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

Abstract

We study the initial-boundary value problem for a nonlinear wave equation given by where 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 (ũ0, ũ1) ∈ H 1 × L 2; k, g ∈ H 1(0, T), η ≥ 0; q ≥ 2, a global existence and uniqueness theorem is proved. The proof is based on the paper by Long and Truong (Nonlinear Anal. Theory Methods Appl. Ser. A) 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, namely g = 0, f(x, t, u) = −|u| p−2 u + F(x, t), p ≥ 2, F ∈ L 1(ℝ+;L 2) ∩ L 2(ℝ+;L 2), , with σ > 0, and (ũ0, ũ1) ∈ H 1 × L 2, k ∈ H 1(ℝ+), 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)‖ + ‖u x (t)‖ decay exponentially to 0 as t → +∞. Finally, we present some numerical results.