On the absence of solutions to damped system of nonlinear wave equations of Kirchhoff-type
Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 4, pp. 44-58 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In higher-order function spaces, some techniques are used to give the nonexistence result to system of wave equations in the Kirchhoff type, to generalize earlier results in the literature.
@article{VMJ_2015_17_4_a3,
     author = {Kh. Zennir and S. Zitouni},
     title = {On the absence of solutions to damped system of nonlinear wave equations of {Kirchhoff-type}},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {44--58},
     year = {2015},
     volume = {17},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2015_17_4_a3/}
}
TY  - JOUR
AU  - Kh. Zennir
AU  - S. Zitouni
TI  - On the absence of solutions to damped system of nonlinear wave equations of Kirchhoff-type
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2015
SP  - 44
EP  - 58
VL  - 17
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VMJ_2015_17_4_a3/
LA  - en
ID  - VMJ_2015_17_4_a3
ER  - 
%0 Journal Article
%A Kh. Zennir
%A S. Zitouni
%T On the absence of solutions to damped system of nonlinear wave equations of Kirchhoff-type
%J Vladikavkazskij matematičeskij žurnal
%D 2015
%P 44-58
%V 17
%N 4
%U http://geodesic.mathdoc.fr/item/VMJ_2015_17_4_a3/
%G en
%F VMJ_2015_17_4_a3
Kh. Zennir; S. Zitouni. On the absence of solutions to damped system of nonlinear wave equations of Kirchhoff-type. Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 4, pp. 44-58. http://geodesic.mathdoc.fr/item/VMJ_2015_17_4_a3/

[1] Abdelli M., Benaissa A., “Energy decay of solutions of degenerate Kirchhoff equation with a weak nonlinear dissipation”, Nonlinear Analysis, 69 (2008), 1999–2008 | DOI | MR | Zbl

[2] Agre K., Rammaha M. A., “Systems of nonlinear wave equations with damping and source terms”, Diff. Int. Equ., 19 (2007), 1235–1270 | MR

[3] Benaissa A., Ouchenane D., Zennir Kh., “Blow up of positive initial-energy solutions to systems of nonlinear wave equations with degenerate damping and source terms”, Nonlinear Studies, 19:4 (2012), 523–535 | MR | Zbl

[4] Benaissa A., Messaoudi S. A., “Blow up of solutions of a nonlinear wave equation”, J. Appl. Math., 2:2 (2002), 105–108 | DOI | MR | Zbl

[5] Benaissa A., Messaoudi S. A., “Blow up of solutions for Kirchhoff equation of $q$-Laplacien type with nonlinear dissipation”, Colloq. Math., 94:1 (2002), 103–109 | DOI | MR | Zbl

[6] Benaissa A., Messaoudi S. A., “Blow up of solutions of a quasilinear wave equation with nonlinear dissipation”, J. Part. Diff. Eq., 15:3 (2002), 61–67 | MR | Zbl

[7] Erhan Piskin, 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., 1:2 (2013), 181–199 | Zbl

[8] Georgiev V., Todorova G., “Existence of a solution of the wave equation with nonlinear damping and source term”, J. Diff. Eq., 109 (1994), 295–308 | DOI | MR | Zbl

[9] Hrusa W. J., Renardy M., “A model equation for viscoelasticity with a strongly singular kernel”, SIAM. J. Math. Anal., 19:2 (1988), 257–259 | DOI | MR | Zbl

[10] Kafini M., Messaoudi S., “A blow-up result in a Cauchy viscoelastic problem”, Appl. Math. Letters, 21 (2008), 549–553 | DOI | MR | Zbl

[11] Kirchhoff G., Vorlesungen uber Mechanik, 3rd ed., Teubner, Leipzig, 1983

[12] Levine H. A., Serrin J., “Global nonexistence theorems for quasilinear evolution equations with dissipation”, Archive for Rational Mechanics and Analysis, 37:4 (1997), 341–361 | DOI | MR

[13] Levine H. A., Park S. R., Serrin J., “Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation”, JMAA, 228:1 (1998), 181–205 | MR | Zbl

[14] Gao Q., Li F., Wang Y., “Blow up of the solution for higher order Kirchhoff type equations with nonlinear dissipation”, Cent. Eur. J. Math., 9:3 (2011), 686–698 | DOI | MR | Zbl

[15] Messaoudi S., “Blow up and global existence in a nonlinear viscoelastic wave equation”, Maths Nachr., 260 (2003), 58–66 | DOI | MR | Zbl

[16] Messaoudi S. A., “On the control of solutions of a viscoelastic equation”, Journal of the Franklin Institute, 344 (2007), 765–776 | DOI | MR | Zbl

[17] Messaoudi S. A., 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., 27 (2004), 1687–1696 | DOI | MR | Zbl

[18] Messaoudi S. A., Said-Houari B., “A blow-up result for a higher-order non-linear Kirchhoff -type hyperbolic equation”, Appl. Math. Letters, 20 (2007), 866–871 | DOI | MR | Zbl

[19] Messaoudi S. A., 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., 365 (2010), 277–287 | DOI | MR | Zbl

[20] Ouchenane D., Zennir Kh., Bayoud M., “Global nonexistence of solutions of a system of nonlinear viscoelastic wave equations with degenerate damping and source terms”, Ukrainian Math. J., 65:7 (2013), 645–669 | MR

[21] Pohozaev S. I., “On a class of quasiunear hyperbolic equations”, Mat. Sbornik, 96(138):1 (1975), 152–166 | MR | Zbl

[22] Rammaha M. A., Sawanya Sakuntasathien, “Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms”, Nonlinear Analysis, 72 (2010), 2658–2683 | DOI | MR | Zbl

[23] Rammaha M. A., Sawanya Sakuntasathien, “Critically and degenerately damped systems of nonlinear wave equations with source terms”, Appl. Anal., 72 (2010), 1201–1227 | DOI | MR

[24] Reed M., Abstract Non Linear Wave Equations, Lect. Notes in Math., 507, Springer-Verlag, Berlin, 1976 | MR | Zbl

[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., 23:1–2 (2010), 79–92 | MR | Zbl

[26] Todorova G., “Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms”, Comptes Rendus de l'académie des Sciences Sér. 1, 326:2 (1998), 191–196 | MR | Zbl

[27] Todorova G., “Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms”, JMAA, 239:2 (1999), 213–226 | MR | Zbl

[28] Vitillaro E., “Global nonexistence theorems for a class of evolution equations with dissipative”, Archive for Rational Mechanics and Analysis, 149:2 (1999), 155–182 | DOI | MR | Zbl

[29] Yang Z., “Blow up of solutions for a class of nonlinear evolution equations with nonlinear damping and source term”, Math. Meth. Ap. Sc., 25:10 (2002), 825–833 | DOI | MR | Zbl

[30] Zennir Kh., “Growth of solutions with positive initial energy to system of degeneratly damped wave equations with memory”, Lobachevskii J. of Math., 35:2 (2014), 147–156 | DOI | MR | Zbl