On the decay rate of solutions of the Bresse system with Gurtin

Abstract

We consider the Cauchy problem for the one-dimensional Bresse system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the solution using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. In fact we prove that the dissipation induced by the heat conduction is very weak and produces very slow decay rates. In addition in some cases, those decay rates are of regularity-loss type. Also, we prove that there is a number (depending on the parameters of the system) that controls the decay rate of the solution and the regularity assumptions on the initial data. In addition, we show that in the absence of the frictional damping, the memory damping term is not strong enough to produce a decay rate for the solution. In fact, we show in this case, despite the fact that the energy is still dissipative, the solution does not decay at all. This result improves and extends several results, such as those in Appl. Math. Optim. (2016), to appear, Communications in Contemporary Mathematics 18(4) (2016), 1550045, Math. Methods Appl. Sci. 38(17) (2015), 3642–3652 and others.

Keywords

The Bresse system decay regularity loss heat conduction Gurtin–Pipkin law Lyapunov functional

1. Introduction

The purpose of this paper is to study the Cauchy problem of the Bresse system with Gurtin–Pipkin heat conduction for the heat flux $\begin{array}{rcl} (1.1) & \begin{matrix} φ_{t t} - {(φ_{x} - ψ - I ω)}_{x} - k_{0}^{2} I (ω_{x} - I φ) = 0, \\ ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ - I ω) + δ θ_{x} = 0, \\ ω_{t t} - k_{0}^{2} {(ω_{x} - I φ)}_{x} - I (φ_{x} - ψ - I ω) + γ ω_{t} = 0, \\ θ_{t} - \frac{1}{β} \int_{0}^{\infty} g (s) θ_{x x} (t - s) d s + δ ψ_{t x} = 0, \end{matrix} \end{array}$ completed by the initial data $\begin{array}{rcl} (1.2) & (φ, φ_{t}, ψ, ψ_{t}, ω, ω_{t}, θ) (x, 0) = (φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, w_{0}, w_{1}, θ_{0}), \end{array}$ where $(x, t) \in R \times R^{+}$ , a, I, δ, $k_{0}$ , and γ are strictly positive fixed constants. The constant $β > 0$ is equal to $1 / κ$ , where κ is the thermal conductivity as defined below. The functions $ω (x, t)$ , $φ (x, t)$ , $ψ (x, t)$ and $θ (x, t)$ are the longitudinal displacements, the vertical displacement of the beam, the rotation angle of the linear filaments material, and the temperature deviation from a reference temperature ϑ along the shear angle displacement respectively. The memory kernel $g (s)$ is a convex summable function on $[0, \infty)$ with a total mass of $\begin{matrix} \int_{0}^{\infty} g (s) d s = 1 . \end{matrix}$ Originally, the Bresse system consists of three wave equations coupled in the following way $\begin{array}{rcl} φ_{t t} - {(φ_{x} - ψ - I ω)}_{x} - k_{0}^{2} I (ω_{x} - I φ) = 0, \\ (1.3) & ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ - I ω) = 0, \\ ω_{t t} - k_{0}^{2} {(ω_{x} - I φ)}_{x} - I (φ_{x} - ψ - I ω) = 0 . \end{array}$

The Cauchy problem associated with (1.3) was first studied by Soufyane and Said-Houari in [11], where they investigated the relationship between the frictional damping terms, the wave speed of propagation, and their influence on the decay rate of the solution.

For the thermoelastic Bresse system, the initial boundary value problem was first investigated by Liu and Rao [6], where they considered the Bresse system with two different thermal dissipative mechanisms. The authors showed that the energy decays exponentially when the wave speeds of the three wave equations are equal, that is to say when $a = k_{0} = 1$ . Otherwise, if two of the wave speeds are different, the authors found polynomial decay rates depending on the boundary conditions.

The coupling between the Fourier law of heat conduction and the Bresse system (Bresse–Fourier) has been investigated by Said-Houari and Soufyane in [9] where they considered the Cauchy problem $\begin{matrix} (1.4) & \begin{matrix} φ_{t t} - {(φ_{x} - ψ - I ω)}_{x} - k_{0}^{2} I (ω_{x} - I φ) = 0, \\ ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ - I ω) + δ θ_{x} = 0, \\ ω_{t t} - k_{0}^{2} {(ω_{x} - I φ)}_{x} - I (φ_{x} - ψ - I ω) + γ ω_{t} = 0, \\ θ_{t} - k_{1} θ_{x x} + δ ψ_{t x} = 0, \end{matrix} \end{matrix}$ completed by the initial data $\begin{array}{rcl} (1.5) & (φ, φ_{t}, ψ, ψ_{t}, ω, ω_{t}, θ) (x, 0) = (φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, w_{0}, w_{1}, θ_{0}), \end{array}$ and showed that the parameter a (which is the wave speed of the second equation) controls the decay rate of the solution. More precisely, they proved that for system (1.4) and when $a \neq 1$ , the solution is of regularity-loss type and decays slowly as: $\begin{matrix} (1.6) & {‖ \partial_{x}^{k} U (t) ‖}_{L^{2}} ⩽ C {(1 + t)}^{- 1 / 12 - k / 6} ‖ U_{0} ‖_{L^{1}} + C {(1 + t)}^{- ℓ / 4} {‖ \partial_{x}^{k + ℓ} U_{0} ‖}_{L^{2}}, \end{matrix}$ while for $a = 1$ the solution is also of regularity-loss type and decays as: $\begin{matrix} (1.7) & {‖ \partial_{x}^{k} U (t) ‖}_{L^{2}} ⩽ C {(1 + t)}^{- 1 / 12 - k / 6} ‖ U_{0} ‖_{L^{1}} + C {(1 + t)}^{- ℓ / 2} {‖ \partial_{x}^{k + ℓ} U_{0} ‖}_{L^{2}}, \end{matrix}$ where k and ℓ in (1.6) and (1.7) are non-negative integers and C is a positive constant and $U = {(φ_{x} - ψ - I w, φ_{t}, a ψ_{x}, ψ_{t}, k_{0} (w_{x} - I φ), w_{t}, θ)}^{T}$ .

Recently, Said-Houari and Hamadouche [7] studied the Bresse system coupled with the Cattaneo law of heat conduction (Bresse–Cattaneo) $\begin{array}{rcl} (1.8) & \begin{matrix} φ_{t t} - {(φ_{x} - ψ - I ω)}_{x} - k_{0}^{2} I (ω_{x} - I φ) = 0, \\ ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ - I ω) + δ θ_{x} = 0, \\ ω_{t t} - k_{0}^{2} {(ω_{x} - I φ)}_{x} - I (φ_{x} - ψ - I ω) + γ ω_{t} = 0, \\ θ_{t} + q_{x} + δ ψ_{t x} = 0, \\ τ_{q} q_{t} + q + κ θ_{x} = 0, \end{matrix} \end{array}$ subjected to the initial data $\begin{array}{rcl} (1.9) & (φ, φ_{t}, ψ, ψ_{t}, ω, ω_{t}, θ, q) (x, 0) = (φ_{0}, φ_{1}, ψ_{0}, ψ_{1}, w_{0}, w_{1}, θ_{0}, q_{0}) . \end{array}$ They showed that the same parameter $\begin{matrix} (1.10) & α = (τ_{q} - 1) (1 - a^{2}) - τ_{q} δ^{2}, \end{matrix}$ obtained for Timoshenko–Cattaneo in [10] and [8] still controls the asymptotic behavior of system (1.8). See the estimates (1.23) and (1.24) below.

By sitting $I = 0$ in system (1.1), it is easy to see that the third equation is decoupled from the other equations and by ignoring that equation, then this leads eventually to the Timoshinko system $\begin{array}{rcl} φ_{t t} - {(φ_{x} - ψ)}_{x} = 0, \\ (1.11) & ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ) + δ θ_{x} = 0, \\ θ_{t} - \frac{1}{β} \int_{0}^{\infty} g (s) θ_{x x} (t - s) d s + δ ψ_{t x} = 0 . \end{array}$ This system has been recently studied by Khader and Said-Houari in [5], where they proved that the same number $α_{g}$ : $\begin{matrix} (1.12) & α_{g} : = (\frac{β}{g (0)} - 1) (1 - a^{2}) - δ^{2} \frac{β}{g (0)}, \end{matrix}$ which was used in [2], to control the behavior of the solution of the Bresse without the frictional damping $γ w_{t}$ , in a bounded domain, also plays a role in unbounded situation and affects the decay rate of the solution. Also, they showed the decay estimates of the solution for $α_{g} = 0$ and $α_{g} \neq 0$ .

The main goal of this paper is to investigate the decay rate of the solution of problem (1.1). In fact, we prove that, as in [5], the same number $α_{g}$ defined in (1.12) controls the behavior of the solution and affects its decay rate. (See Theorem 3.9 below.) More precisely, we show that the energy $\begin{matrix} (1.13) & E_{k} (t) = E_{k} (t) - \frac{1}{β} \int_{0}^{\infty} g^{'} (s) \int_{R} {| \partial_{x}^{k} η (x, t, s) |}^{2} d x d s \end{matrix}$ with $\begin{array}{rcl} E_{k} (t) & : = & \frac{1}{2} \int_{R} {{(\partial_{x}^{k} φ_{t})}^{2} + {(\partial_{x}^{k} w_{t})}^{2} + {(\partial_{x}^{k} ψ_{t})}^{2}} (x, t) d x \\ (1.14) & + \frac{1}{2} \int_{R} {{(\partial_{x}^{k} (φ_{x} - ψ - I w))}^{2} + a^{2} {(\partial_{x}^{k} ψ_{x})}^{2} + k_{0} {(\partial_{x}^{k} (w_{x} - I φ))}^{2}} (x, t) d x, \end{array}$ where $k ⩾ 0$ , and $η (x, t, s)$ is defined in (2.1) and satisfies the estimates:

For $α_{g} = 0$ , then we have $\begin{matrix} (1.15) & E_{k} (t) ⩽ C sup_{| ξ | ⩽ 1} {\hat{E} (ξ, 0)} {(1 + t)}^{- 1 / 6 - k / 3} + C e^{- c t} E_{k} (0) . \end{matrix}$

For $α_{g} \neq 0$ , we obtain $\begin{matrix} (1.16) & E_{k} (t) ⩽ C sup_{| ξ | ⩽ 1} {\hat{E} (ξ, 0)} {(1 + t)}^{- 1 / 6 - k / 3} + C {(1 + t)}^{- ℓ / 5} E_{k + ℓ} (0) . \end{matrix}$

Here

\hat{E} (ξ, 0)

is given in (3.10).

In (1.1), the heat conduction is given by the Gurtin–Pipkin law [3]: $\begin{matrix} (1.17) & q (x, t) = - κ \int_{- \infty}^{t} g (t - s) θ_{x} (s) d s, \end{matrix}$ where $g (s)$ is the heat flux relaxation kernel. The fourth equation in (1.1) (with $δ = 0$ ), can be obtained by combining the heat flux (1.17) with the energy equation of the heat as follows: $\begin{matrix} (1.18) & \{\begin{matrix} θ_{t} + q_{x} = 0, \\ q (x, t) = - κ \int_{- \infty}^{t} g (t - s) θ_{x} (s) d s . \end{matrix} \end{matrix}$ Indeed, it is clear that by taking a change of variables, we have $\begin{matrix} q (x, t) = - κ \int_{- \infty}^{t} g (t - s) θ_{x} (s) d s = - κ \int_{0}^{\infty} g (s) θ_{x} (t - s) d s . \end{matrix}$ This gives, by taking the derivative with respect to x and plugging it into the first equation in (1.18), $\begin{matrix} θ_{t} - \frac{1}{β} \int_{0}^{\infty} g (s) θ_{x x} (t - s) d s = 0, β = \frac{1}{κ} . \end{matrix}$ Many different constitutive models arise from different choices of $g (s)$ . For instance, it is clear that in (1.8), the heat conduction is given by the so-called Cattaneo or Maxwell–Cattaneo law $\begin{matrix} (1.19) & τ_{q} q_{t} (x, t) + q (x, t) = - κ θ_{x} (x, t) . \end{matrix}$ In (1.19), one can directly express $q (x, t)$ in terms of $θ (x, t)$ from this equation, but it becomes a nonlocal (in time) relationship $\begin{matrix} (1.20) & q (x, t) = - \frac{κ}{τ_{q}} \int_{- \infty}^{t} θ_{x} (s) e^{(s - t) / τ_{q}} d s . \end{matrix}$

Equation (1.20) can easily be recovered from (1.17) by assuming that $\begin{matrix} (1.21) & g (s) = \frac{1}{τ_{q}} e^{- s / τ_{q}} . \end{matrix}$

Now, the Cattaneo system $\begin{matrix} (1.22) & \{\begin{matrix} θ_{t} + q_{x} = 0, \\ τ_{q} q_{t} + q = - κ θ_{x}, \end{matrix} \end{matrix}$ gives the last two equations in (1.8). Thus, (1.8) is only a particular case of system (1.1).

