An abstract Volterra equation with applications to linear viscoelasticity
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This article is published in Journal of Differential Equations.The article was published on 1970-05-01 and is currently open access. It has received 313 citations till now. The article focuses on the topics: Volterra integral equation & Viscoelasticity.read more
Citations
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Asymptotic stability in viscoelasticity
TL;DR: In this article, it was shown that the solutions of (1.1) are asymptotically stable provided that g(T) is a nonnegative, monotonic non-increasing and convex function such that
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Energy decay for Timoshenko systems of memory type
TL;DR: The exponential decay is proved for exponential kernels, while polynomial kernels are shown to lead to a polyn coefficients decay, and the optimality of the results is investigated.
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General decay of solutions of a viscoelastic equation
TL;DR: In this paper, the authors considered the following viscoelastic equation: u t t − Δ u + ∫ 0 t g (t − τ ) Δ u ( τ ) d τ = 0, in a bounded domain, and established a general decay result which is not necessarily of exponential or polynomial type.
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Decay estimates for second order evolution equations with memory
TL;DR: In this paper, a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms is presented. But the method is based on integral inequalities and multiplier techniques.
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Decay rates for viscoelastic plates with memory
TL;DR: In this paper, the viscoelastic plate equation is considered and it is shown that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially.
References
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Book
Ordinary differential equations
TL;DR: In this article, the Poincare-Bendixson theory is used to explain the existence of linear differential equations and the use of Implicity Function and fixed point Theorems.
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Volterra integral equations in Banach space
Avner Friedman,Marvin Shinbrot +1 more
TL;DR: In this article, the authors present results on existence, uniqueness, and differentiability of solutions in the context of W. The results on the existence and uniqueness of solutions are discussed.