Abstract
We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given.
Highlights
In the numerical study of the ill-posed Boussinesq equation, utt = uxx + (u2)xx + uxxxx. (1)Darapi and Hua [1] proposed the singularly perturbed Boussinesq equation utt = uxx + (u2)xx + uxxxx + δuxxxxxx (2)as a dispersive regularization of the ill-posed classical Boussinesq equation (1), where δ > 0 is a small parameter
On the basis of far-field analysis and heuristic arguments, Daripa and Dash [3] proved that the traveling wave solutions of (2) are weakly nonlocal solitary waves characterized by small amplitude fast oscillations in the far-field and obtained weakly nonlocal solitary wave solutions of (2)
0. by Theorem 6, we deduce that the solution of initial boundary value problem (62), (64) must blow up in a finite time T1; namely,
Summary
The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given. We employ the energy method and the Jensen inequality to prove that the global solutions of the initial boundary value problem (4), (5) and (4), (6) cease to exist in a finite time, respectively. We show that the global solution of the initial boundary value problem (2), (6) blows up in a finite time.
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