Initial boundary value problem for a class of fourth-order wave equation with viscous damping term

References

• An , L and Peirce , A . 1975 . A weakly nonlinear analysis of elasto-plastic-microstructure models . SIAM J. Appl. Math. , 55 ( 1 ) : 136 – 155 .
   Web of Science ®Google Scholar
• Guenther , RB and Lee , JW . 1988 . Partial Differential Equations of Mathematical Physics and Integral Equations , New Jersey : Prentice Hall .
   Google Scholar
• Varlamov , VV . 1996 . On the Cauchy problem for the damped Boussinesq equation . Differ. Integral Eqns , 9 ( 3 ) : 619 – 634 .
   Google Scholar
• Varlamov , VV . 1996 . Existence and uniqueness of solution to the Cauchy problem for the damped Boussinesq equation . Math. Meth. Appl. Sci. , 19 : 639 – 649 .
   Web of Science ®Google Scholar
• Chen , G and Lu , B . 2009 . The initial-boundary value problems for a class of nonlinear wave equations with damping term . J. Math. Anal. Appl. , 351 : 1 – 15 .
   Web of Science ®Google Scholar
• Sattinger , DH . 1968 . On global solution of nonlinear hyperbolic equations . Arch. Rational Mech. Anal. , 30 : 148 – 172 .
   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
• Levine , HA . 1974 . Instability and nonexistence of global solutions of nonlinear wave equation of the form Pu tt  = Au + F(u) . Trans. Amer. Math. Soc. , 192 : 1 – 21 .
   Web of Science ®Google Scholar
• Ball , J . 1977 . Remarks on blow up and nonexistence theorems for nonlinear evolutions equations . Quart. J. Math. Oxford. , 28 : 437 – 486 .
   Web of Science ®Google Scholar
• Payne , LE and Sattinger , DH . 1975 . Sadle points and instability of nonlinear hyperbolic equations . Israel. J. Math. , 22 : 273 – 303 .
   Web of Science ®Google Scholar
• Levine , HA and Serrin , J . 1997 . Global nonexistence theorems for quasilinear evolution equations with dissipation . Arch. Rational Mech. Anal. , 137 : 341 – 361 .
   Web of Science ®Google Scholar
• Pucci , P and Serrin , J . 1998 . Global nonexistence for abstract evolution equations with positive initial energy . J. Differ. Eqns , 150 : 203 – 214 .
   Web of Science ®Google Scholar
• Levine , HA , Pucci , P and Serrin , J . 1997 . Some remarks on global nonexistence for nonautonomous abstract evolution equations . Contemp. Math. , 208 : 253 – 263 .
   Google Scholar
• Vitillaro , E . 1999 . Global existence theorems for a class of evolution equations with dissipation . Arch. Rational Mech. Anal. , 149 : 155 – 182 .
   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 : 151 – 177 .
   Web of Science ®Google Scholar
• Ohta , M . 1998 . Remarks on blowup of solutions for nonlinear evolution equations of second order . Adv. Math. Sci. Appl. , 8 : 901 – 910 .
   Google Scholar
• 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 : 185 – 207 .
   Web of Science ®Google Scholar
• Gerbi , S and Said-Houari , B . 2012 . Exponential decay for solutions to semilinear damped wave equation . Discr. Contin. Dyn. Syst.-Ser. S , 5 : 559 – 566 .
   Web of Science ®Google Scholar
• Berestycki , H and Cazenave , T . 1981 . Instabilité des États stationnaires dans les sequations de Schrodinger et de Klein–Gordon non-linseairees . C. R. Acad. Sci. Paris. , 293 : 489 – 492 .
   Google Scholar
• Zhang , J . 2002 . Sharp conditions of global existence for nonlinear Schrodinger and Klein–Gordon equations . Nonlinear Anal. TMA. , 48 : 191 – 207 .
   Web of Science ®Google Scholar
• Zhang , J . 2003 . On the standing wave in coupled non-linear Klein–Gordon equations . Math. Methods Appl. Sci. , 26 : 11 – 25 .
   Web of Science ®Google Scholar
• Liu , Y and Junsheng , Z . 2006 . On potential wells and applications to semilinear hyperbolic equations and parabolic equations . Nonlinear Anal. , 64 : 2665 – 2687 .
   Web of Science ®Google Scholar
• Liu , Y and Xu , R . 2007 . Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign . Discr. Contin. Dyn. Syst.-Ser. B , 7 ( 1 ) : 171 – 189 .
   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 . Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation . Physica. D , 237 ( 6 ) : 721 – 731 .
   Web of Science ®Google Scholar
• Liu , Y and Xu , R . 2008 . Potential well method for Cauchy problem of generalized double dispersion equations . J. Math. Anal. Appl. , 338 : 1169 – 1187 .
   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
• Xu , R and Liu , Y . 2009 . Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations . J. Math. Anal. Appl. , 359 : 739 – 751 .
   Web of Science ®Google Scholar
• Xu , R , Liu , Y and Yu , T . 2009 . Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations . Nonlinear Anal. Theory, Methods Appl. , 71 : 4977 – 4983 .
   Web of Science ®Google Scholar
• Xu , R and Shen , J . 2009 . Some generalized results for global well-poseness for wave equations with damping and source terms . Math. Comput. Simul. , 80 : 804 – 807 .
   Web of Science ®Google Scholar
• Xu , R . 2010 . Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein–Gordon equation with dissipative term . Math. Methods Appl. Sci. , 33 : 831 – 844 .
   Web of Science ®Google Scholar
• Xu , R . 2010 . Initial boundary value problem of semilinear hyperbolic equations and parabolic equations with critical initial data . Quar. Appl. Math. , 68 : 459 – 468 .
   Web of Science ®Google Scholar
• Xu , R and Xu , C . 2010 . Nonlinear Schrodinger equation with combined power-type nonlinearities and harmonic potential . Appl. Math. Mech. , 31 : 521 – 528 .
   Web of Science ®Google Scholar
• Xu , R , Liu , Y and Liu , B . 2011 . Cauchy problem for a class of multidimensional Boussinesq type equation . Nonlinear Anal. Theory Methods Appl. , 74 : 2425 – 2437 .
   Web of Science ®Google Scholar