Abstract
This work is concerned with a viscoelastic equation with a nonlinear frictional damping and a relaxation function satisfying g^{prime }(t) le -xi (t) g^{p}(t), t ge 0, 1le p<frac{3}{2}. We establish general decay rate results using the multiplier method and some properties of non-homogeneous ordinary differential inequalities. These results extend and improve many results in the literature.
Highlights
In this paper, we consider the following viscoelastic problem: ⎧ ⎪⎪⎨utt – u++∞ 0 g (s) u(t – s) ds + |ut|m–2ut = 0 in Ω × (0, +∞),⎪⎪⎩uu((xx, t) = 0 –t) =u0(x, t), ut (x, 0) =u1(x) on ∂Ω × (0, +∞), (1)in Ω × (0, +∞), where u denotes the transverse displacement of waves and Ω is a bounded domain of RN (N ≥ 1) with a smooth boundary ∂Ω, g is a positive and decreasing function and m > 1.The study of viscoelastic problems has attracted the attention of many authors, and several decay and blow-up results have been established
In Ω × (0, +∞), where u denotes the transverse displacement of waves and Ω is a bounded domain of RN (N ≥ 1) with a smooth boundary ∂Ω, g is a positive and decreasing function and m > 1
G(s)Bu(t – s) ds = 0 and introduced a new ingenuous approach for proving a more general decay result based on the properties of convex functions and the use of the generalized Young inequality
Summary
We establish general decay rate results using the multiplier method and some properties of non-homogeneous ordinary differential inequalities. The study of viscoelastic problems has attracted the attention of many authors, and several decay and blow-up results have been established. They established an exponential decay result under some restrictions on ω.
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