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Asymptotics for the modified Boussinesq equation in one space dimension. (English) Zbl 1387.35491
The authors consider the Cauchy problem for the modified 1D Boussinesq equation \[ \begin{cases} w_{tt}=a^2\partial^2_xw-\partial^4_xw+\partial^2_x(w^3), \quad (t,x)\in \mathbb{R}^2, \\ w(0,x)=w_0(x), \quad w_t(0,x)=w_1(x), \quad x\in \mathbb{R}, \end{cases} \] where \(a>0\). This equation describes the propagation of long waves in shallow water. By using the so-called factorization technique, the authors investigate the long time behavior of the solutions of the above Cauchy problem. Specifically, they show that for “small” initial data there exists a unique solution on \([0,\infty)\) of the Cauchy problem satisfying the estimate \(|w(t)|_{L^{\infty}}\leq C\varepsilon t^{-1/2}\).

MSC:
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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Full Text: Euclid