Global well-posedness for strongly damped viscoelastic wave equation

References

• Renardy , M , Hrusa , WJ and Nohel , JA . 1987 . Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics , Vol. 35 , New York : John Wiley and Sons .
   Google Scholar
• Kafini , M and Messaoudi , SA . 2008 . A blow-up result in a Cauchy viscoelastic problem . Appl. Math. Lett. , 21 : 549 – 553 .
   Web of Science ®Google Scholar
• Messaoudi , SA . 2003 . Blow up and global existence in a nonlinear viscoelastic wave equation . Math. Nachrichten. , 260 : 58 – 66 .
   Web of Science ®Google Scholar
• Messaoudi , SA . 2006 . Blow up of solutions with positive initial energy in a nonlinear viscoelastic equation . J. Math. Anal. Appl. , 320 : 902 – 915 .
   Web of Science ®Google Scholar
• Jung , A and Han , Y . 2009 . Blow up of solutions of a nonlinear viscoelastic wave equation . Acta Appl. Math. , 11 : 1 – 6 .
   Google Scholar
• Wang , Y . 2009 . A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy . Appl. Math. Lett. , 22 : 1394 – 1400 .
   Web of Science ®Google Scholar
• Messaoudi , SA and Tatar , NE . 2007 . Global existence and uniform stability of solutions for a quasilinear viscoelastic problem . Math. Methods Appl. Sci. , 30 : 665 – 680 .
   Web of Science ®Google Scholar
• Han , X and Wang , M . 2009 . General decay of energy for a viscoelastic equation with nonlinear damping . Math. Methods Appl. Sci. , 32 : 346 – 358 .
   Web of Science ®Google Scholar
• Liu , W . 2010 . General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source . Nonlinear Anal. Theory, Methods Appl. , 73 : 1890 – 1904 .
   Web of Science ®Google Scholar
• Song , H and Zhong , C . 2010 . Blow up of solutions of a nonlinear viscoelastic wave equation . Nonlinear Anal.: Real World Appl. , 11 ( 5 ) : 3877 – 3883 .
   Web of Science ®Google Scholar
• Cavalcanti , MM , Cavalcanti , VND and Ferreira , J . 2001 . Existence and uniform decay for nonlinear viscoelastic equation with strong damping . Math. Methods Appl. Sci. , 24 : 1043 – 1053 .
   Web of Science ®Google Scholar
• Messaoudi , SA and Tatar , NE . 2003 . Global existence and asymptotic behavior for a nonlinear viscoelastic problem . Math. Sci. Res. J. , 7 : 136 – 149 .
   Google Scholar
• Messaoudi , SA and Said-Houari , B . 2010 . 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 : 277 – 287 .
   Web of Science ®Google Scholar
• Payne , L and Sattinger , D . 1975 . Saddle points and instability on nonlinear hyperbolic equations . Israel J. Math. , 22 : 273 – 303 .
   Web of Science ®Google Scholar
• Levine , HA . 1974 . Instability and nonexistence of global solutions of nonlinear wave equation of the form Pu tt  = Au + F(u) . Trans. Am. Math. Soc. , 192 : 1 – 21 .
   Web of Science ®Google Scholar
• Levine , HA . 1974 . Some additional remarks on the nonexistence of global solutions to nonlinear wave equations . SIAM J. Math. Anal. , 5 : 138 – 146 .
   Web of Science ®Google Scholar
• Liu , Y and Xu , R . 2007 . Fourth order wave equations with nonlinear strain and source terms . J. Math. Anal. Appl. , 331 : 585 – 607 .
   Web of Science ®Google Scholar
• Liu , Y and Xu , R . 2008 . A class of fourth order wave equations with dissipative and nonlinear strain terms . J. Differ. Eqns , 244 : 200 – 228 .
   Web of Science ®Google Scholar
• Gazzola , F and Squassina , M . 2006 . Global solutions and finite time blow-up for damped semilinear wave equations . Ann. Inst. Henri Poincaré. Anal. Non Linéaire , 23 : 185 – 207 .
   Web of Science ®Google Scholar
• Liu , Y and Xu , R . 2008 . Global existence and blow-up of solutions for Cauchy problem of generalized Boussinesq equation . Phys. D. Nonlinear Phenomena. , 237 : 721 – 731 .
   Web of Science ®Google Scholar
• Georgiev , V and Todorova , G . 1994 . Existence of solutions of the wave equation with nonlinear damping and source terms . J. Differ. Eqns , 109 : 295 – 308 .
   Web of Science ®Google Scholar