For the Bresse–Cattaneo system, Said-Houari and Hamadouche proved in [7] the following estimates:

For $α = 0$ $\begin{matrix} (1.23) & {‖ \partial_{x}^{k} V (t) ‖}_{L^{2}} ⩽ C {(1 + t)}^{- 1 / 12 - k / 6} ‖ V_{0} ‖_{L^{1}} + C {(1 + t)}^{- ℓ / 2} {‖ \partial_{x}^{k + ℓ} V_{0} ‖}_{L^{2}} . \end{matrix}$

For $α \neq 0$ $\begin{matrix} (1.24) & {‖ \partial_{x}^{k} V (t) ‖}_{L^{2}} ⩽ C {(1 + t)}^{- 1 / 12 - k / 6} ‖ V_{0} ‖_{L^{1}} + C {(1 + t)}^{- ℓ / 10} {‖ \partial_{x}^{k + ℓ} V_{0} ‖}_{L^{2}}, \end{matrix}$

where

V = {(φ_{x} - ψ - I w, φ_{t}, a ψ_{x}, ψ_{t}, k_{0} (w_{x} - I φ), w_{t}, θ, q)}^{T}

with

\begin{matrix} {‖ \partial_{x}^{k} V (t) ‖}_{L^{2}}^{2} = E_{k} (t) + \frac{1}{2} \int_{R} {{(\partial_{x}^{k} θ)}^{2} + τ_{q} {(\partial_{x}^{k} q)}^{2}} d x . \end{matrix}

By sitting

τ_{q} = 0

in Eq. (1.10), we have

α = 0

is equivalent to

a = 1

If we assume that $θ_{x}$ is constant for all time, then (1.17) reduces to the classical Fourier law $\begin{matrix} (1.25) & q (x, t) = - κ θ_{x} (x, t), \end{matrix}$ which gives the heat flux q as a linear function of the temperature gradient $θ_{x}$ . The Fourier law can be also seen as the singular limit when $ε \to 0$ of the Gurtin–Pipkin law (1.17) with the kernel $\begin{matrix} g_{ε} (s) = \frac{1}{ε} g (\frac{s}{ε}), ε > 0 . \end{matrix}$

Also, the heat flux law of Jeffreys’ type $\begin{matrix} q (t) = - κ_{1} θ_{x} (t) - \frac{κ_{2}}{τ_{q}} \int_{- \infty}^{t} θ_{x} (s) e^{(s - t) / τ_{q}} d s, \end{matrix}$ can be seen by letting $\begin{matrix} g (s) = κ_{1} δ (s) + \frac{κ_{2}}{τ_{q}} e^{- s / τ_{q}} \end{matrix}$ in (1.17), where $κ_{1}$ and $κ_{2}$ are two positive constants and δ is the Dirac function. See [4] for more details. Consequently, the result in this paper improves those in [7] and [9], first, by removing the regularity loss in (1.23) and (1.7) and weaken the regularity loss in (1.24). Second, the parameter $α_{g}$ in (1.12) is more general and leads to the constant α defined in (1.10) in the particular case of $g (s)$ as in (1.21). Moreover, this result extends the one obtained recently in [5] (and references therein) to the Bresse system.

In addition, for $γ = 0$ and for $g (s)$ as in (1.21), we show that the solution of our system is not decaying at all. In fact we prove, by using the eigenvalues method, that the damping given by the memory term is not strong enough to produce any decay rate to the solution and we show (see Theorem 4.1), the estimate $\begin{matrix} (1.26) & {‖ \partial_{x}^{k} U (t) ‖}_{L^{2}} ⩽ C {‖ \partial_{x}^{k} U_{0} ‖}_{L^{2}} . \end{matrix}$ The above result is expected since for $I = 0$ , the Bresse system reduces to the Timoshenko system (1.11), with another decoupled free wave equation for ω. Although, we have proved in [5], that system (1.11) decays with the same rate as in (1.15) and (1.16), but since the free wave equation $\begin{matrix} ω_{t t} - k_{0}^{2} ω_{x x} = 0, \end{matrix}$ is undamped and decoupled from the other equations, so, the components $w_{x}$ and $w_{t}$ in the vector solution $V = {(φ_{x} - ψ - I w, φ_{t}, a ψ_{x}, ψ_{t}, k_{0} (w_{x} - I φ), w_{t}, θ, q)}^{T}$ are undamped and therefore, the solution can not decay. In fact we show in Section 4, that there are two pure imaginary eigenvalues of the fundamental matrix associated to the Fourier image of the solution.

This paper is organized as follows: in Section 2 we state the problem. Section 3 is devoted to the energy method in the Fourier space, the construction of the Lyapunov functionals, and to the proof of the main estimates of the solution in the energy space. In Section 4, we show that the solution of our system does not decay. Finally, we conclude in Section 5 by some remarks and open questions that might be worth considering in future works.

2. Statement of the problem

In this section, we state the main problem and introduce some concepts that will be used later. Following [1], we introduce the new variable $\begin{matrix} (2.1) & η (x, t, s) = \int_{0}^{s} θ (x, t - σ) d σ = \int_{t - s}^{t} θ (x, σ) d σ, s ⩾ 0, t > 0 . \end{matrix}$ Differentiating (2.1) with respect to t yields that η satisfies the supplementary equation $\begin{matrix} (2.2) & η_{t} (s) = - η_{s} (s) + θ (t), η (0) = 0, \forall t > 0, \end{matrix}$ which will be added to system (1.1). Then, we define the operator $T η = - η^{'}$ and then from (2.2), we get the following equation: $\begin{matrix} (2.3) & η_{t} = T η + θ . \end{matrix}$ Also, we define $μ (s) = - g^{'} (s)$ and $\begin{matrix} (2.4) & g (0) = \int_{0}^{\infty} μ (s) d s . \end{matrix}$ We also, assume that μ satisfies the following two assumptions:

μ is a non-negative, non-increasing and absolutely continuous function on $R^{+}$ such that $\begin{matrix} μ (0) = lim_{s \to 0} μ (s) \in (0, \infty) . \end{matrix}$

There exists $ν > 0$ such that the differential inequality $\begin{matrix} μ^{'} (s) + ν μ (s) ⩽ 0 \end{matrix}$ holds for almost every $s > 0$ . This last assumption means that $μ (s)$ is decaying exponentially.

With all these new variables, we rewrite system (1.1) as $\begin{array}{rcl} (2.5) & \begin{matrix} φ_{t t} - {(φ_{x} - ψ - I ω)}_{x} - k_{0}^{2} I (ω_{x} - I φ) = 0, \\ ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ - I ω) + δ θ_{x} = 0, \\ ω_{t t} - k_{0}^{2} {(ω_{x} - I φ)}_{x} - I (φ_{x} - ψ - I ω) + γ ω_{t} = 0, \\ θ_{t} - \frac{1}{β} \int_{0}^{\infty} g (s) θ_{x x} (t - s) d s + δ ψ_{t x} = 0, \\ η_{t} = T η + θ . \end{matrix} \end{array}$ To rewrite the above system as a first-order (with respect to t) differential system, we define new variables, as follows: $\begin{matrix} (2.6) & \begin{matrix} v = φ_{x} - ψ - I ω, u = φ_{t}, z = a ψ_{x}, \\ y = ψ_{t}, ϕ = k_{0} (ω_{x} - I φ), ρ = ω_{t} . \end{matrix} \end{matrix}$ Hence, system (2.5) takes the form: $\begin{array}{rcl} v_{t} - u_{x} + y + I ρ = 0, \\ u_{t} - v_{x} - I k_{0} ϕ = 0, \\ z_{t} - a y_{x} = 0, \\ (2.7) & y_{t} - a z_{x} - v + δ θ_{x} = 0, \\ ϕ_{t} - k_{0} ρ_{x} + I k_{0} u = 0, \\ ρ_{t} - k_{0} ϕ_{x} - I v + γ ρ = 0, \\ θ_{t} - \frac{1}{β} \int_{0}^{\infty} μ (s) η_{x x} (s) d s + δ y_{x} = 0, \\ η_{t} = T η + θ . \end{array}$ Now, we define the solution as $\begin{array}{rcl} (2.8) & U (x, t) = {(v, u, z, y, ϕ, ρ, θ, η)}^{T} (x, t) . \end{array}$ Hence, the initial conditions can be written as $\begin{array}{rcl} U (x, 0) & = & U_{0} (x) = {(v_{0}, u_{0}, z_{0}, y_{0}, ϕ_{0}, ρ_{0}, θ_{0}, η_{0})}^{T} (x) \\ (2.9) & = & {((φ_{x 0} - ψ_{0} - I w_{0}), φ_{1}, a ψ_{x 0}, ψ_{1}, k_{0} (w_{x 0} - I φ_{0}), w_{1}, θ_{0}, η_{0})}^{T} (x) . \end{array}$ Before closing this section, we introduce two lemmas, which will be used later.

Lemma 2.1.
The following inequality holds: $\begin{array}{rcl} (2.10) & {| \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s |}^{2} ⩽ g (0) \int_{0}^{\infty} μ (s) {| \hat{η} (t, s) |}^{2} d s . \end{array}$
Proof.
We have by using Holder’s inequality $\begin{array}{l} {| \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s |}^{2} & = {| \int_{0}^{\infty} {(μ (s))}^{\frac{1}{2}} {(μ (s))}^{\frac{1}{2}} \hat{η} (t, s) d s |}^{2} \\ ⩽ {| {(\int_{0}^{\infty} μ (s) d s)}^{\frac{1}{2}} {(\int_{0}^{\infty} μ (s) {(\hat{η} (t, s))}^{2} d s)}^{\frac{1}{2}} |}^{2} \\ = (\int_{0}^{\infty} μ (s) d s) \int_{0}^{\infty} μ (s) {| \hat{η} (t, s) |}^{2} d s \\ = g (0) \int_{0}^{\infty} μ (s) {| \hat{η} (t, s) |}^{2} d s . \end{array}$ This completes the proof of Lemma 2.1. □

The following lemma will be used later and it has been proved in [5]. Lemma 2.2.
For all $k ⩾ 0$ , $c ⩾ 0$ , there exists a constant $C > 0$ such that for all $t ⩾ 0$ the following estimate holds: $\begin{matrix} (2.11) & \int_{| ξ | ⩽ 1} | ξ |^{k} e^{- c | ξ |^{6} t} d ξ ⩽ C {(1 + t)}^{- (k + 1) / 6}, ξ \in R . \end{matrix}$

3. The energy method in the Fourier space

The main goal of this section is to obtain useful decay estimates of the Fourier image of the solution for system (2.7). This method will allow us to find the decay rate of the solution in the energy space by using Plancherel’s theorem together with some integral estimates, such as (2.11). To do so, we use the energy method in the Fourier space and construct appropriate Lyapunov functionals. As, we will see later on in the proof, the parameter $α_{g}$ defined in (1.12) plays a decisive role in the proof.

Applying the Fourier transform to (2.7), we get $\begin{array}{rcl} (3.1) & {\hat{v}}_{t} - i ξ \hat{u} + \hat{y} + I \hat{ρ} = 0, \\ (3.2) & {\hat{u}}_{t} - i ξ \hat{v} - I k_{0} \hat{ϕ} = 0, \\ (3.3) & {\hat{z}}_{t} - i ξ a \hat{y} = 0, \\ (3.4) & {\hat{y}}_{t} - i ξ a \hat{z} - \hat{v} + δ i ξ \hat{θ} = 0, \\ (3.5) & {\hat{ϕ}}_{t} - i ξ k_{0} \hat{ρ} + I k_{0} \hat{u} = 0, \\ (3.6) & {\hat{ρ}}_{t} - i ξ k_{0} \hat{ϕ} - I \hat{v} + γ \hat{ρ} = 0, \\ (3.7) & \hat{θ_{t}} + \frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s + δ i ξ \hat{y} = 0, \\ (3.8) & {\hat{η}}_{t} = T \hat{η} + \hat{θ} . \end{array}$ Together with the initial data, written in terms of the solution vector $\hat{U} (ξ, t) = {(\hat{v}, \hat{u}, \hat{z}, \hat{y}, \hat{ϕ}, \hat{ρ}, \hat{θ}, \hat{η})}^{T} \times (ξ, t)$ , as $\begin{array}{rcl} (3.9) & \hat{U} (ξ, 0) = {\hat{U}}_{0} (ξ) . \end{array}$ The energy functional $\hat{E} (ξ, t)$ associated to the system (3.1)–(3.8) is defined as follows: $\begin{matrix} (3.10) & \hat{E} (ξ, t) = \frac{1}{2} {| \hat{v} |^{2} + | \hat{u} |^{2} + | \hat{z} |^{2} + | \hat{y} |^{2} + | \hat{ϕ} |^{2} + | \hat{θ} |^{2} + | \hat{ρ} |^{2} + \frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) {| \hat{η} (t, s) |}^{2} d s} . \end{matrix}$

