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Applicable Analysis
An International Journal
Volume 91, 2012 - Issue 3
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Original Articles

Global existence and decay of solutions of a nonlinear system of wave equations

Pages 475-489 | Received 17 Nov 2009, Accepted 10 Dec 2010, Published online: 10 Mar 2011

References

  • Gazzola , F and Squassina , M . 2006 . Global solutions and finite time blow up for damped semilinear wave equations . Ann. I. H. Poincar , 23 : 185 – 207 .
  • Yang , Z . 2002 . Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms . Math. Meth. Appl. Sci. , 25 : 795 – 814 .
  • Vitillaro , E . 1999 . Global existence theorems for a class of evolution equations with dissipation . Arch. Rational Mech. Anal. , 149 : 155 – 182 .
  • Messaoudi , SA and Said-Houari , B . 2004 . Global non-existence of solutions of a class of wave equations with non-linear damping and source terms . Math. Methods Appl. Sci. , 27 : 1687 – 1696 .
  • Levine , HA . 1974 . Instability and nonexistence of global solutions to nonlinear wave equations of the form . Trans. Amer. Math. Soc. , 192 : 1 – 21 .
  • Levine , HA . 1974 . Some additional remarks on the nonexistence of global solutions to nonlinear wave equations . SIAM J. Math. Anal. , 5 : 138 – 146 .
  • Ikehata , R and Suzuki , T . 1996 . Stable and unstable sets for evolution equations of parabolic and hyperbolic type . Hiroshima Math. J. , 26 : 475 – 491 .
  • Ikehata , R . 1996 . Some remarks on the wave equations with nonlinear damping and source terms . Nonlinear Anal. , 27 : 1165 – 1175 .
  • Esquivel-Avila , J . 2004 . Qualitative analysis of a nonlinear wave equation . Discr. Cont. Dyn. Syst. , 10 : 787 – 804 .
  • Messaoudi , SA . 2001 . Blow up in a nonlinearly damped wave equation . Math. Nachr. , 231 : 1 – 7 .
  • Haraux , A and Zuazua , E . 1988 . Decay estimates for some semilinear damped hyperbolic problems . Arch. Rational Mech. Anal. , 150 : 191 – 206 .
  • Kopackova , M . 1989 . Remarks on bounded solutions of a semilinear dissipative hyperbolic equation . Comment. Math. Univ. Carolin. , 30 ( 4 ) : 713 – 719 .
  • Ball , J . 1977 . Remarks on blow up and nonexistence theorems for nonlinear evolutions equations . Quart. J. Math. Oxford. , 28 ( 2 ) : 473 – 486 .
  • Kalantarov , VK and Ladyzhenskaya , OA . 1978 . The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type . J. Soviet. Math. , 10 : 53 – 70 .
  • Georgiev , V and Todorova , G . 1994 . Existence of a solution of the wave equation with nonlinear damping and source term . J. Diff. Eqns. , 109 : 295 – 308 .
  • Levine , HA and Serrin , J . 1997 . Global nonexistence theorems for quasilinear evolution equations with dissipation . Arch. Rational Mech. Anal. , 137 ( 4 ) : 341 – 361 .
  • Levine , HA and Park , SR . 1998 . Global existence and global non-existence of solutions of the cauchy problem for a non-linearly damped wave equation . J. Math. Anal. App. , 228 : 181 – 205 .
  • Todorova , G . 1998 . Cauchy problem for a non-linear wave with non-linear damping and source terms . C. R. Acad Sci. Paris Ser. , 326 ( 1 ) : 191 – 196 .
  • Todorova , G . 1999 . Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms . J. Math. Anal. Appl. , 239 : 213 – 226 .
  • Serrin , J , Todorova , G and Vitillaro , E . 2003 . Existence for a nonlinear wave equation with damping and source terms . Diff. Integral Eqns. , 16 ( 1 ) : 13 – 50 .
  • Levine , HA and Todorova , G . 2001 . Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy . Proc. Am. Math. Soc. , 129 ( 2 ) : 793 – 805 .
  • Todorova , G and Vitillaro , E . 2005 . Blow-up for nonlinear dissipative wave equations in ℝ n . J. Math. Anal. Appl. , 303 ( 1 ) : 242 – 257 .
  • Milla Miranda , M and Medeiros , LA . 1987 . On the existence of global solutions of a coupled nonlinear Klein–Gordon equations . Funkcial. Ekvac. , 30 ( 1 ) : 147 – 161 .
  • Segal , I . 1965 . Nonlinear partial differential equations in quantum field theory . Proc. Symp. Appl. Math. AMS , 17 : 210 – 226 .
  • Jorgens , K . Nonlinear wave equations, Tech. Rep., Department of Mathematics, University of Colorado, 1970
  • Agre , K and Rammaha , MA . 2006 . Systems of nonlinear wave equations with damping and source terms . Diff. Integral Eqns. , 19 ( 11 ) : 1235 – 1270 .
  • Said-Houari , B . 2010 . Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms . Diff. Integral Eqns. , 23 ( 1–2 ) : 79 – 92 .
  • Alves , CO , Cavalcanti , MM , Cavalcanti , VND , Rammaha , M 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 . Discr. Cont. Dyn. Syst. Ser. S , 2 : 583 – 608 .
  • Nakao , M . 1977 . Asymptotic stability of the bounded or almost periodic solution of the wave equation . J. Math. Anal. Appl. , 56 : 336 – 343 .

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