Publication Cover
Applicable Analysis
An International Journal
Volume 91, 2012 - Issue 3
278
Views
17
CrossRef citations to date
0
Altmetric
Original Articles

Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping

, &
Pages 575-586 | Received 10 May 2010, Accepted 19 Dec 2010, Published online: 09 Mar 2011

References

  • Reed , M and Simon , B . 1979 . Methods of Modern Mathematical Physics III. Scattering Theory , New York – London : Academic Press .
  • Zauderer , E . 1989 . Partial Differential Equations of Applied Mathematics, Pure and Applied Mathematics, 2nd ed. , New York : John Wiley & Sons, Inc. .
  • Messaoudi , SA . 2002 . Global existence and nonexistence in a system of Petrovsky . J. Math. Anal. Appl. , 265 ( 2 ) : 296 – 308 .
  • Wu , ST and Tsai , LY . 2009 . On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system . Taiwanese J. Math. , 13 ( 2A ) : 545 – 558 .
  • Amroun , NE and Benaissa , A . 2006 . Global existence and energy decay of solutions to a Petrovsky equation with general nonlinear dissipation and source term . Georgian Math. J. , 13 ( 3 ) : 397 – 410 .
  • Guesmia , A . 1998 . Existence globale et stabilisation interne non linéaire d'un système de Petrovsky. [Global existence and nonlinear internal stabilization of a Petrovskii system] . Bull. Belg. Math. Soc. Simon Stevin , 5 ( 4 ) : 583 – 594 . (French)
  • Guesmia , A . 1999 . Energy decay for a damped nonlinear coupled system . J. Math. Anal. Appl. , 239 ( 1 ) : 38 – 48 .
  • Komornik , V and Kouemou-Patcheu , S . 1997 . Well-posedness and decay estimates for a Petrovsky system with internal damping . Adv. Math. Sci. Appl. , 7 ( 1 ) : 245 – 260 .
  • Fang , DY and Xu , J . 2006 . Global existence and uniform decay rates of solutions for a class of strongly damped wave equations . Acta Math. Sci. Ser. A Chin. Ed. , 26 ( 5 ) : 753 – 765 . (Chinese)
  • Gazzola , F and Squassina , M . 2006 . Global solutions and finite time blow up for damped semilinear wave equations . Ann. Inst. H. Poincaré Anal. Non Linéaire , 23 ( 2 ) : 185 – 207 .
  • Xu , J . 2006 . Existence and blow-up of solutions for a class of strongly damped wave equations . Appl. Math. J. Chinese Univ. Ser. A , 21 ( 2 ) : 157 – 164 . (Chinese)
  • Yang , ZF . 2008 . A necessary and sufficient condition for the blow-up solution of a nonlinear Klein–Gordon equation with strong damping . Math. Appl. (Wuhan) , 21 ( 3 ) : 581 – 586 . (Chinese)
  • Yu , SQ . 2009 . On the strongly damped wave equation with nonlinear damping and source terms . Electron. J. Qual. Theory Differ. Eqns , 2009 ( 39 ) : 1 – 18 .
  • Alves , CO , Cavalcanti , MM , Domingos Cavalcanti , VN , Rammaha , MA and Toundykov , D . 2009 . On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms . Discrete Contin. Dyn. Syst. Ser. S , 2 ( 3 ) : 583 – 608 .
  • Benaissa , A and Messaoudi , SA . 2005 . Exponential decay of solutions of a nonlinearly damped wave equation . NoDEA Nonlinear Differ. Eqns Appl. , 12 ( 4 ) : 391 – 399 .
  • Cavalcanti , MM , Domingos Cavalcanti , VN and Martinez , P . 2004 . Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term . J. Differ. Eqns , 203 : 119 – 158 .
  • Cavalcanti , MM , Domingos Cavalcanti , VN and Lasiecka , I . 2007 . Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping – source interaction . J. Differ. Eqns , 236 ( 2 ) : 407 – 459 .
  • Chen , CS . 2002 . Global nonexistence for a nonlinear beam equation . Nanjing Daxue Xuebao Shuxue Bannian Kan , 19 : 247 – 254 . (Chinese)
  • Chen , CS and Ren , L . 2005 . Weak solution for a fourth-order nonlinear wave equation . J. Southeast Univ. (English Ed.) , 21 : 369 – 374 .
  • Georgiev , V and Todorova , G . 1994 . Existence of a solution of the wave equation with nonlinear damping and source terms . J. Differ. Eqns , 109 ( 2 ) : 295 – 308 .
  • Messaoudi , SA . 2001 . Decay of the solution energy for a nonlinearly damped wave equation . Arab. J. Sci. Eng. Sect. A Sci. , 26 ( 1 ) : 63 – 68 .
  • Ye , YJ . 2004 . On the decay of solutions for some nonlinear dissipative hyperbolic equations . Acta Math. Appl. Sin. Engl. Ser. , 20 ( 1 ) : 93 – 100 .
  • Yu , SQ . 2009 . Polynomial stability of solutions for a system of non-linear viscoelastic equations . Appl. Anal. , 88 ( 7 ) : 1039 – 1051 .
  • Adams , RA . 1975 . Sobolev Spaces, Pure and Applied Mathematics , Vol. 65 , New York – London : Academic Press .
  • Payne , L and Sattinger , DH . 1975 . Saddle points and instability of nonlinear hyperbolic equations . Israel J. Math. , 22 ( 3–4 ) : 273 – 303 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.