Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

Trường, Lê Xuân; Ngoc, Le Thi Phuong; Dinh, Alain Pham Ngoc; Long, Nguyễn Thành

doi:10.1016/j.na.2011.07.015

Cited by 15 publications

(31 citation statements)

References 14 publications

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“…The aforementioned works lead to the study of the existence, blow‐up, and exponential decay estimates for a system of nonlinear wave equations associated with initial and Dirichlet boundary conditions and with nonlinear boundary conditions . In this paper, we extend the results of by combination of the methods used in with some appropriate modifications for problem considered here. Our main results are presented in three parts as follows.…”

Section: Introductionmentioning

confidence: 83%

“…Problems of this type arise in material science and physics, which have been studied by many authors. For example, we refer to and the references given therein. In these works, many interesting results about the existence, regularity, and the asymptotic behavior of solutions were obtained.…”

Section: Introductionmentioning

confidence: 99%

“…In , the existence, regularity, blow‐up, and exponential decay estimates of solutions for nonlinear wave equations associated with two‐point boundary conditions have been established. The proofs are based on the Galerkin method associated to a priori estimates, weak convergence, and compactness techniques and also by the construction of a suitable Lyapunov functional.…”

Section: Introductionmentioning

confidence: 99%

“…The proofs are based on the Galerkin method associated to a priori estimates, weak convergence, and compactness techniques and also by the construction of a suitable Lyapunov functional. The authors in proved that any weak solution with negative initial energy will blow up in finite time.…”

Section: Introductionmentioning

confidence: 99%

“…In parts 2 and 3, problems are considered with p > 2 and q = r 1 = r 2 = 2. Under some suitable conditions, by applying techniques as in with some necessary modifications and with some restrictions on the initial data, we prove that the solution of blows up in finite time. We also prove that the solution ( u ( t ), v ( t )) will exponential decay if the initial energy is positive and small via the construction of a suitable Lyapunov functional.…”

Section: Introductionmentioning

confidence: 99%

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Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

Ngoc¹,

Long

2013

Math. Meth. Appl. Sci.

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This paper is devoted to the study of a system of nonlinear equations with nonlinear boundary conditions. First, on the basis of the Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, the exponential decay property of the global solution via the construction of a suitable Lyapunov functional is presented. Copyright © 2013 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

Ngoc¹,

Long

2013

Math. Meth. Appl. Sci.

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Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2N‐point type

Triet

Ngoc

Dinh

et al. 2020

Math Methods in App Sciences

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This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2N-point type, which connect the values of an unknown function u(x,t) at x = 1, x = 0, x = i (t), and x = i (t), where 0 < 1 (t) < 2 (t) < … < N−1 (t) < 1, 0 < 1 (t) < 2 (t) < … < N−1 (t) < 1, for all t ≥ 0. First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u(t) with energy decaying exponentially as t → +∞. Finally, we present numerical results.

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Corrigendum to “Existence, blow‐up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions”

Ngoc

Long

2021

Math Methods in App Sciences

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Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

Cited by 15 publications

References 14 publications

Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2N‐point type

Corrigendum to “Existence, blow‐up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions”

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