Exponential decay of non-linear wave equation with a viscoelastic boundary condition

Rivera, Jaime E. Muñoz; Andrade, Doherty

doi:10.1002/(sici)1099-1476(20000110)23:1<41::aid-mma102>3.0.co;2-b

Cited by 53 publications

(10 citation statements)

References 11 publications

(5 reference statements)

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“…Also, for the homogeneous boundary condition at x = 0 and the initial conditions (1.15), Rivera and Andrade [8] studied the following nonlinear problem:…”

Section: Known Results and Motivationsmentioning

confidence: 99%

See 1 more Smart Citation

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile: Continuous dependence and low-frequency asymptotic expansion

Lê

2011

Nonlinear Analysis: Real World Applications

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“…Also, for the homogeneous boundary condition at x = 0 and the initial conditions (1.15), Rivera and Andrade [8] studied the following nonlinear problem:…”

Section: Known Results and Motivationsmentioning

confidence: 99%

“…In [8], the authors obtained the global existence of smooth solutions and the decay rate of the solutions to the problem. This improves the previous results by Qin [9,10] for the case f = 0.…”

Section: Known Results and Motivationsmentioning

confidence: 99%

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile: Continuous dependence and low-frequency asymptotic expansion

Lê

2011

Nonlinear Analysis: Real World Applications

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“…In these works, existence of solutions and exponential stabilization were proved for linear and for nonlinear equations. In contrast with the large literature for frictional dissipative, for boundary condition with memory, we have only a few works as for example [12,13,14].…”

Section: P(t) = G(t) + Hu(0t) −mentioning

confidence: 99%

On a shock problem involving a nonlinear viscoelastic bar

2005

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We treat an initial boundary value problem for a nonlinear wave equation u tt − u xx + K|u| α u + λ|u t | β u t = f (x,t) in the domain 0 < x < 1, 0 < t < T. The boundary condition at the boundary point x = 0 of the domain for a solution u involves a time convolution term of the boundary value of u at x = 0, whereas the boundary condition at the other boundary point is of the form u x (1,t) + K 1 u(1,t) + λ 1 u t (1,t) = 0 with K 1 and λ 1 given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of α = β = 0, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution (u,P) of this problem up to order N + 1 in two small parameters K, λ.

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“…For frictional dissipative boundary condition, Lasiecka and Tataru [16] investigated problem (1.1) in the absence of the viscoelastic term (g = 0) and, without imposing any growth condition on h, they proved that the energy decays as fast as the solution of an associated differential equation whose coefficients depend on the damping term. A nonlinear wave equation with viscoelastic boundary condition was also studied by Rivera and Andrade [19], and the existence and uniform decay results, under some restriction on the initial data, were established. Santos [21] considered a one-dimensional wave equation with viscoelastic boundary feedback and showed, under some assumptions on both g and g , that the dissipation is strong enough to produce exponential (polynomial) decay of the solution, provided the relaxation function also decays exponentially (polynomially), respectively.…”

Section: Introductionmentioning

confidence: 99%

A general decay result of a viscoelastic equation with past history and boundary feedback

Messaoudi

Al‐Gharabli

2014

Z. Angew. Math. Phys.

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In this paper, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and in the presence of infinite memory term. In the domain as well as on a part of the boundary, we use the multiplier method and some properties of the convex functions to prove an explicit and general decay result.Mathematics Subject Classification. 35L10 · 35L35 · 35B35 · 35B40 · 93D15.

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Exponential decay of non-linear wave equation with a viscoelastic boundary condition

Cited by 53 publications

References 11 publications

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile: Continuous dependence and low-frequency asymptotic expansion

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile: Continuous dependence and low-frequency asymptotic expansion

On a shock problem involving a nonlinear viscoelastic bar

A general decay result of a viscoelastic equation with past history and boundary feedback

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