and Zennir Kh. Blow up of positive initial-energy solutions to systems of nonlinear wave equations with degenerate damping and
source terms // Nonlinear Studies.---2012.---Vol. 19, № 4.---P. 523--535.
4. Benaissa A. and Messaoudi S. A. Blow up of solutions of a nonlinear wave equation // J. Appl. Math.---2002.---Vol. 2(2).---P. 105--108.
5. Benaissa A., Messaoudi S. A.
Blow up of solutions for Kirchhoff equation of \(q\)-Laplacien type with nonlinear dissipation // Colloq. Math.---2002.---94(1).---P. 103--109.
6. Benaissa A., Messaoudi S. A. Blow up of solutions
of a quasilinear wave equation with nonlinear dissipation //
J. Part. Diff. Eq.---2002.---Vol. 15(3).---P. 61--67.
7. Erhan Piskin and Necat Polat
Uniform decay and blow up of solutions for a system of nonlinear higher-order Kirchhoff-type equations with damping and source //
Cont. Anal. Appl. Math.---2013.---Vol. 1, № 2.---P. 181--199.
8. Georgiev V. and Todorova G. Existence of a solution of
the wave equation with nonlinear damping and source term // J. Diff. Eq.---1994.---Vol. 109.---P. 295--308.
9. Hrusa W. J. and Renardy M. A model equation for
viscoelasticity with a strongly singular kernel // SIAM. J. Math. Anal.---1988.---Vol. 19, № 2.
10. Kafini M., Messaoudi S. A blow-up result in a Cauchy
viscoelastic problem // Appl. Math. Letters.---2008.---Vol. 21.---P. 549--553.
11. Kirchhoff G. Vorlesungen uber Mechanik. 3rd ed.---Leipzig: Teubner, 1983.
12. Levine H. A. and Serrin J. Global nonexistence
theorems for quasilinear evolution equations with dissipation //
Archive for Rational Mechanics and Analysis.---1997.---Vol. 37,
№ 4.---P. 341--361.
13. Levine H. A., Park S. R., and Serrin J. Global
existence and global nonexistence of solutions of the Cauchy problem for a
nonlinearly damped wave equation // JMAA.---1998.---Vol. 228, № 1.---P. 181--205.
14. Gao Q., Li F., and Wang Y. Blow up of the solution for higher order Kirchhoff type equations with nonlinear dissipation // Cent. Eur. J. Math.---2011.---Vol. 9(3).---P. 686--698.
15. Messaoudi S. Blow up and global existence in a nonlinear
viscoelastic wave equation // Maths Nachr.---2003.---Vol. 260.---P. 58--66.
16. Messaoudi S. A. On the control of solutions of a
viscoelastic equation // Journal of the Franklin Institute.---2007.---Vol. 344.---P. 765--776.
17. Messaoudi S. A. and Said-Houari B. Global non-existence
of solutions of a class of wave equations with non-linear damping and source
terms // Math. Meth. Appl. Sci.---2004.---Vol. 27.---P. 1687--1696.
18. Messaoudi S. A. and Said-Houari B.
A blow-up result for a higher-order non-linear Kirchhoff -type hyperbolic equation //
Appl. Math. Letters.---2007.---Vol. 20.---P. 866--871.
19. Messaoudi S. A. and Said-Houari B. Global nonexistence
of positive initial-energy solutions of a system of nonlinear viscoelastic
wave equations with damping and source terms // J. Math. Anal. Appl.---2010.---Vol. 365.---P. 277--287.
20. Ouchenane D., Zennir Kh., and Bayoud M. Global nonexistence
of solutions of a system of nonlinear viscoelastic wave equations
with degenerate damping and source terms // Ukrainian Math. J.---2013.---Vol. 65, № 7.
21. Pohozaev S. I. On a class of quasiunear hyperbolic equations //
Mat. Sbornik.---1975.---Vol. 96(138), № 1.
22. Rammaha M. A. and Sawanya Sakuntasathien . Global existence
and blow up of solutions to systems of nonlinear wave equations with
degenerate damping and source terms // Nonlinear Analysis.---2010.---Vol. 72.---P. 2658--2683.
23. Rammaha M. A. and Sawanya Sakuntasathien. Critically and degenerately damped systems of nonlinear wave equations with source terms //
Appl. Anal.---2010.---Vol. 72.---P. 1201--1227.
24. Reed M. Abstract Non Linear Wave Equations //
Lect. Notes in Math.---Berlin: Springer-Verlag, 1976.
25. Said-Houari B. Global nonexistence of positive
initial-energy solutions of a system of nonlinear wave equations with
damping and source terms // Dif. Int. Equ.---2010.---Vol. 23, № 1--2.---P. 79--92.
26. Todorova G. Cauchy problem for a nonlinear wave
equation with nonlinear damping and source terms // Comptes Rendus de l'academie des Sciences.---1998.---Ser. 1, Vol. 326, № 2.---P. 191--196.
27. Todorova G. Stable and unstable sets for the
Cauchy problem for a nonlinear wave equation with nonlinear damping and
source terms // JMAA.---1999.---Vol. 239, № 2.---P. 213--226.
28. Vitillaro E. Global nonexistence theorems for a
class of evolution equations with dissipative // Archive for
Rational Mechanics and Analysis.---1999.---Vol. 149, № 2.---P. 155--182.
29. Yang Z. Blow up of solutions for a class of
nonlinear evolution equations with nonlinear damping and source term //
Math. Meth. Ap. Sc.---2002.---Vol. 25, № 10.---P. 825--833.
30. Zennir Kh. Growth of solutions with positive initial energy to system
of degeneratly damped wave equations with memory //
Lobachevskii J. of Math.---2014.---Vol. 35, № 2.---P. 147--156.
