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Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations. (English) Zbl 1176.35119

Summary: We consider the Cauchy problem of multidimensional generalized double dispersion equations \(u_{tt}-\Delta u- \Delta u_{tt}+\Delta^2u= \Delta f(u)\), where \(f(u)=a|u|^p\). By a potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. Furthermore, we derive some sharp conditions for a global existence and the lack of global existence solution.

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35L82 Pseudohyperbolic equations
35L30 Initial value problems for higher-order hyperbolic equations
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[1] Samsonov, A. M.; Sokurinskaya, E. V., Energy exchange between nonlinear waves in elastic waveguides and external media, (Nonlinear Waves in Active Media (1989), Springer: Springer Berlin), 99-104 · Zbl 0687.73028
[2] Samsonov, A. M., Nonlinear strain waves in elastic waveguide, (Jeffrey, A.; Engelbrecht, J., Nonlinear Waves in Solids. Nonlinear Waves in Solids, CISM Courses and Lectures, vol. 341 (1994), Springer: Springer Wien) · Zbl 0806.73018
[3] Guowang, Chen; Yanping, Wan; Shubin, Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl., 299, 563-577 (2004) · Zbl 1066.35087
[4] Yacheng, Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192, 155-169 (2003) · Zbl 1024.35078
[5] Yacheng, Liu; Junsheng, Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64, 2665-2687 (2006) · Zbl 1096.35089
[6] Yacheng, Liu; Runzhang, Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7, 171-189 (2007) · Zbl 1121.35085
[7] Yacheng, Liu; Runzhang, Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331, 585-607 (2007) · Zbl 1113.35113
[8] Yacheng, Liu; Runzhang, Xu; Tao, Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal., 68, 3332-3348 (2008) · Zbl 1149.35367
[9] Yacheng, Liu; Runzhang, Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244, 200-228 (2008) · Zbl 1138.35066
[10] Yacheng, Liu; Runzhang, Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D, 237, 721-731 (2008) · Zbl 1185.35192
[11] Yacheng, Liu; Runzhang, Xu, Potential well method for Cauchy problem of generalized double dispersion equations, J. Math. Anal. Appl., 338, 1169-1187 (2008) · Zbl 1140.35011
[12] Yacheng, Liu; Runzhang, Xu, Potential well method for initial boundary value problem of the generalized double dispersion equations, Commun. Pure Appl. Anal., 7, 63-81 (2008) · Zbl 1151.35003
[13] Polat, Necat; Ertaş, Abdulkadir, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349, 10-20 (2009) · Zbl 1156.35331
[14] Pasternak, N. L., New Method for Calculation of Foundation on the Elastic Basement (1954), Gosstroiizdat: Gosstroiizdat Moscow, (in Russian)
[15] Shubin, Wang; Guowang, Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64, 159-173 (2006) · Zbl 1092.35056
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