Existence Theory for the Boussinesq Equation in Modulation Spaces
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Existence Theory for the Boussinesq Equation in Modulation Spaces Carlos Banquet1 · Élder J. Villamizar-Roa2 Received: 2 October 2018 / Accepted: 25 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces M sp ,q (Rn ), n ≥ 1. After a decomposition of the Boussinesq equation in a 2 × 2-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in M sp,q × D −1 J M sp,q spaces for suitable p, q and s, including the special case p = 2, q = 1 and s = 0. Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces. Keywords Boussinesq equation · Modulation spaces · Local and global solutions · Scattering · Asymptotic stability Mathematics Subject Classification 35Q53 · 35A01 · 47J35 · 35B40 · 35B35
1 Introduction We consider the initial value problem associated to the generalized Boussinesq equation
B
∂t2 u − u + 2 u + f (u) = 0, (x, t) ∈ Rn+1 , x ∈ Rn , u(x, 0) = u 0 (x), ∂t u(x, 0) = φ(x) = v0 (x),
(1.1)
Carlos Banquet [email protected] Élder J. Villamizar-Roa [email protected]
1
Departamento de Matemáticas y Estadística, Universidad de Córdoba, A.A. 354, Montería, Colombia
2
Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia
123
C. Banquet, É. J. Villamizar-Roa
where u : Rn ×R → R is the unknown, u 0 , v0 : Rn → R are given functions denoting the initial data and the nonlinear term is f (u) = u λ , for some 1 < λ < ∞. Equation (1.1) is physically relevant in the modelling of shallow water waves, ion sound waves in a plasma, the dynamics of stretched string, and other physical phenomena (Cho and Ozawa 2007; Peregrine 1972). The IVP (1.1) is formally equivalent to the following system ⎧ (x, t) ∈ Rn+1 , ⎨ ∂t u = v, (1.2) (x, t) ∈ Rn+1 , ∂ v = u − u − f (u), ⎩ t x ∈ Rn . u(x, 0) = u 0 (x), v(x, 0) = v0 (x), From Duhamel’s principle, the Cauchy problem associated to system (1.2) is equivalent to the integral equation
t
[u(t), v(t)] = B(t)[u 0 , v0 ] −
B(t − τ )[0, f (u(τ ))]dτ,
(1.3)
0
where B is the solution of the linear problem associated to (1.2). More exactly, for initial data [u 0 , v0 ] and t ∈ R, we have B(t)[u 0 , v0 ] = ei x·ξ B 1 (t)u 0 (ξ ) + B 2 (t)v0 (ξ ), B 3 (t)u 0 (ξ ) + B 1 (t)v0 (ξ ) dξ, Rn
(1.4) where B1 (t), B2 (t) and B3 (t) are the multiplier operators with symbols cos(t|ξ |ξ ), −|ξ |ξ −1 sin(t|ξ |ξ ) and |ξ |−1 ξ sin(t|ξ |ξ ) respectively, and ξ = (1+|ξ |2 )1/2 . Note that B(t)[u 0 , v0 ] = [B1 (t)u 0 + B2 (t)v0 , B3 (t)u 0 + B1 (t)v0 ]. Several authors have analyzed the local and global existence, and long time asymptotic behavior of solutions for (1.1) (cf. Bona and Sachs 1988; Cho and Ozawa 2007; Farah 2008, 2009a, b; Ferreira 2011; Kishimoto 2013; Kishimoto and Tsugawa 2010; Linares 1993; Liu 1997; Muñoz et al. 2018; Tsutsumi and Matahashi 1991 and
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