Theoretical and numerical study of the decay in a viscoelastic Bresse System
Jamilu Hassan, Salim Messaoudi, Toufic Arwadi, Mohammad Hindi
TTheoretical and numerical study of the decay in a viscoelastic BresseSystem
Jamilu Hashim Hassan , Salim A. Messaoudi , Toufic El-Arwadi , and Mohamad El Hindi Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O.Box 546, Dhahran 31261, Saudi Arabia. Department of Mathematics, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates.
Department of Mathematics and Computer Science,Beirut Arab University, P.O. Box 11-5020,Beirut, Lebanon. [email protected] [email protected] [email protected] [email protected] Abstract
In this paper, we consider a one-dimensional finite-memory Bresse system with homo-geneous Dirichlet-Neumann-Neumann boundary conditions. We prove some general decayresults for the energy associated with the system in the case of equal and non-equal speedsof wave propagation under appropriate conditions on the relaxation function. In addition,we show by giving an example that in the case of equal speeds of wave propagation and forcertain polynomially decaying relaxation functions, our result gives an optimal decay ratein the sense that the decay rate of the system is exactly the same as that of the relaxationfunction considered. Introduction
Bresse system is a mathematical model that describes the vibration of a planar, linear shearablecurved beam. The model was first derived by Bresse [6] and it consists of three coupled waveequations given by ρ ϕ tt − k ( ϕ x + ψ + lw ) x − lk ( w x − ϕ ) + F = 0 in (0 , L ) × (0 , ∞ ) ,ρ ψ tt − k ψ xx + k ( ϕ x + ψ + lw ) + F = 0 in (0 , L ) × (0 , ∞ ) ,ρ w tt − k ( w x − ϕ ) x + lk ( ϕ x + ψ + lw ) + F = 0 in (0 , L ) × (0 , ∞ ) , (1.1)where ϕ, ψ, w represent the vertical displacement, the shear angle, and the longitudinal dis-placement, respectively; ρ , ρ , k , k , k , l are positive parameters and F , F , F are externalforces.A lot of results dealing with well-posedness and asymptotic behaviour of the above systemhave been published. We start with the work of Santos et al. [28] from 2010, where they studied1 a r X i v : . [ m a t h . A P ] F e b he Bresse system with Dirichlet-Dirichlet-Dirichlet boundary conditions and linear frictionaldamping acting on each equation, that is, ( F , F , F ) = ( γ ϕ t , γ ψ t , γ w t ) , (1.2)where γ , γ , γ > . They established an exponential decay rate for the system using spectraltheory approach developed by Z. Liu and S. Zheng in [18]. They also gave a numerical schemeusing finite difference method to illustrate their theoretical result. Soriano et al. [29] used themethod developed by Lasiecka and Tataru in [16] and proved a uniform decay rate for thesame system with a nonlinear frictional damping acting on the second equation and locallydistributed nonlinear damping acting on the other equations. Precisely, the external forces aregiven by ( F , F , F ) = ( α ( x ) g ( ϕ t ) , g ( ψ t ) , γ ( x ) g ( w t )) with α, γ ∈ L ∞ (0 , L ) and the g i ’s are continuous and monotone increasing functions. Theresults of [28] and [29] were established without imposing any restriction on the speeds of wavepropagation given by s = (cid:115) k ρ , s = (cid:115) k ρ , and s = (cid:115) k ρ . (1.3)Alves et al. [4] used the semigroup and spectral theory to obtain the exponential stability ofthe Bresse system with three controls at the boundary.In the presence of dissipating terms in only one or two of the equations in system (1.1), thedecay rates of the energy associated to the system depend totally on the speeds of the wavepropagation. As illustrated in [2], Alabau-Boussouira et al. studied (1.1) with linear frictionaldamping acting on the second equation; that is, they used (1.2), with γ = γ = 0 and γ > and showed that the system is exponentially stable if and only if it has equal speeds of wavepropagation, k ρ = k ρ = k ρ . (1.4)As mentioned by many authors [2, 3], relation (1.4) is physically unrealistic. In the case ofnon-equal speeds of wave propagation, they proved polynomial stability with rates which canbe improved with the regularity of the initial data. Fatori and Monteiro [8] improved this resultin the case of non-equal speeds of wave propagation by proving optimal decay rate. Soriano et al. [30] established the same exponential stability result as in [2] by replacing the frictionaldamping with indefinite one; that is, they replaced γ in [2] with a function a : (0 , L ) −→ R such that ¯ a = 1 L (cid:90) L a ( x ) dx > and (cid:107) a − ¯ a (cid:107) L (0 ,L ) is small enough. Wehbe and Youcef [31]inspected the situation of two locally distributed dampings acting on the last two equations;that is, ( F , F , F ) = (0 , a ( x ) ψ t , a ( x ) w t ) , where a i : (0 , L ) −→ R are non-negative functions which can take value zero on some part of theinterval (0 , L ) . By using the frequency domain and the multiplier method, they proved that thesystem is exponentially stable if and only if s = s . When s (cid:54) = s they established a polynomialdecay rate which can be improved with the regularity of the initial data. The same result was2stablished by Alves et al. in [3], in the case of non-equal speeds of wave propagation, theyused the recent result of Borichev and Tomilov in [5] to show that the solution is polynomiallystable with optimal decay rate.Concerning the dissipation via heat effect, we mention the work of Liu and Rao [17] wherethe following system ρ ϕ tt − k ( ϕ x + ψ + lw ) x − lk ( w x − ϕ ) + lγχ = 0 in (0 , L ) × (0 , ∞ ) ,ρ ψ tt − k ψ xx + k ( ϕ x + ψ + lw ) + γθ x = 0 in (0 , L ) × (0 , ∞ ) ,ρ w tt − k ( w x − ϕ ) x + lk ( ϕ x + ψ + lw ) + γχ t = 0 in (0 , L ) × (0 , ∞ ) ,ρ θ t − θ xx + γψ xt = 0 in (0 , L ) × (0 , ∞ ) ,ρ χ t − χ xx + γ ( w x − lϕ ) t = 0 in (0 , L ) × (0 , ∞ ) , (1.5)with boundary and initial conditions was considered. They showed that the exponential stabilityof the system is equivalent to the validity of the identity (1.4). In the case where (1.4) doesnot hold, they established a polynomial-type