Lemma 3.1.
Let $\hat{U} (ξ, t)$ be the solution of ( 3.1 )–( 3.9 ), then the energy $\hat{E} (ξ, t)$ given by ( 3.10 ) is a non-increasing function and satisfies, for all $t ⩾ 0$ , $\begin{matrix} (3.11) & \frac{d}{d t} \hat{E} (ξ, t) = \frac{ξ^{2}}{2 β} \int_{0}^{\infty} μ^{'} (s) {| \hat{η} (t, s) |}^{2} d s - γ | \hat{ρ} |^{2} . \end{matrix}$
Proof.
Multiplying Eq. (3.1) by $\bar{\hat{v}}$ , Eq. (3.2) by $\bar{\hat{u}}$ , Eq. (3.3) by $\bar{\hat{z}}$ , Eq. (3.4) by $\bar{\hat{y}}$ , Eq. (3.5) by $\bar{\hat{ϕ}}$ , Eq. (3.6) by $\bar{\hat{ρ}}$ and Eq. (3.7) by $\bar{\hat{θ}}$ , summing up the results and taking the real part, we get $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (| \hat{v} |^{2} + | \hat{u} |^{2} + | \hat{z} |^{2} + | \hat{y} |^{2} + | \hat{ϕ} |^{2} + | \hat{ρ} |^{2} + | \hat{θ} |^{2}) \\ (3.12) & = - Re {\frac{ξ^{2}}{β} \bar{\hat{θ}} (ξ, t) \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) d s} - γ | \hat{ρ} |^{2} . \end{array}$ Taking the conjugate of Eq. (3.8), then multiplying the resulting equation by $μ (s) \hat{η} (ξ, t, s)$ and taking the integration with respect to s we obtain $\begin{matrix} \begin{matrix} \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) {\bar{\hat{η}}}_{t} (ξ, t, s) d s \\ = - \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) {\bar{\hat{η}}}_{s} (ξ, t, s) d s + \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) \bar{\hat{θ}} (ξ, t) d s . \end{matrix} \end{matrix}$ Hence, we have $\begin{array}{rcl} (3.13) & \begin{matrix} - Re {\frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) \bar{\hat{θ}} (ξ, t) d s} \\ = - Re {\frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) {\bar{\hat{η}}}_{t} (ξ, t, s) d s} - Re {\frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) {\bar{\hat{η}}}_{s} (ξ, t, s) d s} \\ = - \frac{1}{2} \frac{d}{d t} {\frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) {| \hat{η} (ξ, t, s) |}^{2} d s} - Re {\frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) \hat{η} (ξ, t, s) {\bar{\hat{η}}}_{s} (ξ, t, s) d s} . \end{matrix} \end{array}$ Integrating the second term on the right-hand side of (3.13) by parts and using the assumption (M1) together with (2.1), we have $\begin{matrix} - \frac{ξ^{2}}{β} \int_{0}^{\infty} μ (s) {\bar{\hat{η}}}_{s} (ξ, t, s) \hat{η} (ξ, t, s) d s = \frac{ξ^{2}}{2 β} \int_{0}^{\infty} μ^{'} (s) {| \hat{η} (ξ, t, s) |}^{2} d s . \end{matrix}$ Hence, collecting (3.12) and (3.13), then (3.11) holds. □
Proposition 3.2.
Let $\hat{U} (ξ, t) = (\hat{v}, \hat{u}, \hat{z}, \hat{y}, \hat{θ}, \hat{η})$ be the solution of ( 3.1 )–( 3.9 ) and let $\begin{matrix} (3.14) & α_{g} = (\frac{β}{g (0)} - 1) (1 - a^{2}) - \frac{β}{g (0)} δ^{2} . \end{matrix}$ Then, there exist two positive constants, C and c, such that for all $t ⩾ 0$ : $\begin{matrix} (3.15) & \hat{E} (ξ, t) ⩽ C \hat{E} (ξ, 0) e^{- c ρ (ξ) t}, \end{matrix}$ where $\begin{matrix} (3.16) & ρ (ξ) = \{\begin{matrix} \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}}, & if α_{g} = 0, \\ \frac{ξ^{6}}{{(1 + ξ^{2})}^{8}}, & if α_{g} \neq 0 . \end{matrix} \end{matrix}$

We are going to prove Proposition 3.2 by means of several lemmas. In order to do this, we need to build some functionals that give dissipation terms for all the first six components in the energy functional (3.10). The dissipation terms of the two last components in (3.10) are given by (3.11) and the assumption (M2).

First, define the functional $\begin{matrix} (3.17) & B_{1} (ξ, t) = Re (I i ξ \hat{y} \bar{\hat{z}}) + Re (i ξ \hat{z} \bar{\hat{ρ}}) . \end{matrix}$ Then, we have the following lemma.
Lemma 3.3.
Let $\hat{U} (ξ, t)$ be the solution of ( 3.1 )–( 3.9 ). Then, the functional $B_{1} (ξ, t)$ satisfies for all $t ⩾ 0$ , $\begin{array}{rcl} \frac{d}{d t} B_{1} (ξ, t) + a I ξ^{2} (| \hat{z} |^{2} - | \hat{y} |^{2}) \\ (3.18) & = Re (I δ ξ^{2} \hat{θ} \bar{\hat{z}}) - Re (a ξ^{2} \hat{y} \bar{\hat{ρ}}) + Re (k_{0} ξ^{2} \hat{ϕ} \bar{\hat{z}}) + Re (γ i ξ \hat{ρ} \bar{\hat{z}}) . \end{array}$
Proof.
Multiplying Eq. (3.4) by $I i ξ \bar{\hat{z}}$ and Eq. (3.3) by $- I i ξ \bar{\hat{y}}$ , adding the resulting equations and taking the real part, we get $\begin{matrix} (3.19) & \frac{d}{d t} Re (I i ξ \hat{y} \bar{\hat{z}}) + I ξ^{2} a (| \hat{z} |^{2} - | \hat{y} |^{2}) = Re (I i ξ \hat{v} \bar{\hat{z}}) + Re (I δ ξ^{2} \hat{θ} \bar{\hat{z}}) . \end{matrix}$ Now, multiplying Eq. (3.3) by $i ξ \bar{\hat{ρ}}$ and Eq. (3.6) by $- i ξ \bar{\hat{z}}$ , adding the resulting equations and taking the real part, we obtain $\begin{matrix} (3.20) & \frac{d}{d t} Re (i ξ \hat{z} \bar{\hat{ρ}}) = - Re (a ξ^{2} \hat{y} \bar{\hat{ρ}}) + Re (k_{0} ξ^{2} \hat{ϕ} \bar{\hat{z}}) - Re (I i ξ \hat{v} \bar{\hat{z}}) + Re (γ i ξ \hat{ρ} \bar{\hat{z}}) . \end{matrix}$ Adding equations (3.19) and (3.20) then (3.18) holds. □

Second, following [8], we define the functional $\begin{matrix} (3.21) & \begin{matrix} B_{2} (ξ, t) = & Re {- \frac{β}{g (0)} \bar{\hat{v}} \hat{y} - \frac{β}{g (0)} a \hat{u} \bar{\hat{z}} + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) δ \hat{θ} \bar{\hat{u}}} \\ + \frac{1 - a^{2}}{δ g (0)} Re (i ξ \int_{0}^{\infty} μ (s) \hat{η} (s) \bar{\hat{v}} d s) . \end{matrix} \end{matrix}$ Then, we have the following lemma.
Lemma 3.4.
Let $\hat{U} (ξ, t)$ be the solution of ( 3.1 )–( 3.9 ). Then, the functional $B_{2} (ξ, t)$ satisfies for all $t ⩾ 0$ , the identity $\begin{array}{rcl} \frac{d}{d t} B_{2} (ξ, t) - \frac{β}{g (0)} | \hat{y} |^{2} + \frac{β}{g (0)} | \hat{v} |^{2} \\ = α_{g} Re (i ξ \bar{\hat{u}} \hat{y}) + \frac{α_{g}}{δ β} Re (ξ^{2} \bar{\hat{u}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) \\ + \frac{β}{g (0)} Re (I \hat{ρ} \bar{\hat{y}}) + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) Re (I k_{0} δ \bar{\hat{θ}} \hat{ϕ}) - \frac{β}{g (0)} Re (I k_{0} a \hat{ϕ} \bar{\hat{z}}) \\ + \frac{1 - a^{2}}{δ g (0)} Re (i ξ \int_{0}^{\infty} μ (s) \hat{y} \bar{\hat{η}} (t, s) d s) + \frac{1 - a^{2}}{δ g (0)} Re (i ξ \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{v}} d s) \\ (3.22) & + \frac{1 - a^{2}}{δ g (0)} Re (i ξ I \int_{0}^{\infty} μ (s) \hat{ρ} \bar{\hat{η}} (t, s) d s), \end{array}$ where $α_{g}$ is defined in ( 1.12 ).
Proof.
Multiplying Eq. (3.1) by $- \bar{\hat{y}}$ and Eq. (3.4) by $- \bar{\hat{v}}$ adding the resulting equations and taking the real part, we get $\begin{matrix} (3.23) & - \frac{d}{d t} Re (\hat{y} \bar{\hat{v}}) - | \hat{y} |^{2} + | \hat{v} |^{2} = - Re (i ξ \hat{u} \bar{\hat{y}}) - Re (i ξ a \hat{z} \bar{\hat{v}}) + Re (δ i ξ \hat{θ} \bar{\hat{v}}) + Re (I \hat{ρ} \bar{\hat{y}}) . \end{matrix}$ Multiplying Eq. (3.2) by $- a \bar{\hat{z}}$ and Eq. (3.3) by $- a \bar{\hat{u}}$ , we find as before $\begin{matrix} (3.24) & - \frac{d}{d t} Re (a \hat{u} \bar{\hat{z}}) = - Re (a i ξ \hat{v} \bar{\hat{z}}) - Re (I k_{0} a \hat{ϕ} \bar{\hat{z}}) - Re (a^{2} i ξ \hat{y} \bar{\hat{u}}) . \end{matrix}$ Furthermore, multiplying Eq. (3.7) by $δ \bar{\hat{u}}$ and Eq. (3.2) by $δ \bar{\hat{θ}}$ and taking the real part, we have $\begin{array}{rcl} \frac{d}{d t} Re (δ \hat{θ} \bar{\hat{u}}) & = & - Re (\frac{δ ξ^{2}}{β} \bar{\hat{u}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) \\ (3.25) & - Re (δ^{2} i ξ \bar{\hat{u}} \hat{y}) + Re (δ i ξ \bar{\hat{θ}} \hat{v}) + Re (I k_{0} δ \bar{\hat{θ}} \hat{ϕ}) . \end{array}$ Now, computing $\frac{β}{g (0)} (3.23) + \frac{β}{g (0)} (3.24) + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) (3.25)$ , we obtain (after collecting the similar terms) $\begin{array}{rcl} \frac{d}{d t} F (ξ, t) - \frac{β}{g (0)} | \hat{y} |^{2} + \frac{β}{g (0)} | \hat{v} |^{2} \\ = - (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) Re (\frac{δ ξ^{2}}{β} \bar{\hat{u}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) \\ + α_{g} Re (i ξ \bar{\hat{u}} \hat{y}) + \frac{1 - a^{2}}{δ} Re (i ξ \bar{\hat{θ}} \hat{v}) + \frac{β}{g (0)} Re (I \hat{ρ} \bar{\hat{y}}) \\ (3.26) & + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) Re (I k_{0} δ \bar{\hat{θ}} \hat{ϕ}) - \frac{β}{g (0)} Re (I k_{0} a \hat{ϕ} \bar{\hat{z}}), \end{array}$ where $\begin{matrix} F (ξ, t) = Re {- \frac{β}{g (0)} \bar{\hat{v}} \hat{y} - \frac{β}{g (0)} a \hat{u} \bar{\hat{z}} + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) δ \bar{\hat{θ}} \hat{u}} \end{matrix}$ and $\begin{matrix} \begin{matrix} α_{g} & = {\frac{β}{g (0)} - \frac{β}{g (0)} a^{2} - δ^{2} (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)})} \\ = (\frac{β}{g (0)} - 1) (1 - a^{2}) - δ^{2} \frac{β}{g (0)} . \end{matrix} \end{matrix}$ Multiplying Eq. (3.1) by $- i ξ μ (s) \bar{\hat{η}} (t, s)$ and Eq. (3.8) by $i ξ μ (s) \bar{\hat{v}}$ respectively, adding the results and taking the integration with respect to s and then taking the real part, we have $\begin{array}{rcl} \frac{d}{d t} Re (i ξ \bar{\hat{v}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) \\ = Re (ξ^{2} \int_{0}^{\infty} μ (s) \hat{u} \bar{\hat{η}} (t, s) d s) + Re (i ξ \int_{0}^{\infty} μ (s) \hat{y} \bar{\hat{η}} (t, s) d s) \\ + Re (i ξ \int_{0}^{\infty} μ (s) T \hat{η} \bar{\hat{v}} d s) + Re (i ξ I \int_{0}^{\infty} μ (s) \hat{ρ} \bar{\hat{η}} (t, s) d s) \\ (3.27) & + Re (i ξ \int_{0}^{\infty} μ (s) \hat{θ} \bar{\hat{v}} d s) . \end{array}$ Furthermore, using (2.4) the last term in (3.27) can be written as $\begin{matrix} \begin{matrix} Re (i ξ \int_{0}^{\infty} μ (s) \hat{θ} \bar{\hat{v}} d s) & = Re (i ξ \hat{θ} \bar{\hat{v}} \int_{0}^{\infty} μ (s) d s) \\ = - g (0) Re (i ξ \bar{\hat{θ}} \hat{v}) . \end{matrix} \end{matrix}$ While, integration by parts leads to $\begin{matrix} Re (i ξ \int_{0}^{\infty} μ (s) T \hat{η} \bar{\hat{v}} d s) = Re (i ξ \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{v}} d s) . \end{matrix}$ Hence, (3.27) can be written as $\begin{array}{rcl} \frac{d}{d t} Re (i ξ \bar{\hat{v}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) \\ = Re (ξ^{2} \int_{0}^{\infty} μ (s) \hat{u} \bar{\hat{η}} (t, s) d s) + Re (i ξ \int_{0}^{\infty} μ (s) \hat{y} \bar{\hat{η}} (t, s) d s) \\ + Re (i ξ \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{v}} d s) + Re (i ξ I \int_{0}^{\infty} μ (s) \hat{ρ} \bar{\hat{η}} (t, s) d s) \\ (3.28) & - g (0) Re (i ξ \bar{\hat{θ}} \hat{v}) . \end{array}$ Computing $(3.26) + \frac{1 - a^{2}}{δ g (0)} (3.28)$ , we have $\begin{matrix} (3.29) & \begin{matrix} \frac{d}{d t} B_{2} (ξ, t) - \frac{β}{g (0)} | \hat{y} |^{2} + \frac{β}{g (0)} | \hat{v} |^{2} \\ = - \frac{δ}{β} (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) Re (ξ^{2} \bar{\hat{u}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) - \frac{β}{g (0)} Re (I k_{0} a \hat{ϕ} \bar{\hat{z}}) \\ + α_{g} Re (i ξ \bar{\hat{u}} \hat{y}) + \frac{β}{g (0)} Re (I \hat{ρ} \bar{\hat{y}}) + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) Re (I k_{0} δ \bar{\hat{θ}} \hat{ϕ}) \\ + \frac{1 - a^{2}}{δ g (0)} Re (ξ^{2} \bar{\hat{u}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) + \frac{1 - a^{2}}{δ g (0)} Re (i ξ \int_{0}^{\infty} μ (s) \hat{y} \bar{\hat{η}} (t, s) d s) \\ + \frac{1 - a^{2}}{δ g (0)} Re (i ξ \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{v}} d s) + \frac{1 - a^{2}}{δ g (0)} Re (i ξ I \int_{0}^{\infty} μ (s) \hat{ρ} \bar{\hat{η}} (t, s) d s) \end{matrix} \end{matrix}$ which is exactly (3.22). This completes the proof of Lemma 3.4. □

