×

The Cauchy problem for a class of the multidimensional Boussinesq-type equation. (English) Zbl 1210.35218

Summary: We study the Cauchy problem for a class of the multidimensional Boussinesq-type equation\(u_{tt}-\Delta u+\Delta^2u+\Delta^2u_{tt}=\Delta f(u)\), where \(f(u)=\pm a|u|^p\) or \(-a|u|^{p-1}u\), \(a>0\) is a constant. First, we establish a local existence theorem for the solution. Then, for \(m=1\) \((n=1)\), \(m=2,3,4\) \((n\leq 3)\), we prove the existence of a global \(H^m\) solution. Finally, we prove the global nonexistence and finite-time blow up of the solution.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B44 Blow-up in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bona, Jerry L.; Sachs, Robert L., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118, 15-29 (1988) · Zbl 0654.35018
[2] Linares, Felipe, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106, 257-293 (1993) · Zbl 0801.35111
[3] Liu, Yue, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26, 1527-1546 (1995) · Zbl 0857.35103
[4] Xue, Ruying, Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation, J. Math. Anal. Appl., 316, 307-327 (2006) · Zbl 1106.35069
[5] Liu, Yacheng; Xu, Runzhang, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Physica D, 237, 721-731 (2008) · Zbl 1185.35192
[6] Lin, Qun; Wu, Yong Hong; Loxton, Ryan, On the Cauchy problem for a generalized Boussinesq equation, J. Math. Anal. Appl., 353, 186-195 (2009) · Zbl 1168.35336
[7] Yang, Zhijian; Guo, Boling, Cauchy problem for the multi-dimensional Boussinesq type equation, J. Math. Anal. Appl., 340, 64-80 (2008) · Zbl 1132.35315
[8] Makhankov, V. G., Dynamics of classical solitons (in non-integrable systems), Phys. Rep., 35, 1-128 (1978)
[9] Wang, S. B.; Chen, G. W., Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl., 264, 846-866 (2002) · Zbl 1136.35425
[10] Wang, S. B.; Chen, G. W., The Cauchy problem for the generalized IMBq equation in \(W^{s, p}(R^n)\), J. Math. Anal. Appl., 266, 38-54 (2002) · Zbl 1043.35118
[11] Chen, Guowang; Wang, Yanping; Wang, Shubin, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl., 299, 563-577 (2004) · Zbl 1066.35087
[12] Wang, Shubin; Chen, Guowang, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64, 159-173 (2006) · Zbl 1092.35056
[13] Liu, Yacheng; Xu, Runzhang, Potential well method for Cauchy problem of generalized double dispersion equations, J. Math. Anal. Appl., 338, 1169-1187 (2008) · Zbl 1140.35011
[14] Polat, Necat; Ertaṣ, 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
[15] Xu, Runzhang; Liu, Yacheng; Yu, Tao, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal., 71, 4977-4983 (2009) · Zbl 1167.35418
[16] Wang, Shubin; Xu, Guixiang, The Cauchy problem for the Rosenau equation, Nonlinear Anal., 71, 456-466 (2009) · Zbl 1171.35424
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.