decay rate. Fatori and Muñoz Rivera [9] obtaineda similar result as in [17] for the thermoelastic Bresse system (1.5) when the fifth equationis omitted. They also showed that the polynomial decay rate is optimal in the case of non-equal speeds of wave propagation. Filippo Dell’Oro [7] gave a detail stability analysis of thethermoelastic Bresse-Gurtin-Pipkin system of the form: ρ ϕ tt − k ( ϕ x + ψ + lw ) x − lk ( w x − ϕ ) = 0 in (0 , L ) × (0 , ∞ ) ,ρ ψ tt − k ψ xx + k ( ϕ x + ψ + lw ) + γθ x = 0 in (0 , L ) × (0 , ∞ ) ,ρ w tt − k ( w x − ϕ ) x + lk ( ϕ x + ψ + lw ) = 0 in (0 , L ) × (0 , ∞ ) ,ρ θ t − k (cid:90) ∞ g ( s ) θ xx ( t − s ) ds + γψ xt = 0 in (0 , L ) × (0 , ∞ ) , (1.6)where g is a bounded convex integrable function on [0 , ∞ ) satisfying (cid:90) ∞ g ( s ) ds = 1 , and there exists a non-increasing absolutely continuous function µ : (0 , ∞ ) −→ [0 , ∞ ) such that µ (0) = lim s → µ ( s ) ∈ (0 , ∞ ) , g ( s ) = (cid:90) ∞ s µ ( τ ) dτ, ∀ s ∈ [0 , ∞ ) and µ (cid:48) ( s ) + νµ ( s ) ≤ ν > a.e. s ∈ (0 , ∞ ) . By introducing a new stability number of the form χ g = (cid:18) ρ ρ k − g (0) k (cid:19) (cid:16) ρ k − ρ b (cid:17) − g (0) k ρ γ ρ bk ,
3e proved that the semigroup generated by (1.6) is exponentially stable if and only if χ g = 0 and k = k . As a special case, he showed that his stability result gave the stability characterization of Bressesystems with Fourier, Maxwell-Cataneo and Coleman-Gurtin thermal dissipation. The readeris referred to [1, 10, 15, 22, 23, 24, 25, 26] and the references therein for more recent results onthermoelastic Bresse system.There are few results that dealt with stabilization of Bresse system via infinite memory. Webegin with the work of Guesmia and Kafini [12] in 2015. They studied the following system ρ ϕ tt − k ( ϕ x + ψ + lw ) x − lk ( w x − lϕ ) + (cid:90) ∞ g ( s ) ϕ xx ( x, t − s ) ds = 0 ,ρ ψ tt − k ψ xx + k ( ϕ x + ψ + lw ) + (cid:90) ∞ g ( s ) ψ xx ( x, t − s ) ds = 0 ,ρ w tt − lk ( w x − lϕ ) x + lk ( ϕ x + ψ + lw ) + (cid:90) ∞ g ( s ) w xx ( x, t − s ) ds = 0 ,ϕ (0 , t ) = ψ (0 , t ) = w (0 , t ) = ϕ ( L, t ) = ψ ( L, t ) = w ( L, t ) = 0 ,ϕ ( x, − t ) = ϕ ( x, t ) , ϕ t ( x,
0) = ϕ ( x ) ,ψ ( x, − t ) = ψ ( x, t ) , ψ t ( x,
0) = ψ ( x ) ,w ( x, − t ) = w ( x, t ) , w t ( x,
0) = w ( x ) , (1.7)where ( x, t ) ∈ (0 , L ) × R + , g i : R + −→ R + are differentiable non-increasing and integrablefunctions, and L, l i , ρ i , k i are positive constants. They proved the well-posedness and theasymptotic stability of (1.7). Later, Guesmia and Kirane [13] used two infinite memories toobtain the same stability result of [12] under the following conditions on the speeds of wavepropagation: k ρ = k ρ in case g = 0 , k ρ = k ρ in case g = 0 , k ρ = k ρ in case g = 0 . Santos et al. [27] discussed the Bresse system with only one infinite memory acting on the shearangle displacement equation. Precisely, they studied problem (1.7) with g = g = 0 and g satisfying : − α g ( t ) ≤ g (cid:48) ( t ) ≤ − α g ( t ) , ∀ t ≥ , for some α , α > . They showed that the solution of the system decays exponentially to zeroif and only if (1.4) holds, otherwise a polynomial stability of the system with an optimal decayrate of type t − / was obtained. Recently, Guesmia [11] analysed the asymptotic stability ofBresse system with one infinite memory in the longitudinal displacement. To the best of our knowledge, there is no result in the literature that deals withthe stability of Bresse system via viscoelastic damping of finite memory-type . In4his paper we will discuss the decay property of the following finite memory-type Bresse system: ρ ϕ tt − k ( ϕ x + ψ + lw ) x − lk ( w x − lϕ ) = 0 , in (0 , L ) × (0 , + ∞ ) ,ρ ψ tt − k ψ xx + k ( ϕ x + ψ + lw ) + (cid:90) t g ( t − s ) ψ xx ( s ) ds = 0 , in (0 , L ) × (0 , + ∞ ) ,ρ w tt − k ( w x − lϕ ) x + lk ( ϕ x + ψ + lw ) = 0 , in (0 , L ) × (0 , + ∞ ) ,ϕ (0 , t ) = ϕ ( L, t ) = ψ x (0 , t ) = ψ x ( L, t ) = w x (0 , t ) = w x ( L, t ) = 0 , for t ≥ ,ϕ ( x,
0) = ϕ ( x ) , ϕ t ( x,
0) = ϕ ( x ) , for x ∈ (0 , L ) ,ψ ( x,
0) = ψ ( x ) , ψ t ( x,
0) = ψ ( x ) , for x ∈ (0 , L ) ,w ( x,
0) = w ( x ) , w t ( x,
0) = w ( x ) , for x ∈ (0 , L ) , ( P )where l, k , k , k , ρ , ρ are positive constants, ϕ , ϕ , ψ , ψ , w , w are given data and g is a relaxation function satisfying some conditions to be specified in the next section. Ourproblem is motivated by the following classical Bresse system ρ ϕ tt − S x − lN = 0 in (0 , L ) × (0 , ∞ ) ,ρ ψ tt − M x + S = 0 in (0 , L ) × (0 , ∞ ) ,ρ w tt − N x − lS = 0 in (0 , L ) × (0 , ∞ ) , where t and x represent the time and space variables, respectively, and N , S and M denote theaxial force, the shear force and the bending moment given by S = k ( ϕ x + ψ + w ) , M = k ψ x − (cid:90) t g ( t − s ) ψ x ( · , s ) ds, N = k ( w x − ϕ ) . We will prove, under a smallness condition on l , generalized energy decay resultsfor the system in the case of equal and different speeds of wave propagation . Thispaper is organized as follows: in Section 2, we state some preliminary results. In Section 3, westate and prove some technical lemmas. The statement and proof of our main results are givenin Sections 4 and 5, while in Section 6 we present some numerical illustrations to validate ourresults. Through out this work we use c to represent a generic positive constant, independentof t but may depend on the initial data. Preliminaries
In this section, we introduce our assumptions, present some useful lemmas and state the exis-tence theorem.
Assumptions:
We assume that the relaxation function g satisfies the following hypotheses:(A1) g : [0 , ∞ ) −→ [0 , ∞ ) is a non-increasing differentiable function such that g (0) > k − (cid:90) + ∞ g ( s ) ds > . (A2) There exists a non-increasing differentiable function ξ : [0 , ∞ ) −→ (0 , ∞ ) and a constant p , with ≤ p < , such that g (cid:48) ( t ) ≤ − ξ ( t ) g p ( t ) , ∀ t ≥ . emma 2.1. Assume that g satisfies hypotheses ( A and ( A . Then, (cid:90) + ∞ ξ ( t ) g − σ ( t ) dt < + ∞ , ∀ < σ < − p. Proof.
