Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping

References

• Torrejón , R and Yong , JM . 1991 . On a quasilinear wave equation with memory . Nonlinear Anal. , 16 ( 1 ) : 61 – 78 .
   Web of Science ®Google Scholar
• Munoz Rivera , JE . 1994 . Global solution on a quasilinear wave equation with memory . Boll. Un. Mat. Ital. B (7) , 8 ( 2 ) : 289 – 303 .
   Google Scholar
• Wu , S-T and Tsai , L-Y . 2006 . On global existence and blow-up of solutions for an integro-differential equation with strong damping . Taiwanese J. Math. , 10 ( 4 ) : 979 – 1014 .
   Web of Science ®Google Scholar
• Wu , S-T . 2010 . Exponential energy decay of solutions for an integro-differential equation with strong damping . J. Math. Anal. Appl. , 364 ( 2 ) : 609 – 617 .
   Web of Science ®Google Scholar
• Matsuyama , T and Ikehata , R . 1996 . On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms . J. Math. Anal. Appl. , 204 ( 3 ) : 729 – 753 .
   Web of Science ®Google Scholar
• Ono , K . 1997 . Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation . Nonlinear Anal. , 30 ( 7 ) : 4449 – 4457 .
   Web of Science ®Google Scholar
• Ono , K . 1997 . Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings . J. Diff. Eqns. , 137 ( 2 ) : 273 – 301 .
   Web of Science ®Google Scholar
• Ono , K . 1997 . On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation . Math. Methods Appl. Sci. , 20 ( 2 ) : 151 – 177 .
   Web of Science ®Google Scholar
• Ono , K . 1997 . On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation . J. Math. Anal. Appl. , 216 ( 1 ) : 321 – 342 .
   Web of Science ®Google Scholar
• Park , JY and Bae , JJ . 1998 . On existence of solutions of nondegenerate wave equations with nonlinear damping terms . Nihonkai Math. J. , 9 ( 1 ) : 27 – 46 .
   Google Scholar
• Benaissa , A and Messaoudi , SA . 2002 . Blow-up of solutions for the Kirchhoff equation of q-Laplacian type with nonlinear dissipation . Colloq. Math. , 94 ( 1 ) : 103 – 109 .
   Google Scholar
• Wu , S-T and Tsai , L-Y . 2007 . On a system of nonlinear wave equations of Kirchhoff type with a strong dissipation . Tamkang J. Math. , 38 ( 1 ) : 1 – 20 .
   Google Scholar
• Li , M-R and Tsai , L-Y . 2003 . Existence and nonexistence of global solutions of some system of semilinear wave equations . Nonlinear Anal. , 54 ( 8 ) : 1397 – 1415 .
   Web of Science ®Google Scholar
• Liang , F and Gao , HJ . 2011 . Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping . Boundary Value Prob. , 2011 ( 22 ) : 19
   Google Scholar
• Han , XS and Wang , MX . 2009 . Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source . Nonlinear Anal. , 71 ( 11 ) : 5427 – 5450 .
   Web of Science ®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 ( 1 ) : 277 – 287 .
   Web of Science ®Google Scholar
• Andrade , D and Fatori , LH . 2004 . The nonlinear transmission problem with memory . Bol. Soc. Parana. Mat. (3) , 22 ( 1 ) : 106 – 118 .
   Google Scholar
• Agre , K and Rammaha , MA . 2006 . Systems of nonlinear wave equations with damping and source terms . Diff. Integral Eqns. , 19 ( 11 ) : 1235 – 1270 .
   Web of Science ®Google Scholar
• Cavalcanti , MM , Cavalcanti , VNDomingos and Soriano , JA . Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Diff. Eqns 2002(44) (2002), 14pp
   Google Scholar
• Cavalcanti , MM , Domingos Cavalcanti , VN , Prates Filho , JS and Soriano , JA . 1998 . Existence and exponential decay for a Kirchhoff–Carrier model with viscosity . J. Math. Anal. Appl. , 226 ( 1 ) : 40 – 60 .
   Web of Science ®Google Scholar
• Cavalcanti , MM , Domingos Cavalcanti , VN and Santos , ML . 2004 . Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary . Appl. Math. Comput. , 150 ( 2 ) : 439 – 465 .
   Web of Science ®Google Scholar
• Cavalcanti , MM , Domingos Cavalcanti , VN and Martinez , P . 2008 . General decay rate estimates for viscoelastic dissipative systems . Nonlinear Anal. , 68 ( 1 ) : 177 – 193 .
   Web of Science ®Google Scholar
• Li , M-R and Tsai , L-Y . 2003 . On a system of nonlinear wave equations . Taiwanese J. Math. , 7 ( 4 ) : 557 – 573 .
   Web of Science ®Google Scholar
• Liu , WJ . 2009 . Uniform decay of solutions for a quasilinear system of viscoelastic equations . Nonlinear Anal. , 71 ( 5–6 ) : 2257 – 2267 .
   Web of Science ®Google Scholar
• Liu , WJ . 2010 . General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source . Nonlinear Anal. , 73 ( 6 ) : 1890 – 1904 .
   Web of Science ®Google Scholar
• Liu , WJ . 2009 . General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms . J. Math. Phys. , 50 ( 11 ) : 113506
   Web of Science ®Google Scholar
• Messaoudi , SA . 2003 . Blow up and global existence in a nonlinear viscoelastic wave equation . Math. Nachr. , 260 : 58 – 66 .
   Web of Science ®Google Scholar
• Park , JY and Bae , JJ . 2002 . Variational inequality for quasilinear wave equations with nonlinear damping terms . Nonlinear Anal. Ser. A: Theory Methods , 50 ( 8 ) : 1065 – 1083 .
   Web of Science ®Google Scholar
• Santos , ML , Raposo , CA and Soares , UR . 2003 . On nonlinear coupled system with nonlocal boundary conditions . Bol. Soc. Parana. Mat. (3) , 21 ( 1–2 ) : 73 – 96 .
   Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Sun , FQ and Wang , MX . 2006 . Global and blow-up solutions for a system of nonlinear hyperbolic equations with dissipative terms . Nonlinear Anal. , 64 : 739 – 761 .
   Web of Science ®Google Scholar
• Vicente , A . 2009 . Wave equation with acoustic/memory boundary conditions . Bol. Soc. Parana. Mat. (3) , 27 ( 1 ) : 29 – 39 .
   Google Scholar
• Wu , S-T . Blow-up of solutions for a system of nonlinear wave equations with nonlinear damping, Electron. J. Diff. Eqns. . 2009(105) (2009), 11pp
   Google Scholar
• Yu , SQ . 2009 . Polynomial stability of solutions for a system of non-linear viscoelastic equations . Appl. Anal. , 88 ( 7 ) : 1039 – 1051 .
   Web of Science ®Google Scholar
• Komornik , V . 1994 . Exact Controllability and Stabilization , Paris : Research in Applied Mathematics .
   Google Scholar
• Vitillaro , E . 1999 . Global nonexistence theorems for a class of evolution equations with dissipation . Arch. Ration. Mech. Anal. , 149 ( 2 ) : 155 – 182 .
   Web of Science ®Google Scholar