Now, we define the functional $\begin{matrix} (3.30) & B_{3} (ξ, t) = a I B_{1} (ξ, t) + ξ^{2} B_{2} (ξ, t) . \end{matrix}$ Then, by taking the derivative of (3.30) with respect to t and using (3.18) together with (3.22), we have $\begin{array}{rcl} \frac{d}{d t} B_{3} (ξ, t) + a^{2} I^{2} ξ^{2} (| \hat{z} |^{2} - | \hat{y} |^{2}) - \frac{β}{g (0)} ξ^{2} | \hat{y} |^{2} + \frac{β}{g (0)} ξ^{2} | \hat{v} |^{2} \\ = Re (a I^{2} δ ξ^{2} \hat{θ} \bar{\hat{z}}) + (\frac{β}{g (0)} - a^{2}) Re (I ξ^{2} \hat{y} \bar{\hat{ρ}}) + Re (a I γ i ξ \hat{ρ} \bar{\hat{z}}) \\ + (1 - \frac{β}{g (0)}) Re (a I k_{0} ξ^{2} \hat{ϕ} \bar{\hat{z}}) + α_{g} Re (i ξ^{3} \bar{\hat{u}} \hat{y}) \\ + \frac{α_{g}}{δ β} Re (ξ^{4} \bar{\hat{u}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) Re (I k_{0} δ ξ^{2} \bar{\hat{θ}} \hat{ϕ}) \\ + \frac{1 - a^{2}}{δ g (0)} Re (i ξ^{3} \int_{0}^{\infty} μ (s) \hat{y} \bar{\hat{η}} (t, s) d s) + \frac{1 - a^{2}}{δ g (0)} Re (i ξ^{3} \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{v}} d s) \\ (3.31) & + \frac{1 - a^{2}}{δ g (0)} Re (i ξ^{3} I \int_{0}^{\infty} μ (s) \hat{ρ} \bar{\hat{η}} (t, s) d s) . \end{array}$ Third, define the functional $\begin{matrix} (3.32) & B_{5} (ξ, t) = Re (- i ξ \hat{ϕ} \bar{\hat{ρ}}) + I Re (- i ξ \hat{y} \bar{\hat{ϕ}}) . \end{matrix}$ Then, we have the following lemma
Lemma 3.5.
Let $\hat{U} (ξ, t)$ be the solution of ( 3.1 )–( 3.9 ). The functional $B_{5} (ξ, t)$ satisfies $\begin{matrix} (3.33) & \begin{matrix} \frac{d}{d t} B_{5} (ξ, t) + (k_{0} - ϵ_{2}) ξ^{2} | \hat{ϕ} |^{2} ⩽ & ϵ_{2}^{'} ξ^{2} | \hat{u} |^{2} + C (ϵ_{2}) ξ^{2} (| \hat{θ} |^{2} + | \hat{z} |^{2}) \\ + C (ϵ_{2}, ϵ_{2}^{'}) (1 + ξ^{2}) (| \hat{ρ} |^{2} + | \hat{y} |^{2}) . \end{matrix} \end{matrix}$
Proof.
Multiplying Eq. (3.5) by $- i ξ \bar{\hat{ρ}}$ and Eq. (3.6) by $i ξ \bar{\hat{ϕ}}$ , and taking the real part then adding the resulting equations, we get $\begin{matrix} (3.34) & \frac{d}{d t} Re (- i ξ \hat{ϕ} \bar{\hat{ρ}}) - ξ^{2} k_{0} | \hat{ρ} |^{2} + ξ^{2} k_{0} | \hat{ϕ} |^{2} = Re (I k_{0} i ξ \hat{u} \bar{\hat{ρ}}) + Re (I i ξ \hat{v} \bar{\hat{ϕ}}) - Re (γ i ξ \hat{ρ} \bar{\hat{ϕ}}) . \end{matrix}$ Multiplying Eq. (3.4) by $- i ξ \bar{\hat{ϕ}}$ and Eq. (3.5) by $i ξ \bar{\hat{y}}$ , then we get as above $\begin{matrix} (3.35) & \frac{d}{d t} Re (- i ξ \hat{y} \bar{\hat{ϕ}}) = Re (a ξ^{2} \hat{z} \bar{\hat{ϕ}}) - Re (i ξ \hat{v} \bar{\hat{ϕ}}) - Re (δ ξ^{2} \hat{θ} \bar{\hat{ϕ}}) - Re (ξ^{2} k_{0} \hat{ρ} \bar{\hat{y}}) - Re (I k_{0} i ξ \hat{u} \bar{\hat{y}}) . \end{matrix}$ Adding Eq. (3.34) to $I \times (3.35)$ , we have $\begin{matrix} (3.36) & \begin{matrix} \frac{d}{d t} B_{5} (ξ, t) - ξ^{2} k_{0} | \hat{ρ} |^{2} + ξ^{2} k_{0} | \hat{ϕ} |^{2} = & Re (I k_{0} i ξ \hat{u} \bar{\hat{ρ}}) - Re (γ i ξ \hat{ρ} \bar{\hat{ϕ}}) + Re (I a ξ^{2} \hat{z} \bar{\hat{ϕ}}) \\ - Re (I δ ξ^{2} \hat{θ} \bar{\hat{ϕ}}) - Re (I ξ^{2} k_{0} \hat{ρ} \bar{\hat{y}}) - Re (I^{2} k_{0} i ξ \hat{u} \bar{\hat{y}}) . \end{matrix} \end{matrix}$ Applying Young’s inequality, we get, for arbitrary $ϵ_{2}, ϵ_{2}^{'} > 0$ , $\begin{matrix} \begin{matrix} | Re (I k_{0} i ξ \hat{u} \bar{\hat{ρ}}) | ⩽ \frac{ϵ_{2}^{'}}{2} ξ^{2} | \hat{u} |^{2} + C (ϵ_{2}^{'}) | \hat{ρ} |^{2}, \\ | Re (γ i ξ \hat{ρ} \bar{\hat{ϕ}}) | ⩽ \frac{ϵ_{2}}{3} ξ^{2} | \hat{ϕ} |^{2} + C (ϵ_{2}) | \hat{ρ} |^{2}, \\ | Re (I a ξ^{2} \hat{z} \bar{\hat{ϕ}}) | ⩽ \frac{ϵ_{2}}{3} ξ^{2} | \hat{ϕ} |^{2} + C (ϵ_{2}) ξ^{2} | \hat{z} |^{2}, \\ | Re (I δ ξ^{2} \hat{θ} \bar{\hat{ϕ}}) | ⩽ \frac{ϵ_{2}}{3} ξ^{2} | \hat{ϕ} |^{2} + C (ϵ_{2}) ξ^{2} | \hat{θ} |^{2}, \\ | Re (I ξ^{2} k_{0} \hat{ρ} \bar{\hat{y}}) | ⩽ ϵ_{2} ξ^{2} | \hat{y} |^{2} + C (ϵ_{2}) ξ^{2} | \hat{ρ} |^{2}, \\ | Re (I^{2} k_{0} i ξ \hat{u} \bar{\hat{y}}) | ⩽ \frac{ϵ_{2}^{'}}{2} ξ^{2} | \hat{u} |^{2} + C (ϵ_{2}^{'}) | \hat{y} |^{2} . \end{matrix} \end{matrix}$ Plugging the above estimates into Eq. (3.36), then Eq. (3.33) is fulfilled and this finishes the proof of lemma 3.5. □

Fourth, define the functional $\begin{matrix} (3.37) & B_{6} (ξ, t) = Re (i ξ \hat{v} \bar{\hat{u}}) - I Re (\hat{v} \bar{\hat{ρ}}) . \end{matrix}$ Then, we have the following lemma.
Lemma 3.6.
Let $\hat{U} (ξ, t)$ be the solution of ( 3.1 )–( 3.9 ). The functional $B_{6} (ξ, t)$ satisfies $\begin{array}{rcl} \frac{d}{d t} B_{6} (ξ, t) + ξ^{2} (1 - ϵ_{3}) | \hat{u} |^{2} + (I^{2} - \frac{3}{2} ϵ_{3}^{'}) | \hat{v} |^{2} \\ (3.38) & ⩽ ξ^{2} | \hat{v} |^{2} + C (ϵ_{3}, ϵ_{3}^{'}) (| \hat{y} |^{2} + | \hat{ρ} |^{2}) + C (ϵ_{3}^{'}) ξ^{2} | \hat{ϕ} |^{2} . \end{array}$
Proof.
Multiplying Eq. (3.1) by $i ξ \bar{\hat{u}}$ and Eq. (3.2) by $- i ξ \bar{\hat{v}}$ , adding the results and taking the real part, we get $\begin{array}{rcl} \frac{d}{d t} Re (i ξ \hat{v} \bar{\hat{u}}) + ξ^{2} (| \hat{u} |^{2} - | \hat{v} |^{2}) \\ (3.39) & = - Re (i ξ \hat{y} \bar{\hat{u}}) - Re (i ξ I \hat{ρ} \bar{\hat{u}}) - Re (i ξ k_{0} I \hat{ϕ} \bar{\hat{v}}) . \end{array}$

