The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization
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The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization Mohammad Akil1,2 · Mouhammad Ghader1,3 · Ali Wehbe1 Received: 17 July 2020 / Accepted: 7 October 2020 © Sociedad Española de Matemática Aplicada 2020
Abstract In this work, we consider a system of two wave equations coupled by velocities in a onedimensional space, with one boundary fractional damping. First, we show that the system is strongly asymptotically stable if and only if the coupling parameter b of the two equations is outside a discrete set of exceptional real values. Next, we show that our system is not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using a frequency domain approach combined with the multiplier method, we prove that the energy decay rate is greatly influenced by the nature of the coupling parameter b, the arithmetic property of the wave propagation speed a and the order of the fractional damping α. Indeed, under the equal speed propagation condition, i.e., a = 1, we establish an optimal polynomial 2 2 energy decay rate of type t − 1−α if the coupling parameter b ∈ / πZ and of type t − 5−α if the coupling parameter b ∈ πZ. Furthermore, when the √ wave propagates with different speeds, √ i.e., a = 1, we prove that, for any rational number a and almost all irrational numbers a, 2 − 5−α the energy . This result still holds if √ of our system decays polynomially to zero like as t a ∈ Q, a ∈ / Q and b small enough. Keywords Coupled wave equations · Fractional boundary damping · Strong stability · Uniform stability · Polynomial stability Mathematics Subject Classification 34G10 · 35B40 · 47D03 · 93D20
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Ali Wehbe [email protected] Mohammad Akil [email protected] Mouhammad Ghader [email protected]
1
Faculty of Sciences 1, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Lebanese University, Hadath-Beirut, Lebanon
2
Insa de Rouen, LMI, 685 Avenue de l’Université, Rouen, France
3
Paris-Saclay University, L2S, 3 Rue Joliot Curie, Gif-sur-Yvette, France
123
M. Akil et al.
1 Introduction In this work, we investigate the energy decay rate of a system of coupled wave equations with only one fractional dissipation law. This system defined on (0, 1) × (0, +∞) takes the following form u tt − u x x + byt = 0, (1.1) ytt − ayx x − bu t = 0, with the following boundary conditions α,η
u(0, t) = y(0, t) = y(1, t) = 0, u x (1, t) + γ ∂t
u(1, t) = 0,
t ∈ (0, +∞) , (1.2)
in addition to the following initial conditions (u(x, 0), u t (x, 0), y(x, 0), yt (x, 0)) = (u 0 (x), u 1 (x), y0 (x), y1 (x)) , x ∈ (0, 1). (1.3) Here γ > 0, η ≥ 0, α ∈]0, 1[, a > 0 and b ∈ R∗ are constant numbers. Fractional calculus includes various extensions of the usual definition of derivative from integer to real order, including the Riemann–Liouville derivative, the Caputo derivative, the Riesz derivative, the α,η Weyl derivative, cf. [36]. In this paper, we consider the Caputo’s fractional derivative ∂t of order α ∈]0, 1[ with respect to time
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