Abstract
In this paper, we consider the wave equation with a weak internal constant delay term: u 00 (x, t) Dxu(x, t) + m1(t) u 0 (x, t) + m2(t) u 0 (x, t t) = 0 in a bounded domain. Under appropriate conditions on m1 and m2, we prove global existence of solutions by the Faedo-Galerkin method and establish a decay rate estimate for the energy using the multiplier method.
Highlights
In this paper we investigate the decay properties of solutions for the initial boundary value problem for the linear wave equation of the form
U0(x), ut(x, 0) = u1(x) ut(x, t − τ) = f0(x, t − τ) on Ω×]0, τ[, where Ω is a bounded domain in IRn, n ∈ IN∗, with a smooth boundary ∂Ω = Γ, τ > 0 is a time delay and the initial data (u0, u1, f0) belong to a suitable function space
The PDEs with time delay effects have become an active area of research since they arise in many practical problems
Summary
The authors of [13] studied the wave equation with a linear internal damping term with constant delay (τ = const in the problem (P) and determined suitable relations between μ1 and μ2, for which the stability or alternatively instability takes place. They showed that the energy is exponentially stable if μ2 < μ1 and they found a sequence of delays for which the corresponding solution of (P) will be unstable if μ2 ≥ μ1. We use the Galerkin approximation scheme and the multiplier technique to prove our results
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