From (A1), we have lim t → + ∞ g ( t ) = 0 . Using (A2), we have (cid:90) + ∞ ξ ( t ) g − σ ( t ) dt = (cid:90) + ∞ ξ ( t ) g p ( t ) g − σ − p ( t ) dt ≤ − (cid:90) + ∞ g (cid:48) ( t ) g − σ − p ( t ) dt = − (cid:20) − σ − p g − σ − p ( t ) (cid:21) t =+ ∞ t =0 < + ∞ , since σ < − p .Now, integrating both sides of the second and third equations in ( P ) over (0 , L ) and usingthe boundary conditions, we get d dt (cid:90) L ψ ( x, t ) dx + k ρ (cid:90) L ψ ( x, t ) dx + lk ρ (cid:90) L w ( x, t ) dx = 0 ∀ t ≥ (2.1)and d dt (cid:90) L w ( x, t ) dx + l k ρ (cid:90) L w ( x, t ) dx + lk ρ (cid:90) L ψ ( x, t ) dx = 0 ∀ t ≥ . (2.2)Solving these ODEs simultaneously yields (cid:90) L ψ ( x, t ) dx = a cos( a t ) + a sin( a t ) + a t + a (2.3)and (cid:90) L w ( x, t ) dx = a l (cid:18) ρ a k − (cid:19) cos( a t ) + a l (cid:18) ρ a k − (cid:19) sin( a t ) − a l t − a l , (2.4)where a = (cid:115) k ρ + l k ρ a = k ρ a (cid:90) L ψ ( x ) dx + lk ρ a (cid:90) L w ( x ) dx,a = k ρ a (cid:90) L ψ ( x ) dx + lk ρ a (cid:90) L w ( x ) dx,a = (cid:18) − k ρ a (cid:19) (cid:90) L ψ ( x ) dx − lk ρ a (cid:90) L w ( x ) dx,a = (cid:18) − k ρ a (cid:19) (cid:90) L ψ ( x ) dx + lk ρ a (cid:90) L w ( x ) dx. (cid:101) ψ = ψ − L (cid:0) a cos( a t ) + a sin( a t ) + a t + a (cid:1)(cid:101) w = w − L (cid:20) a l (cid:18) ρ a k − (cid:19) cos( a t ) + a l (cid:18) ρ a k − (cid:19) sin( a t ) − a l t − a l (cid:21) to get (cid:90) L (cid:101) ψ ( x, t ) dx = (cid:90) L (cid:101) w ( x, t ) dx = 0 , ∀ t ≥ . Furthermore, ( ϕ, (cid:101) ψ, (cid:101) w ) satisfies the equations and the boundary conditions in ( P ) with theinitial data (cid:101) ψ = ψ − L ( a + a ) , (cid:101) ψ = ψ − L ( a a + a ) (cid:101) w = w − L (cid:20) a l (cid:18) ρ a k − (cid:19) − a l (cid:21) , (cid:101) w = w − L (cid:20) a a l (cid:18) ρ a k − (cid:19) − a l (cid:21) . From now on, we work with (cid:101) ψ, (cid:101) w and, respectively, write ψ, w for convenience. We alsointroduce the following spaces, L ∗ (0 , L ) := (cid:26) w ∈ L (0 , L ) : (cid:90) L w ( x ) dx = 0 (cid:27) , H ∗ (0 , L ) := H (0 , L ) ∩ L ∗ (0 , L ) , and H ∗ (0 , L ) := (cid:8) w ∈ H (0 , L ) : w x (0) = w x ( L ) = 0 (cid:9) . Then, Poincaré’s inequality is applicable to the elements of H ∗ (0 , L ) , that is, ∃ c > (cid:90) L v dx ≤ c (cid:90) L v x dx ∀ v ∈ H ∗ (0 , L ) . (2.5)For completeness, we state, without proof the global existence and regularity result whichcan be established by repeating the steps of the proof of the existence result in [21]. Theorem 2.1.
Let ( ϕ , ϕ ) ∈ H (0 , L ) × L (0 , L ) and ( ψ , ψ ) , ( w , w ) ∈ H ∗ (0 , L ) × L ∗ (0 , L ) be given. Assume that g satisfies hypothesis ( A . Then, the problem ( P ) has a unique global(weak) solution ϕ ∈ C ( R + ; H (0 , L )) ∩ C ( R + ; L (0 , L )) , ψ, w ∈ C ( R + ; H ∗ (0 , L )) ∩ C ( R + ; L ∗ (0 , L )) . Moreover, if ( ϕ , ϕ ) ∈ ( H (0 , L ) ∩ H (0 , L )) × H (0 , L ) and ( ψ , ψ ) , ( w , w ) ∈ ( H ∗ (0 , L ) ∩ H ∗ (0 , L )) × H ∗ (0 , L ) , then ϕ ∈ C ( R + ; H (0 , L ) ∩ H (0 , L )) ∩ C ( R + ; H (0 , L )) ∩ C ( R + ; L (0 , L )) , and ψ, w ∈ C ( R + ; H ∗ (0 , L ) ∩ H ∗ (0 , L )) ∩ C ( R + ; H ∗ (0 , L )) ∩ C ( R + ; L (0 , L )) . E ( t ) := 12 (cid:90) L (cid:20) ρ ϕ t + ρ ψ t + ρ w t + (cid:18) k − (cid:90) t g ( s ) ds (cid:19) ψ x + k ( w x − lϕ ) + k ( ϕ x + ψ + lw ) (cid:21) dx + 12 ( g ◦ ψ x )( t ) , ∀ t ≥ , (2.6)where for any v ∈ L loc ([0 , + ∞ ); L (0 , L )) , ( g ◦ v )( t ) := (cid:90) L (cid:90) t g ( t − s ) (cid:0) v ( t ) − v ( s ) (cid:1) dsdx. By multiplying the equations in ( P ) by ϕ t , ψ t , w t , respectively, integrating over (0 , L ) andexploiting the boundary conditions we have the following lemma. Lemma 2.2.
Let ( ϕ, ψ, w ) be the weak solution of ( P ) . Then, E (cid:48) ( t ) = − g ( t ) (cid:90) L ψ x dx + 12 ( g (cid:48) ◦ ψ x )( t ) ≤ , ∀ t ≥ . (2.7)From the Cauchy-Schwarz and Poicaré’s inequalities we have the following lemma. Lemma 2.3 ([19]) . There exists a constant c > such that for any v ∈ L loc ( R + ; H ∗ (0 , L )) , wehave (cid:90) L (cid:18)(cid:90) t g ( t − s )( v ( t ) − v ( s )) ds (cid:19) dx ≤ c ( g ◦ v x )( t ) , ∀ t ≥ . Lemma 2.4 ([19]) . Assume that conditions ( A and ( A hold and let ( ϕ, ψ, w ) be the weaksolution of ( P ) . Then, for any < σ < , we have g ◦ ψ x ≤ c (cid:20)(cid:90) t g − σ ( s ) ds (cid:21) p − p + σ − ( g p ◦ ψ x ) σp + σ − . For σ = , we obtain the following inequality g ◦ ψ x ≤ c (cid:18)(cid:90) t g / ( s ) ds (cid:19) p − p − ( g p ◦ ψ x ) p − . (2.8) Corollary 2.1.
