Global existence, asymptotic stability and blow‐up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

Zhang, Hongwei; Hu, Qingying; Liu, Gongwei

doi:10.1002/mana.201700350

Cited by 3 publications

(2 citation statements)

References 53 publications

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“…First of all, for each m and the corresponding given function f m (x, t), according to the statement of Theorem 4 we have established the existence of a smoother (than in Theorem 3) unique solution u m (x, t) of initial boundary value problem ( 13)- (15) for the corresponding trapezoid Q m xt . We continue functions u m (x, t), f m (x, t) from the trapezoid Q m xt by zero to the entire triangle Q xt and denote them by ũm (x, t), fm (x, t).…”

Section: Proof Of the Theorem 1 Uniquenessmentioning

confidence: 88%

See 1 more Smart Citation

On a boundary value problem for a Boussinesq-type equation in a triangle

Jenaliyev

Kassymbekova²,

Yergaliyev³

2022

JMMCS

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Earlier, we considered an initial-boundary value problem for a one-dimensional Boussinesq-type equation in a domain that is a trapezoid, in which the theorems on its unique weak solvability in Sobolev classes were established by the methods of the theory of monotone operators. In this article, we continue research in this direction and study the issues of correct formulation of the boundary value problem for a one-dimensional Boussinesq-type equation in a degenerate domain, which is a triangle. A scalar product is proposed with the help of which the monotonicity of the main operators is shown, and uniform a priori estimates are obtained. Further, using the methods of the theory of monotone operators and a priori estimates, theorems on its unique weak solvability in Sobolev classes are established. A theorem on increasing the smoothness of a weak solution is established. In proving the smoothness enhancement theorem, we use a generalization of the classical result on compactness in Banach spaces proved by Yu.I. Dubinsky ("Weak convergence in nonlinear elliptic and parabolic equations Sbornik: Mathematics, 67 (109): 4 (1965)) in the presence of a bounded set from a semi-normed space instead of a normed one. It is also shown that the solution may have a singularity at the point of degeneracy of the domain. The order of this feature is determined, and the corresponding theorem is proved.

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Section: Proof Of the Theorem 1 Uniquenessmentioning

confidence: 88%

“…First of all, for each m and the corresponding given function f m (x, t), according to the statement of Theorem 3, we have established the existence of a unique solution u m (x, t) of initial boundary value problem ( 13)- (15).…”

Section: Proof Of Theorem 1 Existencementioning

confidence: 99%

On a boundary value problem for a Boussinesq-type equation in a triangle

Jenaliyev

Kassymbekova²,

Yergaliyev³

2022

JMMCS

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Energy decay and blow-up of solutions for a viscoelastic equation with nonlocal nonlinear boundary dissipation

Zhang

2021

Journal of Mathematical Physics

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In this paper, we consider a nonlinear viscoelastic equation with nonlocal nonlinear boundary damping and two nonlinear source terms (boundary and interior). We obtain the global existence of solution via the potential well method. By introducing suitable Lyapunov functionals, using the multiplier method, and constructing convex differential inequality, we establish an explicit and general decay rate result. This improves previous decay results concerning the problem. We also get a finite time blow-up result of solution with positive initial energy under suitable conditions on the initial data and positive memory function. This is the first result for blow-up of the problem with nonlocal nonlinear boundary damping.

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On boundary value problems for the Boussinesq-type equation with dynamic and non-dynamic boundary conditions

Jenaliyev

KASSYMBEKOVA

Yergaliyev

et al. 2023

Advances in the Theory of Nonlinear Analysis and Its Application

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The work studies boundary value problems with non-dynamic and dynamic boundary conditions for one- and two-dimensional Boussinesq-type equations in domains representing a trapezoid, triangle, "curvilinear" trapezoid, "curvilinear" triangle, truncated cone, cone, truncated "curvilinear" cone, and "curvilinear" cone. Combining the methods of the theory of monotone operators and a priori estimates, in Sobolev classes, we have established theorems on the unique weak solvability of the boundary value problems under study.

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Global existence, asymptotic stability and blow‐up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

Cited by 3 publications

References 53 publications

On a boundary value problem for a Boussinesq-type equation in a triangle

On a boundary value problem for a Boussinesq-type equation in a triangle

Energy decay and blow-up of solutions for a viscoelastic equation with nonlocal nonlinear boundary dissipation

On boundary value problems for the Boussinesq-type equation with dynamic and non-dynamic boundary conditions

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