Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition

Abstract

Abstract. We study the initial-boundary value problem for a nonlinear wave equation given bywhere η ≥ 0, q ≥ 2 are given constants and u 0 , u 1 , g, k, f are given functions.In this paper, we consider two main parts. In Part 1, under a certain local Lipschitzian condition on f with (e u 0 , ea global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part 2, the asymptotic behavior of the solution u as t → +∞ is studied, under more restrictive conditions, namelye σt F (t) 2 dt < +∞, with σ > 0, and (e u 0 , e, and some others ( · denotes the L 2 (0, 1) norm). It is proved that under these conditions, a unique solution u(t) exists on R + such that u / (t) + ux(t) decay exponentially to 0 as t → +∞. Finally, we present some numerical results.

High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

“…Finally, the exact solution and the approximate solution were illustrated numerically. In [14], Ngoc et al also obtained the same results given in [31] for the following nonlinear wave equation associated with nonlocal boundary conditions:…”

“…Although there are many studies of solution properties of viscoelastic problems, however, it seems that few works related to numerical algorithms for this type were published. In [31], Long et al proved the global existence and exponential decay of equation ( 1) associated with a mixed nonhomogeneous condition…”

A High-Order Iterative Scheme for a Nonlinear Pseudoparabolic Equation and Numerical Results

“…. , M and approximate the integral t 0 g(t − s)C u →(m) (s)ds by the trapezoidal formula (see [24], page 56), and we remark that this technique was also used in [26,27,30]. en, we obtain the following algorithm to determine the finite-difference approximate solutions of u (m) given by the 2-order iterative scheme formula (71)…”

General decay and blow-up results of solutions for a viscoelastic wave equation with Robin-Dirichlet conditions

“…It is well known that the single viscoelastic wave equation of the form with initial and boundary conditions, where Ω ⊂ R n is bounded domains with a smooth boundary ∂ Ω, has been extensively studied and many results concerning existence, nonexistence, exponential decay and blow-up in finite time have been proved (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] and references therein).…”

“…In [8], Long et al considered the following initial-boundary value problem      u tt − u xx + t 0 k(t − s)u xx ds + |u t | q−2 u t = f (x,t, u), (x,t) ∈ (0, 1) × (0, T ), u x (0,t) = u(0,t), u x (1,t) + ηu(1,t) = g(t), u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), where η ≥ 0, q ≥ 2 are given constants and u 0 , u 1 , g, k, f are given functions. The first result obtained in [8] is the unique existence of a weak solution u(t). On the other hand, in case of f (x,t, u) = −|u| p−2 u + F(x,t), the solution u(t) is exponentially decay to zero as t → +∞.…”