Assume that g satisfies ( A , ( A and ( ϕ, ψ, w ) is the weak solution of ( P ) .Then, ξ ( t )( g ◦ ψ x )( t ) ≤ c ( − E (cid:48) ( t )) p − , ∀ t ≥ . Proof.
Multiplying both sides of the inequality (2.8) by ξ ( t ) and using Lemmas 2.1 and 2.2, weget ξ ( t )( g ◦ ψ x )( t ) ≤ cξ p − p − ( t ) (cid:18)(cid:90) t g / ( s ) ds (cid:19) p − p − ( ξg p ◦ ψ x ) p − ( t ) ≤ c (cid:18)(cid:90) t ξ ( s ) g / ( s ) ds (cid:19) p − p − ( − g (cid:48) ◦ ψ x ) p − ≤ c ( − E (cid:48) ( t )) p − . emma 2.5 (Jensen’s inequality) . Let G : [ a, b ] −→ R be a concave function. Assume that thefunctions f : Ω −→ [ a, b ] and h : Ω −→ R are integrable such that h ( x ) ≥ , for any x ∈ Ω and (cid:90) Ω h ( x ) dx = k > . Then, k (cid:90) Ω G ( f ( x )) h ( x ) dx ≤ G (cid:18) k (cid:90) Ω f ( x ) h ( x ) dx (cid:19) . In particular, for G ( y ) = y p , y ≥ , p > , we have k (cid:90) Ω f /p ( x ) h ( x ) dx ≤ (cid:18) k (cid:90) Ω f ( x ) h ( x ) dx (cid:19) /p . Technical Lemmas
In this section, we state and prove some lemmas needed to establish our main results. All thecomputations are done for regular solutions but they still hold for weak and strong solutionsby a density argument.
Lemma 3.1.
Assume that conditions ( A and ( A hold. Then, the functional I defined by I ( t ) := − ρ (cid:90) L ψ t (cid:90) t g ( t − s )( ψ ( t ) − ψ ( s )) dsdx satisfies, along the solution of ( P ) , the estimates I (cid:48) ( t ) ≤ − ρ (cid:18)(cid:90) t g ( s ) ds − δ (cid:19) (cid:90) L ψ t dx + δ (cid:90) L ( ϕ x + ψ + lw ) dx + cδ (cid:90) L ψ x dx + cδ ( g ◦ ψ x − g (cid:48) ◦ ψ x ) , ∀ δ > . (3.1) Proof.
Differentiating I , using equations in ( P ) and integrating by parts, we get I (cid:48) ( t ) = − ρ (cid:90) L ψ t (cid:90) t g (cid:48) ( t − s )( ψ ( t ) − ψ ( s )) dsdx − ρ (cid:18)(cid:90) t g ( s ) ds (cid:19) (cid:90) L ψ t dx + k (cid:90) L ψ x (cid:90) t g ( t − s )( ψ x ( t ) − ψ x ( s )) dsdx + k (cid:90) L ( ϕ x + ψ + lw ) (cid:90) t g ( t − s )( ψ ( t ) − ψ ( s )) dsdx − (cid:90) L (cid:18)(cid:90) t g ( t − s ) ψ x ( s ) ds (cid:19) (cid:18)(cid:90) t g ( t − s )( ψ x ( t ) − ψ x ( s )) ds (cid:19) dx. Next, we estimate the terms on the right-hand side of the above equation.Using Young’s inequality and Lemma 2.3 for ( − g (cid:48) ) , we obtain, for any δ > , − ρ (cid:90) L ψ t (cid:90) t g (cid:48) ( t − s )( ψ ( t ) − ψ ( s )) dsdx ≤ δρ (cid:90) L ψ t dx − cδ ( g (cid:48) ◦ ψ x ) . k (cid:90) L ψ x (cid:90) t g ( t − s )( ψ x ( t ) − ψ x ( s )) dsdx ≤ δ (cid:90) L ψ x + cδ ( g ◦ ψ x ) ,k (cid:90) L ( ϕ x + ψ + lw ) (cid:90) t g ( t − s )( ψ ( t ) − ψ ( s )) dsdx ≤ k δ (cid:90) L ( ϕ x + ψ + lw ) dx + cδ ( g ◦ ψ x ) , and − (cid:90) L (cid:18)(cid:90) t g ( t − s ) ψ x ( s ) ds (cid:19) (cid:18)(cid:90) t g ( t − s )( ψ x ( t ) − ψ x ( s )) ds (cid:19) dx ≤ cδ (cid:90) L ψ x dx + c (cid:18) δ + 1 δ (cid:19) ( g ◦ ψ x ) . A combination of these estimates gives the desired result.
Lemma 3.2.
Assume that the hypotheses ( A and ( A hold. Then, for any ε , δ > , thefunctional I defined by I ( t ) := − ρ k (cid:90) L ( w x − lϕ ) (cid:90) x w t ( y, t ) dydx − ρ k (cid:90) L ϕ t (cid:90) x ( ϕ x + ψ + lw )( y, t ) dydx satisfies, along the solution of ( P ) , the estimate I (cid:48) ( t ) ≤ k (cid:90) L ( ϕ x + ψ + lw ) dx − k (cid:90) L ( w x − lϕ ) dx + cε (cid:90) L ψ t dx + (cid:18) ε − ρ k + lρ | k − k | δ (cid:19) (cid:90) L ϕ t dx (3.2) + ρ (cid:18) k + c l | k − k | δ (cid:19) (cid:90) L w t dx. Proof.
Differentiation of I , using equations in ( P ) and integration by parts yield I = ρ k (cid:90) L w t dx + lρ k (cid:90) L ϕ t (cid:90) x w t ( y, t ) dydx − k (cid:90) L ( w x − lϕ ) dx + k (cid:90) L ( ϕ x + ψ + lw ) dx − ρ k (cid:90) L ϕ t dx − ρ k (cid:90) L ϕ t (cid:90) x ( ψ t + lw t )( y, t ) dydx. Using Young’s inequality, we get, for any ε , δ > , I ≤ k (cid:90) L ( ϕ x + ψ + lw ) dx − k (cid:90) L ( w x − lϕ ) dx + cε (cid:90) L ψ t dx + (cid:18) ε − ρ k + lρ | k − k | δ (cid:19) (cid:90) L ϕ t dx + ρ (cid:18) k + c l | k − k | δ (cid:19) (cid:90) L w t dx. emma 3.3. Under the conditions ( A and ( A , the functional I defined by I ( t ) := − ρ (cid:90) L ( ϕ x + ψ + lw ) w t dx − k ρ k (cid:90) L ( w x − lϕ ) ϕ t dx satisfies, along the solution of ( P ) and for any ε > , the estimate I (cid:48) ( t ) ≤ lk (cid:90) L ( ϕ x + ψ + lw ) dx − lk k (cid:90) L ( w x − lϕ ) dx + cε (cid:90) L ψ t dx + lρ k k (cid:90) L ϕ t dx + ( ε − lρ ) (cid:90) L w t dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx. (3.3) Proof.
