Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions

Doronin, Gleb G.; Larkin, Nikolai A.

doi:10.1016/s0362-546x(01)00804-5

Cited by 15 publications

(19 citation statements)

References 5 publications

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“…T and ξ is defined by (9) or (11). Therefore, if V 0 ∈ H and V ∈ H , the problem (15) is formally equivalent to the following abstract evolution equation in the Hilbert space H :…”

Section: Semigroup Formulation Of the Problemmentioning

confidence: 99%

Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term

Gerbi¹,

Said‐Houari²

2012

Applied Mathematics and Computation

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In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions related to the Kelvin-Voigt damping and a delay term acting on the boundary. If the weight of the delay term in the feedback is less than the weight of the term without delay or if it is greater under an assumption between the damping factor, and the difference of the two weights, we prove the global existence of the solutions. Under the same assumptions, the exponential stability of the system is proved using an appropriate Lyapunov functional. More precisely, we show that even when the weight of the delay is greater than the weight of the damping in the boundary conditions, the strong damping term still provides exponential stability for the system.

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“…T and ξ is defined by (9) or (11). Therefore, if V 0 ∈ H and V ∈ H , the problem (15) is formally equivalent to the following abstract evolution equation in the Hilbert space H :…”

Section: Semigroup Formulation Of the Problemmentioning

confidence: 99%

Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term

Gerbi¹,

Said‐Houari²

2012

Applied Mathematics and Computation

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“…We mention only a few particular results in the one dimensional space and for a linear damping i.e. (m = 2) [16,28,7,18]. A problem related to (1) is the following:…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

Gerbi

Said‐Houari

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…In addition, G.G. Doronin et al [6] studied the following nonlinear hyperbolic problem with nonlinear second-order boundary conditions: u tt − a(u)u xx + g(u t ) = f (x, t), (x, t) ∈ (0, 1) × (0, T ), (1.4) u(0, t) = 0, u x + K(u)u tt + h(u t ) (1, t) = 0, (1.5) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ (0, 1). (1.6) It was proved that the problem (1.4)-(1.6) has global strong solution without any restrictions on the size of initial data and f by the method of Galerkin and compactness arguments.…”

Section: Introductionmentioning

confidence: 99%

Decay rate for a class of viscoelasticity equation with nonlinear damping on the boundary

Han

Xue

2008

Journal of Mathematical Analysis and Applications

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In this paper we study the equation of viscoelasticitywith the nonlinear damping term on the boundary. Making use of the so-called energy perturbation method and a comparison inequality we prove the relationship between decay rate of solutions and f (x, t), that is, they have the same exponential decay or algebraic decay for the case F (s) = s + |s| α−2 (α 2), and for the case F (s) = |s| α−2 (α > 2), they have the same algebraic decay.

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Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions

Cited by 15 publications

References 5 publications

Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term

Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

Decay rate for a class of viscoelasticity equation with nonlinear damping on the boundary

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