Abstract
We consider in this paper the problem of asymptotic behavior of solutions for two viscoelastic wave equations with infinite memory. We show that the stability of the system holds for a much larger class of kernels and get better decay rate than the ones known in the literature. More precisely, we consider infinite memory kernels satisfying, where and are given functions. Under this very general assumption on the behavior of g at infinity and for each viscoelastic wave equation, we provide a relation between the decay rate of the solutions and the growth of g at infinity, which improves the decay rates obtained in [15, 16, 17, 19, 40]. Moreover, we drop the boundedness assumptions on the history data considered in [15, 16, 17, 40].
Highlights
In this paper, we consider the following two viscoelastic problems: +∞utt(x, t) − ∆u(x, t) +g(s)∆u(x, t − s)ds = 0u(x, t) = 0 u(x, −t) = u0(x, t), ut(x, 0) = u1(x) in Ω × R∗+, on ∂Ω × R∗+, in Ω × R+ (1.1)Copyright c 2020 The Author(s)
We consider infinite memory kernels g : R+ := [0, +∞[→ R∗+ :=]0, +∞[ satisfying g (t) ≤ −ξ(t)G(g(t)), ∀t ∈ R+, where ξ are given functions. Under this very general assumption on the behavior of g at infinity and for each viscoelastic wave equation, we provide a relation between the decay rate of the solutions and the growth of g at infinity, which improves the decay rates obtained in [15, 16, 17, 19, 40]
Where u denotes the transverse displacement of waves, ∆ is the Laplacian operator with respect to the space variable x, ut and utt denote, respectively, the first and second derivatives with respect to the time variable t, g : R+ → R∗+ is a given function representing the infinite memory kernel and satisfying some hypotheses, Ω is a bounded domain of RN, N ∈ N∗ := {1, 2, . . .}, with a smooth boundary ∂Ω, and u0 and u1 are fixed history and initial data in a suitable Hilbert space
Summary
We consider the following two viscoelastic problems:. u(x, t) = 0 u(x, −t) = u0(x, t), ut(x, 0) = u1(x) in Ω × R∗+, on ∂Ω × R∗+, in Ω × R+. U(x, t) = 0 u(x, −t) = u0(x, t), ut(x, 0) = u1(x) in Ω × R∗+, on ∂Ω × R∗+, in Ω × R+ Where u denotes the transverse displacement of waves, ∆ is the Laplacian operator with respect to the space variable x, ut and utt denote, respectively, the first and second derivatives with respect to the time variable t, g : R+ → R∗+ is a given function representing the infinite memory kernel and satisfying some hypotheses, Ω is a bounded domain of RN , N ∈ N∗ := {1, 2, . Where u denotes the transverse displacement of waves, ∆ is the Laplacian operator with respect to the space variable x, ut and utt denote, respectively, the first and second derivatives with respect to the time variable t, g : R+ → R∗+ is a given function representing the infinite memory kernel and satisfying some hypotheses, Ω is a bounded domain of RN , N ∈ N∗ := {1, 2, . . .}, with a smooth boundary ∂Ω, and u0 and u1 are fixed history and initial data in a suitable Hilbert space
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