Differentiating I , using equations in ( P ) and integrating by parts, we have I = − ρ (cid:90) L ψ t w t dx − lρ (cid:90) L w t dx + lk (cid:90) L ( ϕ x + ψ + lw ) dx + lρ k k (cid:90) L ϕ t dx − lk k (cid:90) L ( w x − lϕ ) dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx. Use of Young’s inequality for the first term in the right-hand side gives (3.3).
Lemma 3.4.
Assume that conditions ( A and ( A hold. Then for any δ > , the functional I defined by I ( t ) := − (cid:90) L ( ρ ϕϕ t + ρ ψψ t + ρ ww t ) dx satisfies, along the solution of ( P ) , the estimate I (cid:48) ( t ) ≤ − (cid:90) L ( ρ ϕ t + ρ ψ t + ρ w t ) dx + k (cid:90) L ( ϕ x + ψ + lw ) dx + k (cid:90) L ( w x − lϕ ) dx + (cid:18) k + δ − (cid:90) t g ( s ) ds (cid:19) (cid:90) L ψ x dx + cδ ( g ◦ ψ x ) . (3.4) Proof.
Differentiation of I , using equations of ( P ) gives I (cid:48) ( t ) = − (cid:90) L ( ρ ϕ t + ρ ψ t + ρ w t ) dx + k (cid:90) L ( ϕ x + ψ + lw ) dx + k (cid:90) L ( w x − lϕ ) dx + (cid:18) k − (cid:90) t g ( s ) ds (cid:19) (cid:90) L ψ x dx − (cid:90) L ψ x (cid:90) t g ( t − s )( ψ x ( t ) − ψ x ( s )) dsdx. Repeating the above computations yields the desired result.
Lemma 3.5.
Assume that conditions ( A and ( A hold. Then for any δ, δ > , the func-tional I defined by I ( t ) := − ρ (cid:90) L ψ x (cid:90) x ψ t ( y, t ) dydx satisfies, along the solution of ( P ) , the estimate I (cid:48) ( t ) ≤ ρ (cid:90) L ψ t dx + (cid:18) k δ + (cid:90) t g ( s ) ds + δ − k (cid:19) (cid:90) L ψ x dx + c k δ (cid:90) L ( ϕ x + ψ + lw ) dx + cδ ( g ◦ ψ x ) . (3.5)11 roof. Using equations of ( P ) and repeating similar computations as above, we arrive at I (cid:48) ( t ) = ρ (cid:90) L ψ t dx − k (cid:90) L ψ x dx + k (cid:90) L ψ x (cid:90) x ( ϕ x + ψ + lw )( y, t ) dydx + (cid:90) L ψ x (cid:90) t g ( t − s ) ψ x ( s ) dsdx ≤ ρ (cid:90) L ψ t dx + (cid:18) k δ + (cid:90) t g ( s ) ds + δ − k (cid:19) (cid:90) L ψ x dx + k δ (cid:90) L (cid:18)(cid:90) x ( ϕ x + ψ + lw )( y, t ) dy (cid:19) dx + cδ ( g ◦ ψ x ) . Poincaré’s inequality for the third term yields (3.5).
Lemma 3.6.
Assume that the hypotheses ( A and ( A hold. Then, for any ε , ε , ε , δ > ,the functional I defined by I ( t ) := ρ (cid:90) L ψ t ( ϕ x + ψ + lw ) dx + bρ k (cid:90) L ϕ t ψ x dx − ρ k (cid:90) L ϕ t (cid:90) t g ( t − s ) ψ x ( s ) dsdx satisfies, along the solution of ( P ) , the estimate I (cid:48) ( t ) ≤ − k (cid:90) L ( ϕ x + ψ + lw ) dx + (cid:18) lk k ε k + lk ε k (cid:90) t g ( s ) ds + δ (cid:19) (cid:90) L ( w x − lϕ ) dx + δ (cid:90) L ϕ t dx + (cid:18) lk k k ε + lk k ε (cid:90) t g ( s ) ds + cδ g ( t ) (cid:19) (cid:90) L ψ x dx (3.6) + ε (cid:90) L w t dx + cε (cid:90) L ψ t dx + cδ ( g ◦ ψ x − g (cid:48) ◦ ψ x ) + (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx. Proof.
Use of equations of ( P ) and integration by parts lead to I (cid:48) ( t ) = − k (cid:90) L ( ϕ x + ψ + lw ) dx + ρ (cid:90) L ψ t dx + lρ (cid:90) L ψ t w t dx + lk k k (cid:90) L ( w x − lϕ ) ψ x dx − lk k (cid:90) L ( w x − lϕ ) (cid:90) t g ( t − s ) ψ x ( s ) dsdx − ρ k g ( t ) (cid:90) L ϕ t ψ x dx + ρ k (cid:90) L ϕ t (cid:90) t g (cid:48) ( t − s )( ψ x ( t ) − ψ x ( s )) dsdx + (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ x ψ xt dx. Next, we estimate the terms in the right-hand side of the above equation.Exploiting Young’s inequality, we get lρ (cid:90) L ψ t w t dx ≤ ε (cid:90) L w t dx + cε (cid:90) L ψ t dx, ∀ ε > . ε , ε , δ > , lk k k (cid:90) L ( w x − lϕ ) ψ x dx − lk k (cid:90) L ( w x − lϕ ) (cid:90) t g ( t − s ) ψ x ( s ) dsdx = lk k (cid:18) k − (cid:90) t g ( s ) ds (cid:19) (cid:90) L ( w x − lϕ ) ψ x dx + lk k (cid:90) L ( w x − lϕ ) (cid:90) t g ( t − s )( ψ x ( t ) − ψ x ( s )) dsdx ≤ (cid:18) lk k ε k + lk ε k (cid:90) t g ( s ) ds + δ (cid:19) (cid:90) L ( w x − lϕ ) dx + (cid:18) lk k k ε + lk k ε (cid:90) t g ( s ) ds (cid:19) (cid:90) L ψ x dx + cδ ( g ◦ ψ x ) and − ρ k g ( t ) (cid:90) L ϕ t ψ x dx + ρ k (cid:90) L ϕ t (cid:90) t g (cid:48) ( t − s )( ψ x ( t ) − ψ x ( s )) dsdx ≤ δ (cid:90) L ϕ t dx + cδ g ( t ) (cid:90) L ψ x dx − cδ ( g (cid:48) ◦ ψ x ) . A combination of these estimates gives the desired result. General Decay Rates for Equal Speeds of Wave Propagation
In this section, we state and prove a general decay result under equal speeds of wave propagationcondition. The exponential and polynomial decay results are only special cases.
Theorem 4.1.