Multiplying Eq. (3.1) by $- \bar{\hat{ρ}}$ and Eq. (3.6) by $- \bar{\hat{v}}$ , then taking the real part after adding the two results, we obtain $\begin{array}{rcl} - \frac{d}{d t} Re (\hat{v} \bar{\hat{ρ}}) + I | \hat{v} |^{2} - I | \hat{ρ} |^{2} \\ (3.40) & = - Re (i ξ \bar{\hat{ρ}} \hat{u}) + Re (\hat{y} \bar{\hat{ρ}}) - Re (i ξ k_{0} \hat{ϕ} \bar{\hat{v}}) + γ Re (\bar{\hat{v}} \hat{ρ}) . \end{array}$ Summing up $(3.39) + I \times (3.40)$ , we get $\begin{array}{rcl} \frac{d}{d t} B_{6} (ξ, t) + ξ^{2} (| \hat{u} |^{2} - | \hat{v} |^{2}) + I^{2} | \hat{v} |^{2} - I^{2} | \hat{ρ} |^{2} \\ = - Re (i ξ \hat{y} \bar{\hat{u}}) - 2 Re (i ξ k_{0} I \hat{ϕ} \bar{\hat{v}}) + I Re (\hat{y} \bar{\hat{ρ}}) + I γ Re (\bar{\hat{v}} \hat{ρ}) . \end{array}$ Applying Young’s inequality, we get for any $ϵ_{3}, ϵ_{3}^{'} > 0$ , $\begin{matrix} \begin{matrix} | Re (i ξ \hat{y} \bar{\hat{u}}) | ⩽ ϵ_{3} ξ^{2} | \hat{u} |^{2} + C (ϵ_{3}) | \hat{y} |^{2}, \\ | Re (i ξ k_{0} I \hat{ϕ} \bar{\hat{v}}) | ⩽ \frac{ϵ_{3}^{'}}{2} | \hat{v} |^{2} + C (ϵ_{3}^{'}) ξ^{2} | \hat{ϕ} |^{2}, \\ | Re (\hat{y} \bar{\hat{ρ}}) | ⩽ ϵ_{3}^{'} | \hat{ρ} |^{2} + C (ϵ_{3}^{'}) | \hat{y} |^{2}, \\ | Re (\hat{ρ} \bar{\hat{v}}) | ⩽ \frac{ϵ_{3}^{'}}{2} | \hat{v} |^{2} + C (ϵ_{3}^{'}) | \hat{ρ} |^{2}, \end{matrix} \end{matrix}$ then (3.38) holds by plugging the above inequalities into the last identity. □

Fifth, define the functional $\begin{matrix} (3.41) & B_{7} (ξ, t) = - Re (i ξ \hat{y} \bar{\hat{θ}}) . \end{matrix}$ Then, we have the following lemma.
Lemma 3.7.
Let $\hat{U} (ξ, t)$ be the solution of ( 3.1 )–( 3.9 ). The functional $B_{7} (ξ, t)$ satisfies $\begin{matrix} (3.42) & \begin{matrix} \frac{d}{d t} B_{7} (ξ, t) + (δ - ϵ_{4}) ξ^{2} | \hat{y} |^{2} ⩽ & ϵ_{4}^{'} ξ^{2} | \hat{z} |^{2} + C (ϵ_{4}^{'}) (1 + ξ^{2}) | \hat{θ} |^{2} + ϵ_{4}^{'} ξ^{2} | \hat{v} |^{2} \\ + C (ϵ_{4}) ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s . \end{matrix} \end{matrix}$
Proof.
Multiplying Eq. (3.4) by $i ξ \bar{\hat{θ}}$ and Eq. (3.7) by $- i ξ \bar{\hat{y}}$ , and taking the real part then add the resulting equations, we get $\begin{matrix} (3.43) & \begin{matrix} - \frac{d}{d t} Re (i ξ \bar{\hat{y}} \hat{θ}) - δ ξ^{2} | \hat{θ} |^{2} + δ ξ^{2} | \hat{y} |^{2} = & - Re (a ξ^{2} \hat{z} \bar{\hat{θ}}) + Re (i ξ \hat{v} \bar{\hat{θ}}) \\ + Re (i \frac{ξ^{3}}{β} \bar{\hat{y}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) . \end{matrix} \end{matrix}$ Applying Young’s inequality we obtain for arbitrary $ϵ_{4}^{'} > 0$ , $\begin{array}{l} (3.44) & \begin{matrix} | - Re (a ξ^{2} \hat{z} \bar{\hat{θ}}) | ⩽ ϵ_{4}^{'} ξ^{2} | \hat{z} |^{2} + C (ϵ_{4}^{'}) ξ^{2} | \hat{θ} |^{2}, \\ | Re (i ξ \hat{v} \bar{\hat{θ}}) | ⩽ ϵ_{4}^{'} ξ^{2} | \hat{v} |^{2} + C (ϵ_{4}^{'}) | \hat{θ} |^{2} . \end{matrix} \end{array}$ On the other hand, using Young’s inequality and Lemma 2.1, we have $\begin{array}{l} | Re (i \frac{ξ^{3}}{β} \bar{\hat{y}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) | & ⩽ ϵ_{4} ξ^{2} | \hat{y} |^{2} + C (ϵ_{4}) ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s, \end{array}$ where $ϵ_{4}$ is an arbitrary positive constant. Therefore, plugging the above estimates into (3.43), then (3.42) is fulfilled. □

Sixth, define the functional $\begin{matrix} (3.45) & B_{8} (ξ, t) = - Re (ξ^{2} \hat{θ} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) d s) . \end{matrix}$ Then we have the following lemma.
Lemma 3.8.
Let $\hat{U} (ξ, t)$ be the solution of ( 3.1 )–( 3.9 ). The functional $B_{8} (ξ, t)$ satisfies the following estimates: $\begin{matrix} (3.46) & \begin{matrix} \frac{d}{d t} B_{8} (ξ, t) + (g (0) - ϵ_{5}) ξ^{2} | \hat{θ} |^{2} \\ ⩽ C (ϵ_{5}^{'}) (1 + ξ^{2}) g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s + ϵ_{5}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{y} |^{2} \\ + C (ϵ_{5}) g^{'} (0) \int_{0}^{\infty} ξ^{2} μ^{'} (s) {| \hat{η} (t, s) |}^{2} d s, \end{matrix} \end{matrix}$ where $ϵ_{5}$ and $ϵ_{5}^{'}$ are arbitrary positive constants.
Proof.
Multiplying Eq. (3.8) by $- ξ^{2} μ (s) \bar{\hat{θ}}$ and Eq. (3.7) by $- ξ^{2} μ (s) \bar{\hat{η}} (t, s)$ , adding the results and taking the integration with respect to s for the real parts, we have $\begin{matrix} \begin{matrix} - \frac{d}{d t} Re (ξ^{2} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) \hat{θ} d s) + ξ^{2} \int_{0}^{\infty} μ (s) | \hat{θ} |^{2} d s \\ = Re (\frac{ξ^{4}}{β} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s d s) - Re (ξ^{2} \int_{0}^{\infty} μ (s) \bar{\hat{θ}} T \hat{η} d s) \\ + Re (δ i ξ^{3} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) \hat{y} d s) . \end{matrix} \end{matrix}$ Now, integrating by parts the second term on the right hand side, we obtain $\begin{matrix} (3.47) & \begin{matrix} - \frac{d}{d t} Re (ξ^{2} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) \hat{θ} d s) + ξ^{2} g (0) | \hat{θ} |^{2} \\ = Re (\frac{ξ^{4}}{β} {| \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s |}^{2}) - Re (ξ^{2} \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{θ}} d s) \\ + Re (δ i ξ^{3} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) \hat{y} d s) . \end{matrix} \end{matrix}$ Applying Young’s inequality, we have for any $ϵ_{5}$ , $ϵ_{5}^{'} > 0$ , $\begin{array}{l} | Re (δ i ξ^{3} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) \hat{y} d s) | ⩽ C (ϵ_{5}^{'}) (1 + ξ^{2}) g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s + ϵ_{5}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{y} |^{2}, \\ | Re (ξ^{2} \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{θ}} d s) | ⩽ ϵ_{5} ξ^{2} | \hat{θ} |^{2} + C (ϵ_{5}) g^{'} (0) \int_{0}^{\infty} ξ^{2} μ^{'} (s) {| \hat{η} (t, s) |}^{2} d s \end{array}$ and $\begin{matrix} | Re (ξ^{4} {| \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s |}^{2}) | ⩽ ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s . \end{matrix}$ Hence, inserting the above estimates into (3.47), then (3.46) is fulfilled. □

Now we are ready to prove Proposition 3.2.
Proof of Proposition 3.2.
Case one: $α_{g} = 0$ . Substituting $α_{g} = 0$ into Eq. (3.31), we obtain $\begin{array}{rcl} (3.48) & \begin{matrix} \frac{d}{d t} B_{3} (ξ, t) + a^{2} I^{2} ξ^{2} (| \hat{z} |^{2} - | \hat{y} |^{2}) - \frac{β}{g (0)} ξ^{2} | \hat{y} |^{2} + \frac{β}{g (0)} ξ^{2} | \hat{v} |^{2} \\ = Re (a I^{2} δ ξ^{2} \hat{θ} \bar{\hat{z}}) + (\frac{β}{g (0)} - a^{2}) Re (I ξ^{2} \hat{y} \bar{\hat{ρ}}) \\ + Re (a I γ i ξ \hat{ρ} \bar{\hat{z}}) + (1 - \frac{β}{g (0)}) Re (a I k_{0} ξ^{2} \hat{ϕ} \bar{\hat{z}}) \\ + (\frac{1}{δ^{2}} - \frac{a^{2}}{δ^{2}} + \frac{β}{g (0)}) Re (I k_{0} δ ξ^{2} \bar{\hat{θ}} \hat{ϕ}) + \frac{1 - a^{2}}{δ g (0)} Re (i ξ^{3} \int_{0}^{\infty} μ (s) \hat{y} \bar{\hat{η}} (t, s) d s) \\ + \frac{1 - a^{2}}{δ g (0)} Re (i ξ^{3} \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{v}} d s) + \frac{1 - a^{2}}{δ g (0)} Re (i ξ^{3} I \int_{0}^{\infty} μ (s) \hat{ρ} \bar{\hat{η}} (t, s) d s) . \end{matrix} \end{array}$

Now, applying Young’s inequality, we find for any $ϵ_{1}$ , $ϵ_{1}^{'} > 0$ , $\begin{matrix} \begin{matrix} | Re (a I^{2} δ ξ^{2} \hat{θ} \bar{\hat{z}}) | ⩽ \frac{ϵ_{1}}{3} ξ^{2} | \hat{z} |^{2} + C (ϵ_{1}) ξ^{2} | \hat{θ} |^{2}, \\ | Re (I ξ^{2} \hat{y} \bar{\hat{ρ}}) | ⩽ \frac{ϵ_{1}}{2} ξ^{2} | \hat{y} |^{2} + C (ϵ_{1}) ξ^{2} | \hat{ρ} |^{2}, \\ | Re (a I γ i ξ \hat{ρ} \bar{\hat{z}}) | ⩽ \frac{ϵ_{1}}{3} ξ^{2} | \hat{z} |^{2} + C (ϵ_{1}) | \hat{ρ} |^{2}, \\ | Re (a I k_{0} ξ^{2} \hat{ϕ} \bar{\hat{z}}) | ⩽ \frac{ϵ_{1}}{3} ξ^{2} | \hat{z} |^{2} + C (ϵ_{1}) ξ^{2} | \hat{ϕ} |^{2}, \\ | Re (I k_{0} δ ξ^{2} \bar{\hat{θ}} \hat{ϕ}) | ⩽ ϵ_{1}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{ϕ} |^{2} + C (ϵ_{1}^{'}) (1 + ξ^{2}) | \hat{θ} |^{2}, \\ | Re (i ξ^{3} \int_{0}^{\infty} μ (s) \hat{y} \bar{\hat{η}} (t, s) d s) | ⩽ \frac{ϵ_{1}}{2} ξ^{2} | \hat{y} |^{2} + \frac{C (ϵ_{1})}{2} ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s, \\ | Re (i ξ^{3} I \int_{0}^{\infty} μ (s) \hat{ρ} \bar{\hat{η}} (t, s) d s) | ⩽ ϵ_{1} ξ^{2} | \hat{ρ} |^{2} + \frac{C (ϵ_{1})}{2} ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s \end{matrix} \end{matrix}$ and $\begin{matrix} | Re (i ξ^{3} \int_{0}^{\infty} μ^{'} (s) \hat{η} (t, s) \bar{\hat{v}} d s) | ⩽ ϵ_{1} ξ^{2} | \hat{v} |^{2} + C (ϵ_{1}) ξ^{2} g^{'} (0) \int_{0}^{\infty} ξ^{2} μ^{'} (s) {| \hat{η} (t, s) |}^{2} d s . \end{matrix}$ Plugging the above estimates into (3.48) we have, $\begin{array}{rcl} \frac{d}{d t} B_{3} (ξ, t) + (a^{2} I^{2} - ϵ_{1}) ξ^{2} | \hat{z} |^{2} + (\frac{β}{g (0)} - ϵ_{1}) ξ^{2} | \hat{v} |^{2} \\ ⩽ C (ϵ_{1}) ξ^{2} | \hat{y} |^{2} + C (ϵ_{1}, ϵ_{1}^{'}) (1 + ξ^{2}) | \hat{θ} |^{2} \\ + C (ϵ_{1}) (1 + ξ^{2}) | \hat{ρ} |^{2} + ϵ_{1}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{ϕ} |^{2} \\ + C (ϵ_{1}) ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s \\ (3.49) & + C (ϵ_{1}) ξ^{2} g^{'} (0) \int_{0}^{\infty} ξ^{2} μ^{'} (s) {| \hat{η} (t, s) |}^{2} d s . \end{array}$

Define the functional $L_{1} (ξ, t)$ as follows: $\begin{matrix} (3.50) & \begin{matrix} L_{1} (ξ, t) = & \frac{ξ^{4}}{{(1 + ξ^{2})}^{2}} [B_{5} (ξ, t) + λ_{1} B_{6} (ξ, t)] + λ_{2} B_{8} (ξ, t) \\ + λ_{3} \frac{ξ^{2}}{1 + ξ^{2}} B_{7} (ξ, t) + λ_{4} \frac{ξ^{2}}{1 + ξ^{2}} B_{3} (ξ, t), \end{matrix} \end{matrix}$ where $λ_{1}$ , $λ_{2}$ , $λ_{3}$ and $λ_{4}$ are positive constants to be fixed later.

