Blow-up for Generalized Boussinesq Equation with Double Damping Terms
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Blow-up for Generalized Boussinesq Equation with Double Damping Terms Jianghao Hao and Aiyuan Gao Abstract. In this paper, we consider the Cauchy problem for a generalized Boussinesq equation with double damping terms. By using improved convexity method combined with potential well method and Fourier transform, we show the finite time blow-up of the solution with arbitrarily high initial energy while many similar results require the corresponding energy to be less than some certain numbers. Mathematics Subject Classification. 35L20, 35B35, 93D20. Keywords. Generalized Boussinesq equation, cauchy problem, damping, blow-up, high initial energy.
1. Introduction In this paper, we study the Cauchy problem of the generalized Boussinesq equation with double damping terms utt − uxx + uxxxx + (f (u))xx + αut − βuxxt = 0, x ∈ R, t > 0, (1.1) u(x, 0) = u0 (x) ut (x, 0) = u1 (x), x ∈ R, where α > 0, β > 0 are constants and the function f (u) satisfies the assumptions f (u) = ±a|u|p
or
f (u) = a|u|p−1 u,
a > 0, p > 1,
(1.2)
u0 (x) and u1 (x) are the given initial value functions. There are two damping terms, a strong damping and a weak damping. In the 1870’s, Boussinesq derived some model equations for the propagation of small amplitude, long waves on the surface of water. These equations possess special, travelling-wave solution called solitary waves. Scott-Russell [1] studied Boussinesq’s theory, and give a satisfactory, scientific explanation of the phenomenon of solitary waves. It is well known that Boussinesq equation can be written in the following basic form utt − uxx + μuxxxx − (u2 )xx = 0. 0123456789().: V,-vol
(1.3)
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J. Hao and A. Gao
MJOM
Equation 1.3 depends on the sign of μ, indeed, the case μ > 0 is called the good Boussinesq equation since it is linearly stable and governs small nonlinear transverse oscillations of an elastic beam (see [2]), while Eq. 1.3 with μ < 0 received the name of the bad Boussinesq equation since it possesses the linear instability. In terms of Boussinesq equation, the mainly effects of small nonlinearity and dispersion are taken into consideration. Polat and Ertas [3] studied the equation utt + auxxxx − 2buxxt − uxx − γ(u2 )xx = 0,
(1.4)
where uxxt is the strong damping term and a > 0, b > 0, γ ∈ R are constants. However, in many situation, damping effects are compare in strength to the nonlinear and dispersive one. The Cauchy problem of the generalized Boussinesq equation utt − uxx + uxxxx + f (u)xx = 0
(1.5)
has been considered in many papers, for example [4–7], and there are many results about global existence, nonexistence and finite time blow up of solutions. Polat et al. [8] established the blow-up result of the solutions for the initial-boundary value problem of the damped Boussinesq equation utt − buxx + δuxxxx − γuxxt − (f (u))xx = 0.
(1.6)
Wang and Chen [9] studied the existence and blow-up of the solution for the Cauchy problem of the generalized double dispersion equation utt − uxx + uxxtt + uxxxx − αuxxt = (g(u))xx .
(1.7)
Xu et al. [10] consi
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