Let ( ϕ , ϕ ) ∈ H (0 , L ) × L (0 , L ) and ( ψ , ψ ) , ( w , w ) ∈ H ∗ (0 , L ) × L ∗ (0 , L ) .Assume that ( A and ( A hold and that k ρ = k ρ and k = k . (4.1) Then for l small enough and for any t > , the solution of ( P ) satisfies, for t > t , E ( t ) ≤ C exp (cid:18) − λ (cid:90) tt ξ ( s ) ds (cid:19) , for p = 1 , (4.2) and E ( t ) ≤ C (cid:32)
11 + (cid:82) tt ξ p − ( s ) ds (cid:33) p − , for 1 < p < , (4.3) where C > is a constant independent of t but may depend on the initial data and λ > is aconstant independent of both t and the initial data. Moreover, if (cid:90) + ∞ t (cid:32)
11 + (cid:82) tt ξ p − ( s ) ds (cid:33) p − dt < + ∞ , for 1 < p < , (4.4)13 hen E ( t ) ≤ C (cid:32)
11 + (cid:82) tt ξ p ( s ) ds (cid:33) p − , for 1 < p < . (4.5) Remark 4.1.
Inequalities (4.3) and (4.4) together give (cid:90) + ∞ E ( t ) dt < + ∞ . Remark 4.2.
The smallness condition on l makes the Bresse system close to Timoshenkosystem and, hence, inherits some of its stability properties. Proof of Theorem 4.1.
Define a functional L by L := N E + (cid:88) j =1 N j I j , where N, N j > for j = 1 , , . . . , with N = N = 1 . Then from (3.1) − (3.6) we have L (cid:48) ( t ) ≤ (cid:20) − ρ ( k N + N ) + lρ | k − k | δ N lρ k k + ε N + δ (cid:21) (cid:90) L ϕ t dx + (cid:20) − ρ (cid:18) N (cid:90) t g ( s ) ds + N − N (cid:19) + ρ δN + cε (1 + N ) (cid:21) (cid:90) L ψ t dx + (cid:20) − lρ + ρ ( k N − N ) + c lρ | k − k | N δ + ε (cid:21) (cid:90) L w t dx + (cid:20) ( N − N ) (cid:90) t g ( s ) ds + k ( N − N ) + k N δ + lk k k ε + lk k ε (cid:90) t g ( s ) ds + δ ( cN + N + N ) + cδ g ( t ) (cid:21) (cid:90) L ψ x dx + (cid:20) − lk k − k ( k N − N ) + lk k ε k + lk k ε k (cid:90) t g ( s ) ds + δ (cid:21) (cid:90) L ( w x − lϕ ) dx + (cid:20) − k (cid:18) − k N − l − N − c δ N (cid:19) + δN (cid:21) (cid:90) L ( ϕ x + ψ + lw ) dx + cδ (1 + N + N + N ) g ◦ ψ x − cδ (1 + N ) g (cid:48) ◦ ψ x + N E (cid:48) ( t )+ (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx. By setting δ = 1 , N = k N , N = 4 k N , δ = k k − g , ε = k k , and ε = k g , where g = (cid:90) ∞ g ( s ) ds , we arrive at 14 (cid:48) ( t ) ≤ − ρ (cid:20) ( k + k ) N − l (cid:18) | k − k | N + k k (cid:19)(cid:21) (cid:90) L ϕ t dx − ρ (cid:18) N (cid:90) t g ( s ) ds − k N (cid:19) (cid:90) L ψ t dx − lρ (cid:18) − c | k − k | N (cid:19) (cid:90) L w t dx − (cid:20) ( k − g ) k N − lk (cid:18) k g (cid:19)(cid:21) (cid:90) L ψ x dx − lk k (cid:90) L ( w x − lϕ ) dx − k (cid:20) − (cid:18) k + k + 2 c k k k − g (cid:19) N − l (cid:21) (cid:90) L ( ϕ x + ψ + lw ) dx +(1 + N ) ε (cid:90) L ( ϕ t + w t ) dx + cε (1 + N ) (cid:90) L ψ t dx + N E (cid:48) ( t )+ δ (cid:90) L (cid:16) ϕ t + ρ N ψ t + c ( N + 5 k N ) ψ x + N ( ϕ x + ψ + lw ) (cid:17) dx + cδ (1 + N + 5 k N ) g ◦ ψ x + cδ (1 + N ) (cid:20) g ( t ) (cid:90) L ψ x dx − g (cid:48) ◦ ψ x (cid:21) + (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx. Now, we set ε = lρ N ) , to get L (cid:48) ( t ) ≤ − ρ (cid:20) ( k + k ) N − l (cid:18)
12 + k k + | k − k | N (cid:19)(cid:21) (cid:90) L ϕ t dx − ρ (cid:18) N (cid:90) t g ( s ) ds − k N − c (1 + N ) lρ ρ (cid:19) (cid:90) L ψ t dx − lk k (cid:90) L ( w x − lϕ ) dx − lρ − c | k − k | N ) (cid:90) L w t dx − (cid:20) ( k − g ) k N − lk (cid:18) k g (cid:19)(cid:21) (cid:90) L ψ x dx − k (cid:20) − (cid:18) k + k + 2 c k k k − g (cid:19) N − l (cid:21) (cid:90) L ( ϕ x + ψ + lw ) dx + δc N ,N E ( t ) + (cid:104) N − cδ (1 + N ) (cid:105) E (cid:48) ( t ) + cδ (1 + N + 5 k N ) g ◦ ψ x + (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx. Fix t > and choose N so small that − c | k − k | N > − (cid:18) k + k + 2 c k k k − g (cid:19) N > . Next, we select l small enough so that ( k + k ) N − l (cid:18)
12 + k k + | k − k | N (cid:19) > , ( k − g ) k N − lk (cid:18) k g (cid:19) > , − (cid:18) k + k + 2 c k k k − g (cid:19) N − l > . After that, we pick N very large so that N (cid:90) t g ( s ) ds − k N − c (1 + N ) lρ ρ > . Therefore, we have L (cid:48) ( t ) ≤ − ( β − cδ ) E ( t ) + (cid:16) N − cδ (cid:17) E (cid:48) ( t ) + cδ g ◦ ψ x + (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx, for some β > . At this point, we take δ < βc . Consequently, we obtain, for some k > , L (cid:48) ( t ) ≤ − kE ( t ) + ( N − c ) E (cid:48) ( t ) + c ( g ◦ ψ x )+ (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx, ∀ t ≥ t . (4.6)Finally, we choose N so large that N > c and