Using assumption (M2), we may write $\begin{matrix} (3.51) & \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s ⩽ \frac{1}{ν} \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (t, s) |}^{2} d s . \end{matrix}$ Now, taking the derivative of $L_{1} (ξ, t)$ with respect to t and using (3.33), (3.38), (3.42), (3.46) and (3.49), together with (3.51) we find $\begin{matrix} (3.52) & \begin{matrix} \frac{d}{d t} L_{1} (ξ, t) + \frac{ξ^{4}}{1 + ξ^{2}} [λ_{4} (a^{2} I^{2} - ϵ_{1}) - C (ϵ_{2}) - λ_{3} ϵ_{4}^{'}] | \hat{z} |^{2} \\ + \frac{ξ^{4}}{1 + ξ^{2}} [λ_{4} (\frac{β}{g (0)} - ϵ_{1}) - λ_{1} - λ_{3} ϵ_{4}^{'}] | \hat{v} |^{2} + λ_{1} (I^{2} - \frac{3}{2} ϵ_{3}^{'}) \frac{ξ^{4}}{{(1 + ξ^{2})}^{2}} | \hat{v} |^{2} \\ + \frac{ξ^{6}}{{(1 + ξ^{2})}^{2}} [λ_{1} (1 - ϵ_{3}) - ϵ_{2}^{'}] | \hat{u} |^{2} \\ + \frac{ξ^{6}}{{(1 + ξ^{2})}^{2}} [(k_{0} - ϵ_{2}) - λ_{1} C (ϵ_{3}^{'}) - λ_{4} ϵ_{1}^{'}] | \hat{ϕ} |^{2} \\ + ξ^{2} [λ_{2} (g (0) - ϵ_{5}) - C (ϵ_{2}) - λ_{3} C (ϵ_{4}^{'}) - λ_{4} C (ϵ_{1}, ϵ_{1}^{'})] | \hat{θ} |^{2} \\ + \frac{ξ^{4}}{1 + ξ^{2}} [λ_{3} (δ - ϵ_{4}) - C (ϵ_{2}, ϵ_{2}^{'}) - λ_{1} C (ϵ_{3}, ϵ_{3}^{'}) - λ_{2} ϵ_{5}^{'} - λ_{4} C (ϵ_{1})] | \hat{y} |^{2} \\ + ξ^{2} [- C (ϵ_{2}, ϵ_{2}^{'}) - λ_{1} C (ϵ_{3}, ϵ_{3}^{'}) - λ_{4} C (ϵ_{1})] | \hat{ρ} |^{2} \\ + C (1 + ξ^{2}) g^{'} (0) \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (t, s) |}^{2} d s ⩽ 0, \end{matrix} \end{matrix}$ where C is a generic positive constant that depends on $ϵ_{i}$ , $λ_{j}$ and ν, yet is independent on t and ξ. In the above estimate we made use of some trivial inequalities such as $\frac{ξ^{2}}{1 + ξ^{2}} ⩽ min (1, ξ^{2})$ .

Now we choose the constants in (3.52) very carefully in order to make all the coefficients (except the last two terms) in (3.52) positive. Indeed, let us fix $ϵ_{1}$ , $ϵ_{2}$ , $ϵ_{3}$ , $ϵ_{3}^{'}$ , $ϵ_{4}$ and $ϵ_{5}$ small enough such that $\begin{matrix} ϵ_{1} < min {a^{2} I^{2}, \frac{β}{g (0)}}, ϵ_{2} < k_{0}, ϵ_{3} < 1, ϵ_{3}^{'} < \frac{2}{3} I^{2}, ϵ_{4} < δ and ϵ_{5} < g (0) . \end{matrix}$ Now, we choose $λ_{1}$ small enough such that $\begin{matrix} λ_{1} < \frac{k_{0} - ϵ_{2}}{C (ϵ_{3}^{'})} . \end{matrix}$ Next, we take $λ_{4}$ large enough such that $\begin{matrix} λ_{4} > max {\frac{C (ϵ_{2})}{a^{2} I^{2} - ϵ_{1}}, \frac{λ_{1}}{\frac{β}{g_{0}} - ϵ_{1}}} . \end{matrix}$ Then, we choose $ϵ_{2}^{'}$ small enough such that $\begin{matrix} ϵ_{2}^{'} < λ_{1} (1 - ϵ_{3}) . \end{matrix}$ Then, we pick $λ_{3}$ large enough such that $\begin{matrix} λ_{3} > \frac{C (ϵ_{2}, ϵ_{2}^{'}) + λ_{1} C (ϵ_{3}, ϵ_{3}^{'}) + λ_{4} C (ϵ_{1})}{δ - ϵ_{4}} . \end{matrix}$ Moreover, we select $ϵ_{4}^{'}$ small enough such that $\begin{matrix} ϵ_{4}^{'} < min (\frac{λ_{4} (\frac{β}{g (0)} - ϵ_{1}) - λ_{1}}{λ_{3}}, \frac{λ_{4} (a^{2} I^{2} - ϵ_{1}) - C (ϵ_{2})}{λ_{3}}) . \end{matrix}$ Now, we choose $ϵ_{1}^{'}$ small enough such that $\begin{matrix} ϵ_{1}^{'} < \frac{(k_{0} - ϵ_{2}) - λ_{1} C (ϵ_{3}^{'})}{λ_{4}} . \end{matrix}$ Furthermore, we select $λ_{2}$ large enough such that $\begin{matrix} λ_{2} > \frac{C (ϵ_{2}) + λ_{3} C (ϵ_{4}^{'}) + λ_{4} C (ϵ_{1}, ϵ_{1}^{'})}{g (0) - ϵ_{5}} . \end{matrix}$ Finally, we take $ϵ_{5}^{'}$ small enough such that $\begin{matrix} ϵ_{5}^{'} < \frac{λ_{3} (δ - ϵ_{4}) - λ_{4} C (ϵ_{1}) - C (ϵ_{2}, ϵ_{2}^{'}) - λ_{1} C (ϵ_{3}, ϵ_{3}^{'})}{λ_{2}} . \end{matrix}$ Consequently, we deduce that there exists a positive constant $η_{1} > 0$ such that $\begin{matrix} (3.53) & \frac{d}{d t} L_{1} (ξ, t) + η_{1} Q_{1} (ξ, t) ⩽ C (1 + ξ^{2}) \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (t, s) |}^{2} d s + C_{1} ξ^{2} | \hat{ρ} |^{2}, \end{matrix}$ where $C_{1}$ is a generic positive constant and $\begin{array}{rcl} Q_{1} (ξ, t) & = & \frac{ξ^{4}}{1 + ξ^{2}} (| \hat{z} |^{2} + | \hat{v} |^{2} + | \hat{y} |^{2}) + \frac{ξ^{6}}{{(1 + ξ^{2})}^{2}} (| \hat{u} |^{2} + | \hat{ϕ} |^{2}) \\ (3.54) & + ξ^{2} | \hat{θ} |^{2} + \frac{ξ^{4}}{{(1 + ξ^{2})}^{2}} | \hat{v} |^{2} . \end{array}$

It is not hard to see that $\begin{matrix} (3.55) & Q_{1} (ξ, t) ⩾ \frac{ξ^{6}}{{(1 + ξ^{2})}^{2}} (| \hat{v} |^{2} + | \hat{z} |^{2} + | \hat{ϕ} |^{2} + | \hat{u} |^{2} + | \hat{y} |^{2} + | \hat{θ} |^{2}) . \end{matrix}$

Now, we define the Lyapunov functional $\begin{matrix} (3.56) & L_{1} (ξ, t) = N (1 + ξ^{2}) \hat{E} (ξ, t) + L_{1} (ξ, t), \end{matrix}$ where N is a large positive constant that we will choose it later. Now, using (3.11) and (3.53), the functional $L_{1} (ξ, t)$ satisfies the estimate $\begin{matrix} (3.57) & \begin{matrix} \frac{d}{d t} L_{1} (ξ, t) + (1 + ξ^{2}) [(N / 2 β - C) \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (t, s) |}^{2} d s + (N γ - C_{1}) | \hat{ρ} |^{2}] \\ + η_{1} Q_{1} (ξ, t) ⩽ 0 . \end{matrix} \end{matrix}$

By choosing N large enough such that $\begin{matrix} N > max {2 β C, \frac{C_{1}}{γ}}, \end{matrix}$ then we deduce from (3.10) and (3.55) that there exists a positive constant $η_{2}$ such that $\begin{matrix} (3.58) & \frac{d}{d t} L_{1} (ξ, t) + η_{2} \frac{ξ^{6}}{{(1 + ξ^{2})}^{2}} \hat{E} (ξ, t) ⩽ 0, \forall t ⩾ 0 . \end{matrix}$

Now, exploiting (3.56) and (3.50) together with the definitions of all the functionals involved in (3.50) then, we deduce that there exist two positive numbers $β_{1}$ and $β_{2}$ such that, for all $t ⩾ 0$ , $\begin{matrix} (3.59) & β_{1} (1 + ξ^{2}) \hat{E} (ξ, t) ⩽ L_{1} (ξ, t) ⩽ β_{2} (1 + ξ^{2}) \hat{E} (ξ, t) . \end{matrix}$

Combining (3.58) and (3.59), we find that for all $t ⩾ 0$ , we have $\begin{matrix} (3.60) & \frac{d}{d t} L_{1} (ξ, t) ⩽ - \frac{η_{2}}{β_{2}} \frac{ξ^{6}}{{(1 + ξ^{2})}^{3}} L_{1} (ξ, t) . \end{matrix}$ Now, applying Gronwall’s lemma and using (3.59) once again, then (3.15) holds.

Case two: $α_{g} \neq 0$ . In this case from (3.31), we estimate the terms involving $α_{g}$ as follows: using Young’s inequality, for any $ϵ_{1}^{'} > 0$ , we have $\begin{array}{rcl} | α_{g} Re (i ξ^{3} \bar{\hat{u}} \hat{y}) | ⩽ \frac{ϵ_{1}^{'}}{2} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{u} |^{2} + C (ϵ_{1}^{'}) ξ^{2} (1 + ξ^{2}) | \hat{y} |^{2} \end{array}$ and Lamma 2.1, gives $\begin{array}{rcl} | \frac{α_{g}}{δ β} Re (ξ^{4} \bar{\hat{u}} \int_{0}^{\infty} μ (s) \hat{η} (t, s) d s) | \\ ⩽ \frac{ϵ_{1}^{'}}{2} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{u} |^{2} + C (ϵ_{1}^{'}) ξ^{4} (1 + ξ^{2}) g (0) \int_{0}^{\infty} μ (s) {| \hat{η} (t, s) |}^{2} d s . \end{array}$ Hence, (3.31) can be written as $\begin{array}{rcl} \frac{d}{d t} B_{3} (ξ, t) + (a^{2} I^{2} - ϵ_{1}) ξ^{2} | \hat{z} |^{2} + (\frac{β}{g (0)} - ϵ_{1}) ξ^{2} | \hat{v} |^{2} \\ ⩽ C (ϵ_{1}, ϵ_{1}^{'}) ξ^{2} (1 + ξ^{2}) | \hat{y} |^{2} + C (ϵ_{1}, ϵ_{1}^{'}) (1 + ξ^{2}) | \hat{θ} |^{2} \\ + C (ϵ_{1}) (1 + ξ^{2}) | \hat{ρ} |^{2} + ϵ_{1}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{ϕ} |^{2} + ϵ_{1}^{'} \frac{ξ^{4}}{1 + ξ^{2}} | \hat{u} |^{2} \\ + C (ϵ_{1}, ϵ_{1}^{'}) ξ^{4} (1 + ξ^{2}) g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s + C (ϵ_{1}) ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s \\ (3.61) & + C (ϵ_{1}) ξ^{2} g^{'} (0) \int_{0}^{\infty} ξ^{2} μ^{'} (s) {| \hat{η} (t, s) |}^{2} d s . \end{array}$

Also, we modify the estimate (3.46) by making some changes on the estimate as follows: $\begin{matrix} \begin{matrix} | Re (δ i ξ^{3} \int_{0}^{\infty} μ (s) \bar{\hat{η}} (t, s) \hat{y} d s) | \\ ⩽ C (ϵ_{5}^{'}) {(1 + ξ^{2})}^{3} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s + ϵ_{5}^{'} \frac{ξ^{4}}{{(1 + ξ^{2})}^{3}} | \hat{y} |^{2} . \end{matrix} \end{matrix}$

Hence, $\begin{matrix} (3.62) & \begin{matrix} \frac{d}{d t} B_{8} (ξ, t) + (g (0) - ϵ_{5}) ξ^{2} | \hat{θ} |^{2} \\ ⩽ C (ϵ_{5}^{'}) {(1 + ξ^{2})}^{3} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s + ϵ_{5}^{'} \frac{ξ^{4}}{{(1 + ξ^{2})}^{3}} | \hat{y} |^{2} \\ + C (ϵ_{5}) g^{'} (0) \int_{0}^{\infty} ξ^{2} μ^{'} (s) {| \hat{η} (t, s) |}^{2} d s, \end{matrix} \end{matrix}$ where $ϵ_{5}$ and $ϵ_{5}^{'}$ are arbitrary positive constants.

