Small amplitude solutions of the generalized IMBq equation

On small amplitude solutions to the generalized Boussinesq equations

“…Theorem 1.1 is applicable to the cases p i > 4 for n = 1, p 1 > 3, p 2 > 4 for n = 2 and p 1 > n, p 2 > 2n for n ≥ 3. This improves the results in [7,24,39].…”

“…For the global existence and scattering of small amplitude solutions, it is necessary to study the dispersion of the operators ∂ t S i , S i and T i with respect to time, and to compare them with nonlinearity, especially to compare the time decay rate with power p. To get a time decay dispersive estimate, Linares [21], and Linares and Scialom [22] [24] for (1.1) with n = 1, and p > 9 2 of Cho and Ozawa [7] for (1.2) with n = 1, and integer p greater than 2 + 1 θ(n,s) of Wang and Chen [39] for (1.2) with n ≥ 2, where θ = In this paper, we improve all the known results under some vanishing condition of initial data at the zero frequency in one dimensional case and extend the results not only on existence and scattering but dispersive estimates to the high dimensional case. Moreover, we also provide a non-existence of nontrivial asymptotically free solutions in the case of small power p, which is a high dimensional version of Theorem 1.3 of [7].…”

“…3 Actually, Wang and Chen in [39] obtained n-dimensional estimate but their estimate was the same as 1-dimensional one because they integrated only in radial radial direction by using spherical coordinate. …”

On properties of solutions to the improved modified Boussinesq equation

“…Equation (1.3) and its generalized form in n space dimensions u tt − ∆u − ∆u tt = ∆φ(u) (1.4) can also describe the dynamical and thermodynamical properties of an harmonic monatomic and diatomic chains (see [14,15]). Existence and nonexistence of global solutions, the global existence of small amplitude solutions for the Cauchy problem for (1.4) were obtained by Wang et al [13,19,20]. Cho and Ozawa [4] studied the existence and scattering of global small amplitude solutions to (1.4).…”

A Note on “The Cauchy Problem for Coupled Imbq Equations”

“…The system (1.1), (1.2) arises from DNA (see [3][4][5][6][7][8][9][10][11][12]). The system (1.1), (1.2) are closely related to the IMBq equations and generalized IMBq equations (see [13][14][15][16][17][18][19][20] and see the reference in [1]). The detailed cases can be found in the introduction of the reference [1].…”