L ∼ E , therefore we have, ∀ t ≥ t , L (cid:48) ( t ) ≤ − kE ( t ) + c ( g ◦ ψ x ) + (cid:18) k ρ k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx + ρ (cid:18) k k − (cid:19) (cid:90) L ϕ xt w t dx. (4.7)Note that from this point, the proof goes similarly as in [20]. But we will continue for thesake of completeness.By recalling (4.1) and multiplying both sides of (4.7) by ξ ( t ) and using Corollary 2.1, wearrive at ξ ( t ) L (cid:48) ( t ) ≤ − kξ ( t ) E ( t ) + cξ ( t )( g ◦ ψ x )( t ) ≤ − kξ ( t ) E ( t ) + c ( − E (cid:48) ( t )) p − , ∀ t ≥ t . (4.8)For p = 1 , it follows from non-increasing property of ξ and (4.8) that (cid:0) ξ ( t ) L ( t ) + cE ( t ) (cid:1) (cid:48) ≤ ξ ( t ) L (cid:48) ( t ) + cE (cid:48) ( t ) ≤ − kξ ( t ) E ( t ) , ∀ t ≥ t . Using the fact that F = ξ L + cE ∼ E , there exists a λ > such that F (cid:48) ( t ) ≤ − λξ ( t ) F ( t ) , ∀ t ≥ t . A simple integration over ( t, t ) leads to E ( t ) ≤ C exp (cid:18) − λ (cid:90) tt ξ ( s ) ds (cid:19) , ∀ t ≥ t . For < p < , we multiply both sides of (4.8) by ( ξE ) α ( t ) , with α = 2 p − , to obtain ξ α +1 ( t ) E α ( t ) L (cid:48) ( t ) ≤ − k ( ξE ) α +1 ( t ) + c ( ξE ) α ( t )( − E (cid:48) ( t )) p − . q = α + 1 α and q (cid:48) = α + 1 , we get ξ α +1 ( t ) E α ( t ) L (cid:48) ( t ) ≤ − ( k − cγ )( ξE ) α +1 ( t ) − c γ E (cid:48) ( t ) , ∀ γ > . We choose γ such that λ := k − cγ > and use the non-increasing property of ξ and E , tohave ( ξ α +1 E α L ) (cid:48) ( t ) ≤ ξ α +1 ( t ) E α ( t ) L (cid:48) ( t ) ≤ − λ ( ξE ) α +1 ( t ) − cE (cid:48) ( t ) , this entails that ( ξ α +1 E α L + cE ) (cid:48) ( t ) ≤ − λ ( ξE ) α +1 ( t ) . Let F = ξ α +1 E α L + cE ∼ E , then F (cid:48) ( t ) ≤ − λξ α +1 ( t ) F α +1 ( t ) , for some λ > . Integration over ( t , t ) gives E ( t ) ≤ C (cid:32)
11 + (cid:82) tt ξ p − ( s ) ds (cid:33) p − , ∀ t ≥ t . This establishes (4.3).To prove (4.5), we treat (4.8) as follows ξ ( t ) L (cid:48) ( t ) ≤ − kξ ( t ) E ( t ) + cξ ( t )( g ◦ ψ x )( t ) ≤ − kξ ( t ) E ( t ) + c η ( t ) η ( t ) (cid:90) t (cid:0) ξ p ( s ) g p ( s ) (cid:1) p (cid:107) ψ x ( t ) − ψ x ( t − s ) (cid:107) ds, (4.9)for any t ≥ t , where η ( t ) = (cid:90) t (cid:107) ψ x ( t ) − ψ x ( t − s ) (cid:107) ds ≤ (cid:90) t ( (cid:107) ψ x ( t ) (cid:107) + (cid:107) ψ x ( t − s ) (cid:107) ) ds ≤ (cid:90) t ( E ( t ) + E ( t − s )) ds ≤ (cid:90) t E ( t − s ) ds = 8 (cid:90) t E ( s ) ds ≤ (cid:90) + ∞ E ( s ) ds < + ∞ , by Remark 4.1. Applying Jensen’s inequality to the second term in the right-hand side of (4.9),with G ( y ) = y p , y > , f ( s ) = ξ p ( s ) g p ( s ) and h ( s ) = (cid:107) ψ x ( t ) − ψ x ( t − s ) (cid:107) , we obtain ξ ( t ) L (cid:48) ( t ) ≤ − kξ ( t ) E ( t ) + cη ( t ) (cid:18) η ( t ) (cid:90) t ξ p ( s ) g p ( s ) (cid:107) ψ x ( t ) − ψ x ( t − s ) (cid:107) ds (cid:19) p , where we assume that η ( t ) > , otherwise we get, from (4.7), E ( t ) ≤ C exp( − kt ) , ∀ t ≥ t . ξ ( t ) L (cid:48) ( t ) ≤ − kξ ( t ) E ( t ) + cη p − p ( t ) (cid:18) ξ p − (0) (cid:90) t ξ ( s ) g p ( s ) (cid:107) ψ x ( t ) − ψ x ( t − s ) (cid:107) ds (cid:19) p ≤ − kξ ( t ) E ( t ) + c ( − g (cid:48) ◦ ψ x ) p ( t ) ≤ − kξ ( t ) E ( t ) + c ( − E (cid:48) ( t )) p . Multiplying both sides of the above inequality by ( ξE ) α ( t ) , for α = p − , and repeating theabove computations, we arrive at E ( t ) ≤ C (cid:32)
11 + (cid:82) tt ξ p ( s ) ds (cid:33) p − , ∀ t > t , which establishes (4.5). Example 4.1.
Let g ( t ) = a (1 + t ) q with q > , and a > is to be chosen so that ( A issatisfied. Then g (cid:48) ( t ) = − a (cid:18) a (1 + t ) q (cid:19) q +1 q = − ξ ( t ) g p ( t ) , with ξ ( t ) = a = qa /q and p = q + 1 q < , we have, for any fixed t > , (cid:90) + ∞ t (cid:32)
11 + (cid:82) tt ξ p − ( s ) ds (cid:33) p − dt = (cid:90) + ∞ t (cid:18)
11 + c ( t − t ) (cid:19) p − dt < + ∞ . Therefore, inequality (4.5) entails that there exists
C > such that E ( t ) ≤ C (cid:32)
11 + (cid:82) tt ξ p ( s ) ds (cid:33) p − = c (1 + t ) q , with the optimal decay rate q . For more examples, see [20]. General Decay Rate for Different Speeds of Wave Propagation
In this section, we state and prove a generalized decay result in the case of non-equal speedsof wave propagation. We start by differentiating both sides of the differential equations in ( P )with respect to t and use the fact that ∂∂t (cid:20)(cid:90) t g ( t − s ) ψ xx ( s ) ds (cid:21) = ∂∂t (cid:20)(cid:90) t g ( s ) ψ xx ( t − s ) ds (cid:21) = g ( t ) ψ xx (0) + (cid:90) t g ( s ) ψ xxt ( t − s ) ds = (cid:90) t g ( t − s ) ψ xxt ( s ) ds + g ( t ) ψ xx ,
18o obtain the following system ρ ϕ ttt − k ( ϕ xt + ψ t + lw t ) x − lk ( w xt − lϕ t ) = 0 ,ρ ψ ttt − k ψ xxt + k ( ϕ xt + ψ t + lw t ) + (cid:90) t g ( t − s ) ψ xxt ( s ) ds + g ( t ) ψ xx = 0 ,ρ w ttt − k ( w xt − lϕ t ) x + lk ( ϕ xt + ψ t + lw t ) = 0 . ( P ∗ )The energy functional associated to ( P ∗ ) is given by E ∗ ( t ) := 12 (cid:90) L (cid:20) ρ ϕ tt + ρ ψ tt + ρ w tt + (cid:18) k − (cid:90) t g ( s ) ds (cid:19) ψ xt + k ( w xt − lϕ t ) + k ( ϕ xt + ψ t + lw t ) (cid:21) dx + 12 ( g ◦ ψ xt )( t ) , ∀ t ≥ , (5.1)Using similar arguments as in [14, Lemma 3.11] we have the following result. Lemma 5.1.