Moreover, we modify the estimate (3.42) as follows: $\begin{matrix} (3.63) & \begin{matrix} \frac{d}{d t} B_{7} (ξ, t) + (δ - ϵ_{4}) ξ^{2} | \hat{y} |^{2} ⩽ & ϵ_{4}^{'} \frac{ξ^{2}}{{(1 + ξ^{2})}^{2}} | \hat{z} |^{2} + C (ϵ_{4}^{'}) {(1 + ξ^{2})}^{3} | \hat{θ} |^{2} + ϵ_{4}^{'} \frac{ξ^{2}}{{(1 + ξ^{2})}^{2}} | \hat{v} |^{2} \\ + C (ϵ_{4}) ξ^{2} g (0) \int_{0}^{\infty} ξ^{2} μ (s) {| \hat{η} (t, s) |}^{2} d s . \end{matrix} \end{matrix}$ where we have used the estimates $\begin{array}{l} | - Re (a ξ^{2} \hat{z} \bar{\hat{θ}}) | ⩽ ϵ_{4}^{'} \frac{ξ^{2}}{{(1 + ξ^{2})}^{2}} | \hat{z} |^{2} + C (ϵ_{4}^{'}) ξ^{2} {(1 + ξ^{2})}^{2} | \hat{θ} |^{2}, \\ | Re (i ξ \hat{v} \bar{\hat{θ}}) | ⩽ ϵ_{4}^{'} \frac{ξ^{2}}{{(1 + ξ^{2})}^{2}} | \hat{v} |^{2} + C (ϵ_{4}^{'}) {(1 + ξ^{2})}^{2} | \hat{θ} |^{2}, \end{array}$ instead of (3.44).

Now, we define the functional $L_{2} (ξ, t)$ as follows $\begin{matrix} (3.64) & \begin{matrix} L_{2} (ξ, t) = & \frac{ξ^{2}}{{(1 + ξ^{2})}^{3}} [\frac{ξ^{2}}{{(1 + ξ^{2})}^{2}} B_{5} (ξ, t) + γ_{1} \frac{ξ^{2}}{{(1 + ξ^{2})}^{2}} B_{6} (ξ, t) \\ + \frac{γ_{4}}{1 + ξ^{2}} B_{3} (ξ, t) + γ_{3} B_{7} (ξ, t)] + γ_{2} B_{8} (ξ, t), \end{matrix} \end{matrix}$ where $γ_{1}$ , $γ_{4}$ , $γ_{3}$ and $γ_{2}$ are positive constants to be fixed later. Finding the first derivative of $L_{2} (ξ, t)$ with respect to t and using the following inequalities (3.33), (3.38), (3.61), (3.63) and (3.62), together with (3.51), we have $\begin{matrix} (3.65) & \begin{matrix} \frac{d}{d t} L_{2} (ξ, t) + \frac{ξ^{4}}{{(1 + ξ^{2})}^{4}} [γ_{4} (a^{2} I^{2} - ϵ_{1}) - C (ϵ_{2}) - γ_{3} ϵ_{4}^{'}] | \hat{z} |^{2} \\ + \frac{ξ^{4}}{{(1 + ξ^{2})}^{5}} [γ_{1} (I^{2} - \frac{3}{2} ϵ_{3}^{'}) - γ_{3} ϵ_{4}^{'}] | \hat{v} |^{2} + \frac{ξ^{4}}{{(1 + ξ^{2})}^{4}} [γ_{4} (\frac{β}{g (0)} - ϵ_{1}) - γ_{1}] | \hat{v} |^{2} \\ + \frac{ξ^{6}}{{(1 + ξ^{2})}^{5}} [γ_{1} (1 - ϵ_{3}) - ϵ_{2}^{'} - γ_{4} ϵ_{1}^{'}] | \hat{u} |^{2} \\ + \frac{ξ^{6}}{{(1 + ξ^{2})}^{5}} [(k_{0} - ϵ_{2}) - γ_{1} C (ϵ_{3}^{'}) - γ_{4} ϵ_{1}^{'}] | \hat{ϕ} |^{2} \\ + ξ^{2} [γ_{2} (g (0) - ϵ_{5}) - C (ϵ_{2}) - γ_{3} C (ϵ_{4}^{'}) - γ_{4} C (ϵ_{1}, ϵ_{1}^{'})] | \hat{θ} |^{2} \\ + \frac{ξ^{4}}{{(1 + ξ^{2})}^{3}} [γ_{3} (δ - ϵ_{4}) - C (ϵ_{2}, ϵ_{2}^{'}) - γ_{1} C (ϵ_{3}, ϵ_{3}^{'}) - γ_{2} ϵ_{5}^{'} - γ_{4} C (ϵ_{1}, ϵ_{1}^{'})] | \hat{y} |^{2} \\ + [- C (ϵ_{2}, ϵ_{2}^{'}) - γ_{1} C (ϵ_{3}, ϵ_{3}^{'}) - γ_{4} C (ϵ_{1})] | \hat{ρ} |^{2} \\ + C {(1 + ξ^{2})}^{3} g^{'} (0) \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (t, s) |}^{2} d s ⩽ 0 . \end{matrix} \end{matrix}$ Here C is a generic positive constant that depends on $ϵ_{i}$ , $γ_{i}$ and ν, yet is independent on t and ξ. As we did in case one, we fix $ϵ_{1}$ , $ϵ_{2}$ , $ϵ_{3}$ , $ϵ_{3}^{'}$ , $ϵ_{4}$ and $ϵ_{5}$ as follows: $\begin{matrix} ϵ_{1} < min {a^{2} I^{2}, \frac{β}{g (0)}}, ϵ_{2} < k_{0}, ϵ_{3} < 1, ϵ_{3}^{'} < \frac{2}{3} I^{2}, ϵ_{4} < δ, ϵ_{5} < g (0) . \end{matrix}$ Also, we take $γ_{i}$ as we did for $λ_{i}$ . We choose $γ_{1}$ small enough such that $\begin{matrix} γ_{1} < \frac{k_{0} - ϵ_{2}}{C (ϵ_{3}^{'})} . \end{matrix}$ Next, we choose $γ_{4}$ large enough such that $\begin{matrix} γ_{4} > {\frac{C (ϵ_{2})}{a^{2} I^{2} - ϵ_{1}}, \frac{γ_{1}}{\frac{β}{g (0)} - ϵ_{1}}} . \end{matrix}$ Then, we select $ϵ_{2}^{'}$ small enough such that $\begin{matrix} ϵ_{2}^{'} < γ_{1} (1 - ϵ_{3}) . \end{matrix}$ Now, we pick $ϵ_{1}^{'}$ small enough such that $\begin{matrix} ϵ_{1}^{'} < min (\frac{γ_{1} (1 - ϵ_{3}) - ϵ_{2}^{'}}{γ_{4}}, \frac{(k_{0} - ϵ_{2}) - γ_{1} C (ϵ_{3}^{'})}{γ_{4}}) . \end{matrix}$ Next, we choose $γ_{3}$ large enough such that $\begin{matrix} γ_{3} > \frac{C (ϵ_{2}, ϵ_{2}^{'}) + γ_{1} C (ϵ_{3}, ϵ_{3}^{'}) + γ_{4} C (ϵ_{1}, ϵ_{1}^{'})}{δ - ϵ_{4}} . \end{matrix}$ Moreover, we determine $ϵ_{4}^{'}$ small enough such that $\begin{matrix} ϵ_{4}^{'} < min (\frac{γ_{4} (a^{2} I^{2} - ϵ_{1}) - C (ϵ_{2})}{γ_{3}}, \frac{γ_{1} (I^{2} - ϵ_{3}^{'})}{γ_{3}}) . \end{matrix}$ Now, we fix $γ_{2}$ large enough such that $\begin{matrix} γ_{2} > \frac{C (ϵ_{2}) + γ_{3} C (ϵ_{4}^{'}) + γ_{4} C (ϵ_{1}, ϵ_{1}^{'})}{g (0) - ϵ_{5}} . \end{matrix}$ Finally, we choose $ϵ_{5}^{'}$ small enough such that $\begin{matrix} ϵ_{5}^{'} < \frac{γ_{3} (δ - ϵ_{4}) - C (ϵ_{2}, ϵ_{2}^{'}) - γ_{1} C (ϵ_{3}, ϵ_{3}^{'}) - γ_{4} C (ϵ_{1}, ϵ_{1}^{'})}{γ_{2}} . \end{matrix}$ Consequently, we deduce that there exist a positive constant $η_{3} > 0$ such that $\begin{matrix} (3.66) & \frac{d}{d t} L_{2} (ξ, t) + η_{3} Q_{2} (ξ, t) ⩽ C {(1 + ξ^{2})}^{3} \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (t, s) |}^{2} d s + C_{2} | \hat{ρ} |^{2}, \end{matrix}$ where $\begin{array}{rcl} Q_{2} (ξ, t) & = & \frac{ξ^{4}}{{(1 + ξ^{2})}^{4}} (| \hat{z} |^{2} + | \hat{v} |^{2}) + \frac{ξ^{4}}{{(1 + ξ^{2})}^{3}} | \hat{y} |^{2} \\ (3.67) & + \frac{ξ^{6}}{{(1 + ξ^{2})}^{5}} (| \hat{u} |^{2} + | \hat{ϕ} |^{2}) + ξ^{2} | \hat{θ} |^{2} . \end{array}$

It is straightforward to see that $\begin{matrix} (3.68) & Q_{2} (ξ, t) ⩾ \frac{ξ^{6}}{{(1 + ξ^{2})}^{5}} (| \hat{v} |^{2} + | \hat{z} |^{2} + | \hat{ϕ} |^{2} + | \hat{u} |^{2} + | \hat{y} |^{2} + | \hat{θ} |^{2}) . \end{matrix}$

Now, we define the Lyapunov functional $\begin{matrix} (3.69) & L_{2} (ξ, t) = M {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) + L_{2} (ξ, t), \end{matrix}$ where M is a large positive number that we will choose it later. Now, using (3.11) and (3.66) the functional $L_{2} (ξ, t)$ satisfies the estimate $\begin{matrix} (3.70) & \begin{matrix} \frac{d}{d t} L_{2} (ξ, t) + {(1 + ξ^{2})}^{3} [(M / 2 β - C) \int_{0}^{\infty} ξ^{2} (- μ^{'} (s)) {| \hat{η} (t, s) |}^{2} d s + (M γ - C_{2}) | \hat{ρ} |^{2}] \\ + η_{3} Q_{2} (ξ, t) ⩽ 0 . \end{matrix} \end{matrix}$

By choosing M large enough such that $\begin{matrix} M > max {2 β C, \frac{C_{2}}{γ}}, \end{matrix}$ then we deduce from (3.10) and (3.68) that there exists a positive constant $η_{2}$ such that $\begin{matrix} (3.71) & \frac{d}{d t} L_{2} (ξ, t) + η_{4} \frac{ξ^{6}}{{(1 + ξ^{2})}^{5}} \hat{E} (ξ, t) ⩽ 0, \forall t ⩾ 0 . \end{matrix}$

Now, using (3.69), (3.64) together with the definitions of all the functionals involved in (3.64) then, we deduce that there exist two positive constants $β_{3}$ and $β_{4}$ such that, for all $t ⩾ 0$ , $\begin{matrix} (3.72) & β_{3} {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) ⩽ L_{2} (ξ, t) ⩽ β_{4} {(1 + ξ^{2})}^{3} \hat{E} (ξ, t) . \end{matrix}$

Combining (3.66), (3.70) and (3.72), we find that $\begin{matrix} (3.73) & \frac{d}{d t} L_{2} (ξ, t) ⩽ - \frac{η_{4}}{β_{2}} \frac{ξ^{6}}{{(1 + ξ^{2})}^{8}} L_{2} (ξ, t), \forall t ⩾ 0 . \end{matrix}$ Now, applying Gronwall’s lemma and using (3.72) once again, then (3.15) holds. □

