• PDF / 426,201 Bytes
• 19 Pages / 439.642 x 666.49 pts Page_size
• 23 Downloads / 240 Views

DOWNLOAD

REPORT

---

Decay Results for a Viscoelastic Problem with Nonlinear Boundary Feedback and Logarithmic Source Term Mohammad M. Al-Gharabli1

· Adel M. Al-Mahdi1 · Salim A. Messaoudi2

Received: 30 June 2020 / Revised: 30 June 2020 / Accepted: 17 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, g  (t) ≤ −ξ(t)G(g(t)). We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = s p and p covers the full admissible range [1, 2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature. Keywords Viscoelastic · Stability · Logarithmic Sobolev inequality · Boundary feedback · Convex functions Mathematics Subject Classification (2010) 35B35 · 35L55 · 75D05 · 74D10 · 93D20

1 Introduction In this paper, we consider the following viscoelastic problem: ⎧ t utt (t) − u(t) + u + 0 g(t − s)u(s)ds = ku ln |u|, ⎪ ⎪ ⎨ ∂u t ∂u ∂ν (t) − 0 g(t − s) ∂ν (s)ds + h(ut (t)) = 0, ⎪ u(t) = 0, ⎪ ⎩ u(x, 0) = u0 (x), ut (x, 0) = u1 (x),

in  × R+ on 1 × R+ on 0 × R+ in 

(1.1)

where u denotes the transverse displacement of waves,  is a bounded domain of RN (N ≥ 1) with a smooth boundary ∂ = 0 ∪ 1 such that 0 and 1 are closed and disjoint, with meas.(0 ) > 0, ν is the unit outer normal to ∂, the constant k is a small positive real number, and g and h are specific functions.

 Mohammad M. Al-Gharabli

[email protected]

Extended author information available on the last page of the article.

Mohammad M. Al-Gharabli, Adel M. Al-Mahdi and Salim A. Messaoudi

The importance of the viscoelastic properties of materials has been realized because of the rapid developments in rubber and plastics industry. Many advances in the studies of constitutive relations, failure theories, and life prediction of viscoelastic materials and structures were reported and reviewed in the last two decades [1]. In the absence of the logarithmic source term (k = 0), problem (1.1) has been investigated by many authors and several stability results were established. Cavalcanti et al. [2] studied (1.1) (with k = 0) and proved a global existence result for weak and strong solutions. Moreover, they gave some uniform decay rate results under some restrictive assumptions on both the kernel g and the damping function h. These restrictions had been relaxed by Cavalcanti et al. [3]; and furthermore, they established a uniform stability depending on the behavior of h near the origin and on the behavior of g at infinity. Al-Gharabli et al. [4] dis