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On local existence of solutions of initial boundary value problems for the “bad” Boussinesq-type equation. (English) Zbl 1022.35052

The paper deals with the differential equation \[ u_{tt}-u_{xx}-bu_{xxxx}= \sigma(u)_{xx} \] together with certain appropriate initial and boundary conditions and where \(\sigma(s)\) is a given nonlinear function. In the case \(b < 0\) this equation is called the “bad” Boussinesq-equation since in this case even the linear part behaves bad. There are examples of not well-posed problems having this linear part. H. A. Levine and B. D. Sleeman [J. Math. Anal. Appl. 107, 206-210 (1985; Zbl 0591.35010)] showed the nonexistence of global positive solutions.
In this interesting paper the existence of local generalized solutions is shown under rather weak conditions on \(\sigma(u)\) and the initial data, like \(\sigma\) twice continuously differentiable and \( \sigma'' (s) \) is locally Lipschitz continuous. If \(\sigma(s)\) is a concave function then, under certain conditions on the initial data, the solutions have a finite escape time (blow up).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 0591.35010
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References:

[1] Berryman, J., Stability of solitary waves in shallow water, Phys. Fluids, 19, 771-777 (1976) · Zbl 0351.76023
[2] Bona, J. L.; Sachs, R. 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
[3] Bona, J. L.; Smith, R. A., A model for the two-way propagation of water waves in a channel, Math. Proc. Cambridge Philos. Soc., 79, 167-182 (1976) · Zbl 0332.76007
[4] Boussinesq, J., Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en comuniquant au liquide contene dans ce canal des vitesses sunsiblement parielles de la surface au fond, J. Math. Pures Appl. Ser., 2, 17, 55-108 (1872) · JFM 04.0493.04
[5] Guowang, Chen; Zhijian, Yang, Existence and non-existence of global solutions for a class of non-linear wave equation, Math. Methods Appl. Sci., 23, 615-631 (2000) · Zbl 1007.35046
[6] Clarkson, P. A., New exact solutions of the Boussinesq equations, European J. Appl. Math., 1, 279-300 (1990) · Zbl 0721.35074
[7] Clarkson, P. A.; Kruskal, M. D., New similarity reductions of the Boussinesq equation, J. Math. Phys., 30, 2201-2213 (1983) · Zbl 0698.35137
[8] Defrutos, J.; Ortega, T.; Sanz-Serna, J. M., Pseudospectral method for the “good” Boussinesq equation, Math. Comp., 57, 109-122 (1991) · Zbl 0735.65089
[9] Deift, P.; Tomei, C.; Trubowitz, E., Inverse scattering and the Boussinesq equation, Comm. Pure. Appl. Math., 35, 567-628 (1982) · Zbl 0479.35074
[10] Kalantarov, V. K.; Ladyzhenskaya, O. A., Occurrence of blow-up for quasi-linear partial differential equations of parabolic and hyperbolic types, Zap. Nauk. Som. Lomi., 69, 77-102 (1977) · Zbl 0354.35054
[11] Kalantarov, V. K.; Ladyzhenskaya, O. A., The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math., 10, 53-70 (1978) · Zbl 0388.35039
[12] Levine, H. A.; Sleeman, B. D., A note on the non-existence of global solutions of initial boundary value problems for the Boussinesq equation \(u_{ tt }=3u_{ xxxx }+u_{ xx } \)−\(12(u^2)_{ xx } \), J. Math. Anal. Appl., 107, 206-210 (1985) · Zbl 0591.35010
[13] Linares, F., Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106, 257-293 (1993) · Zbl 0801.35111
[14] Makhankov, V. G., Dynamics of classical solitons, Phys. Rev. Lett., 35, c, 1-128 (1978)
[15] Pani, A. K.; Sarange, H., Finite element Galerkin method for the “good” Boussinesq equation, Nolinear Anal. TMA, 29, 937-956 (1997) · Zbl 0880.35097
[16] Sachs, R. L., On the blow-up of certain solutions of the “good” Boussinesq equation, Appl. Anal., 36, 145-152 (1990) · Zbl 0674.35082
[17] Varlamov, V., Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation, Math. Methods Appl. Sci., 19, 639-649 (1996) · Zbl 0847.35111
[18] Yang Zhijian, Chen Guowang, Blowup of solutions for a class of generalized Boussinesq equations, Acta Math. Sci. 16 (1996) 31-40 (in Chinese).; Yang Zhijian, Chen Guowang, Blowup of solutions for a class of generalized Boussinesq equations, Acta Math. Sci. 16 (1996) 31-40 (in Chinese). · Zbl 0960.35090
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