Now, we state and prove the main result of this paper. Theorem 3.9.
Let s be a nonnegative integer, $α_{g} = (\frac{β}{g (0)} - 1) (1 - a^{2}) - δ^{2} \frac{β}{g (0)}$ as in ( 1.12 ), and assume that $E_{s} (0)$ and ${sup}_{| ξ | ⩽ 1} {\hat{E} (ξ, 0)}$ are bounded. Then, the energy $E_{k} (t)$ , defined in ( 1.13 ), satisfies the following decay estimates:
If $α_{g} = 0$ , then $\begin{matrix} (3.74) & E_{k} (t) ⩽ C sup_{| ξ | ⩽ 1} {\hat{E} (ξ, 0)} {(1 + t)}^{- 1 / 6 - k / 3} + C e^{- c t} E_{k} (0) . \end{matrix}$

If $α_{g} \neq 0$ , then $\begin{matrix} (3.75) & E_{k} (t) ⩽ C sup_{| ξ | ⩽ 1} {\hat{E} (ξ, 0)} {(1 + t)}^{- 1 / 6 - k / 3} + C {(1 + t)}^{- ℓ / 5} E_{k + ℓ} (0), \end{matrix}$
where k and ℓ are nonnegative integers satisfying $k + ℓ ⩽ s$ and C and c are positive constants.
Proof.
Case one $α_{g} = 0$ . In this case $ρ (ξ)$ satisfies $\begin{matrix} (3.76) & ρ ξ) ⩾ \{\begin{matrix} c ξ^{6}, & for | ξ | ⩽ 1, \\ c, & for | ξ | ⩾ 1 . \end{matrix} \end{matrix}$ Applying the Plancherel theorem together with the estimate (3.15), we have $\begin{matrix} \begin{matrix} E_{k} (t) & = \int_{R} | ξ |^{2 k} \hat{E} (ξ, t) d ξ \\ ⩽ C \int_{R} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ \\ = C \int_{| ξ | ⩽ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ + C \int_{| ξ | ⩾ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ \\ = I_{1} (t) + I_{2} (t) . \end{matrix} \end{matrix}$ Here we split the integral into two parts, $I_{1} (t)$ the low-frequency part where $| ξ | ⩽ 1$ and $I_{2} (t)$ the high-frequency part where $| ξ | ⩾ 1$ . Using the first inequality in (3.76) we can estimate $I_{1} (t)$ as $\begin{array}{rcl} I_{1} (t) & = & C \int_{| ξ | ⩽ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ \\ ⩽ & C \int_{| ξ | ⩽ 1} | ξ |^{2 k} e^{- c ξ^{6} t} \hat{E} (ξ, 0) d ξ \\ (3.77) & ⩽ & C sup_{| ξ | ⩽ 1} {\hat{E} (ξ, 0)} \int_{| ξ | ⩽ 1} | ξ |^{2 k} e^{- c ξ^{6} t} d ξ . \end{array}$ Finally, using (2.11), hence, we obtain $\begin{matrix} (3.78) & I_{1} (t) ⩽ C sup_{| ξ | ⩽ 1} {\hat{E} (ξ, 0)} {(1 + t)}^{- 1 / 6 - k / 3} . \end{matrix}$ Using the second inequality of (3.76) we can find the estimate for $I_{2} (t)$ , as follows: $\begin{array}{rcl} I_{2} (t) & = & C \int_{| ξ | ⩾ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ \\ = & C e^{- c t} \int_{| ξ | ⩾ 1} | ξ |^{2 k} \hat{E} (ξ, 0) d ξ \\ (3.79) & ⩽ & C e^{- c t} E_{k} (0) . \end{array}$ Now adding estimates (3.78) and (3.79), so that estimate (3.74) holds.

Case two $α_{g} \neq 0$ . In this case $ρ (ξ)$ verifies $\begin{matrix} (3.80) & ρ (ξ) ⩾ \{\begin{matrix} c ξ^{6}, & for | ξ | ⩽ 1, \\ c ξ^{- 10}, & for | ξ | ⩾ 1 . \end{matrix} \end{matrix}$ As before, applying the Plancherel theorem together with inequality (3.15), we have $\begin{matrix} \begin{matrix} E_{k} (t) & = \int_{R} | ξ |^{2 k} \hat{E} (ξ, t) d ξ \\ ⩽ C \int_{R} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ \\ = C \int_{| ξ | ⩽ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ + C \int_{| ξ | ⩾ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ \\ = J_{1} (t) + J_{2} (t) . \end{matrix} \end{matrix}$ As we did above, we split the integral into two parts, $J_{1} (t)$ the low-frequency part where $| ξ | ⩽ 1$ and $J_{2} (t)$ the high-frequency part where $| ξ | ⩾ 1$ . Using the first inequality in (3.80), then we may estimate $J_{1} (t)$ exactly as we did for $I_{1} (t)$ .

Now, $J_{2} (t)$ can be estimated using the second inequality in (3.80) as $\begin{array}{rcl} J_{2} (t) & = & C \int_{| ξ | ⩾ 1} | ξ |^{2 k} e^{- c ρ (ξ) t} \hat{E} (ξ, 0) d ξ \\ ⩽ & C \int_{| ξ | ⩾ 1} | ξ |^{2 k} e^{c ξ^{- 10} t} \hat{E} (ξ, 0) d ξ \\ ⩽ & C sup_{| ξ | ⩾ 1} {| ξ |^{- 2 ℓ} e^{c ξ^{- 10} t}} \int_{R} | ξ |^{2 (ℓ + k)} \hat{E} (ξ, 0) d ξ \\ (3.81) & ⩽ & C {(1 + t)}^{- ℓ / 5} E_{k + ℓ} (0), \end{array}$ where we have used the estimate $\begin{matrix} sup_{| ξ | ⩾ 1} | ξ |^{- 2 ℓ} e^{- c | ξ |^{- 10} t} ⩽ C {(1 + t)}^{- ℓ / 5} . \end{matrix}$

Now, adding estimates (3.78) and (3.81), then, (3.75) is fulfilled. □

4. No decay rate for $γ = 0$

In this section, we consider the problem when $γ = 0$ and show that even if $g (s)$ has an exponential decay rate as in (1.21), the dissipation given by the memory term is not strong enough to produce any decay rate of the solution. To do so, we take $\begin{matrix} g (s) = \frac{1}{τ_{q}} e^{- s / τ_{q}} . \end{matrix}$ Then, in this case system (1.1) reduces to the Bresse–Cattaneo system $\begin{array}{rcl} φ_{t t} - {(φ_{x} - ψ - I ω)}_{x} - k_{0}^{2} I (ω_{x} - I φ) = 0, \\ ψ_{t t} - a^{2} ψ_{x x} - (φ_{x} - ψ - I ω) + δ θ_{x} = 0, \\ (4.1) & ω_{t t} - k_{0}^{2} {(ω_{x} - I φ)}_{x} - I (φ_{x} - ψ - I ω) = 0, \\ θ_{t} + q_{x} + δ ψ_{t x} = 0, \\ τ_{q} q_{t} + q + κ θ_{x} = 0 . \end{array}$ Using the same change of variables (2.6) hence, system (4.1) takes the form: $\begin{matrix} (4.2) & \begin{matrix} v_{t} - u_{x} + y + I ρ = 0, \\ u_{t} - v_{x} - I k_{0} ϕ = 0, \\ z_{t} - a y_{x} = 0, \\ y_{t} - a z_{x} - v + δ θ_{x} = 0, \\ ϕ_{t} - k_{0} ρ_{x} + I k_{0} u = 0, \\ ρ_{t} - k_{0} ϕ_{x} - I v = 0, \\ θ_{t} + q_{x} + δ y_{x} = 0, \\ τ_{q} q_{t} + q + κ θ_{x} = 0 . \end{matrix} \end{matrix}$ For $τ_{q} \neq 0$ , then system (4.2) together with the initial data takes the form $\begin{matrix} (4.3) & \{\begin{matrix} U_{t} + A U_{x} + L U = 0, \\ U (x, 0) = U_{0} (x), \end{matrix} \end{matrix}$ where $U = {(v, u, z, y, ϕ, ρ, θ, q)}^{T}$ and $\begin{array}{rcl} A = (\begin{matrix} 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - a & 0 & 0 & 0 & 0 \\ 0 & 0 & - a & 0 & 0 & 0 & δ & 0 \\ 0 & 0 & 0 & 0 & 0 & - k_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & - k_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & δ & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{κ}{τ_{q}} & 0 \end{matrix}), \\ L = (\begin{matrix} 0 & 0 & 0 & 1 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & - I k_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & I k_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ - I & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{τ_{q}} \end{matrix}) . \end{array}$ By taking the Fourier transform of (4.3) we obtain the following Cauchy problem of a first order (ODE) system $\begin{matrix} (4.4) & \{\begin{matrix} {\hat{U}}_{t} + i ξ A \hat{U} + L \hat{U} = 0, \\ \hat{U} (ξ, 0) = {\hat{U}}_{0} . \end{matrix} \end{matrix}$ The solution of (4.4) is given by $\begin{matrix} \hat{U} (ξ, t) = e^{Φ (i ξ) t} {\hat{U}}_{0} (ξ), \end{matrix}$ where $\begin{matrix} (4.5) & Φ (ζ) = - (L + ζ A), ζ = i ξ \in C . \end{matrix}$ The characteristic polynomial of the matrix $Φ (ζ)$ is given by $\begin{matrix} P (λ) = \frac{- ζ^{2} k_{0}^{2} + k_{0}^{2} I^{2} + λ^{2}}{τ_{q}} Q (λ), \end{matrix}$ where $Q (λ)$ is a polynomial of degree 6 given by $\begin{array}{rcl} Q (λ) & = & τ_{q} λ^{6} + λ^{5} + (τ_{q} α_{0} - ζ^{2} κ) λ^{4} + α_{0} λ^{3} + (α_{1} τ_{q} - κ (α_{0} + ζ^{2} δ^{2})) λ^{2} \\ + α_{1} λ - a^{2} ζ^{4} κ (ζ^{2} - I^{2}) \end{array}$ with $\begin{matrix} α_{0} = (I^{2} + 1 - ζ^{2} (a^{2} + δ^{2} + 1)) and α_{1} = ζ^{2} (a^{2} + δ^{2}) (ζ^{2} - I^{2}) . \end{matrix}$

It is clear that for $| ζ | < I$ , then $P (λ)$ has two pure imaginary roots. This means that the matrix $Φ (ζ)$ has two eigenvalues with real parts equal to zero. Consequently, in this case and for small frequencies, the Fourier image of the solution does not decay. So, using, (3.11), we can only show the estimate $\begin{matrix} (4.6) & {| \hat{U} (ξ, t) |}^{2} ⩽ {| \hat{U} (ξ, 0) |}^{2}, \forall t ⩾ 0 . \end{matrix}$ Hence, using Plancherel’s theorem together with (4.6), we deduce the following theorem.

Theorem 4.1 (No decay rates).

Let s be a nonnegative integer. Let $U (x, t)$ be the solution of ( 4.3 ). Assume that $U_{0} \in H^{s} (R)$ , then the following estimate holds: $\begin{matrix} (4.7) & {‖ \partial_{x}^{k} U (t) ‖}_{L^{2}} ⩽ C {‖ \partial_{x}^{k} U_{0} ‖}_{L^{2}}, k = 0, 1, \dots, s, \end{matrix}$ where C is a positive constant.

5. Concluding remarks

In this section, we conclude with some remarks and list some open questions for the interested reader.

Remark 5.1.

By replacing $a^{2} ψ_{x}$ in the second equation of (1.1) with a nonlinear function of the form $σ (ψ_{x})$ where σ is a smooth function with $σ^{'} (0) = a^{2}$ , then, it is not obvious how to show the global existence of the solution with small initial data, due to the regularity loss and the slow decay rate. Even, if someone succeeded to prove it, then certainly some higher regularity assumption is needed, especially for the case $α_{g} \neq 0$ .

Remark 5.2.

It has been shown in [11], that the assumption on the wave speed of the solution of the Bresse system (without heat conduction) is related to the position of the damping terms in the system. So, if we consider, instead of $γ w_{t}$ , a damping term of the form $γ_{0} φ_{t}$ ( $γ_{0} > 0$ ), acting on the first equation in (1.1), then we expect to get a new stability number ${\bar{α}}_{g}$ instead of $α_{g}$ .

Remark 5.3.

In (M2), we assumed that the function $μ (s)$ is decaying exponentially. Some modification in the proof presented here might be possible to treat the case where $μ (s)$ is decaying polynomially and satisfying, instead of (M2), the following assumption: $\begin{matrix} μ^{'} (s) + ν μ^{p} (s) ⩽ 0, p > 1 . \end{matrix}$

Footnotes

Acknowledgements

The authors want to thank the referee for his (her) careful reading of the proofs and for his (her) suggestions and comments which improved the content of the paper.

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On the decay rate of solutions of the Bresse system with Gurtin–Pipkin thermal law

Abstract

Keywords

1. Introduction

2. Statement of the problem

Theorem 4.1 (No decay rates).

5. Concluding remarks

Footnotes

Acknowledgements

References