On small amplitude solutions to the generalized Boussinesq equations

Cho, Yonggeun; Ozawa, Tohru

doi:10.3934/dcds.2007.17.691

Cited by 52 publications

(32 citation statements)

References 32 publications

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“…Applying respectively Lemma 2.1 (with η = k = C g 2 V−3 , δ = κ, γ = 1, t 0 = η + K) and the Gronwall lemma to (17) gives estimate (7). Letting = 0 in (16) and integrating the resulting expression over (0, t) give estimate (8).…”

Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 99%

“…(1) was paid to the case where the dissipative term (−∆) α u t is either absent (cf. [2,3,8,16,23]) or α = 0 (weak damping, cf. [10,11]), α = 1 (structural damping, cf.…”

mentioning

confidence: 99%

“…−α−γ dτ. (57) By (7)-(8) and(18), u α0 t ∈ L 2 (0, t; V α0−1 ) ∩ C w ([0, t]; V −1 ), then, for each τ ∈ [0, t], Aw j = λ j w j , j = 1, 2, . .…”

mentioning

confidence: 99%

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Attractors and their stability on Boussinesq type equations with gentle dissipation

Yang¹,

Ding²,

Liu³

2019

Communications on Pure &Amp; Applied Analysis

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The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation: utt + ∆ 2 u + (−∆) α ut − ∆f (u) = g(x), with α ∈ (0, 1). For general bounded domain Ω ⊂ R N (N ≥ 1), we show that there exists a critical exponent pα ≡ N +2(2α−1) (N −2) + depending on the dissipative index α such that when the growth p of the nonlinearity f (u) is up to the range: 1 ≤ p < pα, (i) the weak solutions of the equations are of additionally global smoothness when t > 0; (ii) the related dynamical system possesses a global attractor Aα and an exponential attractor A α exp in natural energy space for each α ∈ (0, 1), respectively; (iii) the family of global attractors {Aα} is upper semicontinuous at each point α 0 ∈ (0, 1], i.e., for any neighborhood U of Aα 0 , Aα ⊂ U when |α − α 0 | 1. These results extend those for structural damping case: α ∈ [1, 2) in [31, 32].

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Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 99%

“…(1) was paid to the case where the dissipative term (−∆) α u t is either absent (cf. [2,3,8,16,23]) or α = 0 (weak damping, cf. [10,11]), α = 1 (structural damping, cf.…”

mentioning

confidence: 99%

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Attractors and their stability on Boussinesq type equations with gentle dissipation

Yang¹,

Ding²,

Liu³

2019

Communications on Pure &Amp; Applied Analysis

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“…On the other hand, asymptotic behavior and scattering of solutions were also extensively investigated. We refer to Linares and Scialom and Liu for results in one dimension and Cho and Ozawa for arbitrary dimension. The reciprocal problem of the scattering theory was analyzed in Farah by constructing a solution with a given scattering state.…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness and scattering of solutions to the sixth‐order Boussinesq type equation in the framework of modulation spaces

Wang

2020

Math Methods in App Sciences

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In this paper, we investigate the initial value problem for the sixth order Boussinesq type equation in the framework of modulation spaces. Under suitable conditions, we first prove that the problem has a unique local solutions and global solutions. Then scattering and stability of solutions are also discussed. The proof is mainly based on the decay properties of the solutions operator in modulation spaces and the contraction mapping principle.

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“…Later, extensive research has been carried out to study the Boussinesq equation by different views. Cho and Ozawa [3] established the global existence and the scattering of a small amplitude solution to the Cauchy problem of the equation (4). Considering the effect of damping, Varlamov [15,16] considered the following damped Boussinesq equation…”

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confidence: 99%

The initial-boundary value problems for a class of sixth order nonlinear wave equation

Xu¹,

Zhang²,

Chen³

et al. 2017

Discrete &Amp; Continuous Dynamical Systems - A

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This paper considers the initial boundary value problem of solutions for a class of sixth order 1-D nonlinear wave equations. We discuss the probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and non-global existence of solutions at three different initial energy levels, i.e., sub-critical level, critical level and sup-critical level. 1. Introduction. In this paper, we consider the initial boundary value problem (IBVP) for the following 1-D nonlinear wave equation of sixth order u tt − au xx + u xxxx + u xxxxtt + f (u x) x = 0, (x, t) ∈ (0, 1) × (0, ∞), (1) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ [0, 1], (2)

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On small amplitude solutions to the generalized Boussinesq equations

Abstract: Abstract. We study the existence and scattering of global small amplitude solutions to generalized Boussinesq (Bq) and improved modified Boussinesq (imBq) equations with nonlinear term f (u) behaving as a power u p as u → 0 in R n , n ≥ 1.

Cited by 52 publications

References 32 publications

Attractors and their stability on Boussinesq type equations with gentle dissipation

Attractors and their stability on Boussinesq type equations with gentle dissipation

Well‐posedness and scattering of solutions to the sixth‐order Boussinesq type equation in the framework of modulation spaces

The initial-boundary value problems for a class of sixth order nonlinear wave equation

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