Let ( ϕ, ψ, w ) be the strong solution of ( P ) . Then, the energy of ( P ∗ ) satisfies,for all t ≥ , E (cid:48)∗ ( t ) = − g ( t ) (cid:90) L ψ xt dx + 12 ( g (cid:48) ◦ ψ xt ) − g ( t ) (cid:90) L ψ tt ψ xx dx (5.2) and E ∗ ( t ) ≤ c (cid:18) E ∗ (0) + (cid:90) L ψ xx dx (cid:19) . (5.3) Lemma 5.2.
Assume that hypotheses ( A and ( A hold and let ( ϕ, ψ, w ) be the strong solutionof ( P ) . Then, for any < σ < , we have g ◦ ψ xt ≤ (cid:20) c (cid:18) E ∗ (0) + (cid:90) L ψ xx dx (cid:19) (cid:90) t g − σ ( s ) ds (cid:21) p − p + σ − (cid:0) g p ◦ ψ xt (cid:1) σp + σ − . In particular, for σ = , we get the following inequality g ◦ ψ xt ≤ c (cid:18)(cid:90) t g / ( s ) ds (cid:19) p − p − ( g p ◦ ψ xt ) p − . (5.4) Proof.
By setting r = p + σ − p − and q = ( p − − σ ) p + σ − , we have rr − p + σ − σ and19 − q = σpp + σ − . Then exploiting Hölder’s inequality and (5.3), we obtain g ◦ ψ xt = (cid:90) L (cid:90) t g ( t − s )( ψ xt ( t ) − ψ xt ( s )) dsdx = (cid:90) L (cid:90) t (cid:104) g q ( t − s )( ψ xt ( t ) − ψ xt ( s )) r (cid:105) (cid:104) g − q ( t − s )( ψ xt ( t ) − ψ xt ( s )) r − r (cid:105) dsdx ≤ (cid:20)(cid:90) L (cid:90) t g qr ( t − s )( ψ xt ( t ) − ψ xt ( s )) dsdx (cid:21) r × (cid:20)(cid:90) L (cid:90) t g (1 − q ) rr − ( t − s )( ψ xt ( t ) − ψ xt ( s )) dsdx (cid:21) r − r ≤ (cid:20)(cid:90) L (cid:90) t g − σ ( t − s )( ψ xt ( t ) − ψ xt ( s )) dsdx (cid:21) p − p + σ − ( g p ◦ ψ xt ) σp + σ − ≤ (cid:20) (cid:90) L (cid:90) t g − σ ( s )( ψ xt ( t ) + ψ xt ( t − s )) dsdx (cid:21) p − p + σ − ( g p ◦ ψ xt ) σp + σ − ≤ (cid:20) k − g (cid:90) t g − σ ( s )( E ∗ ( t ) + E ∗ ( t − s )) ds (cid:21) p − p + σ − ( g p ◦ ψ xt ) σp + σ − ≤ (cid:20) c (cid:18) E ∗ (0) + (cid:90) L ψ xx dx (cid:19) (cid:90) t g − σ ( s ) ds (cid:21) p − p + σ − ( g p ◦ ψ xt ) σp + σ − . For σ = , we get (5.4). This completes the proof. Corollary 5.1.
Assume that conditions ( A and ( A hold and let ( ϕ, ψ, w ) be the strongsolution of ( P ) . Then, ξ ( t )( g ◦ ψ xt )( t ) ≤ c (cid:0) − E (cid:48)∗ ( t ) + c g ( t ) (cid:1) p − , ∀ t ≥ , for some positive constant c .Proof. From equation (5.2) and inequality (5.3) we have ≤ − g (cid:48) ◦ ψ xt = − E (cid:48)∗ ( t ) − g ( t ) (cid:90) L ψ xt dx − g ( t ) (cid:90) L ψ tt ψ xx dx ≤ − E (cid:48)∗ ( t ) − g ( t ) (cid:90) L ψ tt ψ xx dx ≤ − E (cid:48)∗ ( t ) + g ( t ) (cid:90) L ( ψ tt + ψ xx ) dx ≤ − E (cid:48)∗ ( t ) + g ( t ) (cid:18) ρ E ∗ ( t ) + (cid:90) L ψ xx dx (cid:19) ≤ c ( − E (cid:48)∗ ( t ) + c g ( t )) , (5.5)20or some positive constant c . Multiplication of both sides of (5.4) by ξ ( t ) and use of Lemma2.1 and inequality (5.5) give ξ ( t )( g ◦ ψ xt )( t ) ≤ c (cid:18) ξ ( t ) (cid:90) t g / ( s ) ds (cid:19) p − p − (cid:0) ξg p ◦ ψ xt (cid:1) p − ( t ) ≤ c (cid:18)(cid:90) t ξ ( s ) g / ( s ) ds (cid:19) p − p − (cid:0) − g (cid:48) ◦ ψ xt (cid:1) p − ( t ) ≤ c (cid:0) − E (cid:48)∗ ( t ) + c g ( t ) (cid:1) p − . Now we estimate the third term in the right-hand side of (4.7) as in [14].
Lemma 5.3.
Let ( ϕ, ψ, w ) be the strong solution of ( P ) . Then, for any ε > , we have (cid:18) ρ k k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx ≤ εE ( t ) + cε ( g ◦ ψ xt − E (cid:48) ( t ) + g ( t )) , ∀ t ≥ t . (5.6) Proof. (cid:18) ρ k k − ρ (cid:19) (cid:90) L ϕ t ψ xt dx = (cid:16) ρ k k − ρ (cid:17)(cid:82) t g ( s ) ds (cid:90) L ϕ t (cid:90) t g ( t − s )( ψ xt ( t ) − ψ xt ( s )) dsdx + (cid:16) ρ k k − ρ (cid:17)(cid:82) t g ( s ) ds (cid:90) L ϕ t (cid:90) t g ( t − s ) ψ xt ( s ) dsdx. (5.7)By observing that (cid:90) t g ( s ) ds ≥ (cid:90) t g ( s ) ds , for all t ≥ t and exploiting Young’s inequality andLemma 2.3 (for ψ xt ), we get, for ε > and t ≥ t , (cid:16) ρ k k − ρ (cid:17)(cid:82) t g ( s ) ds (cid:90) L ϕ t (cid:90) t g ( t − s )( ψ xt ( t ) − ψ xt ( s )) dsdx ≤ ε ρ (cid:90) L ϕ t dx + cε ( g ◦ ψ xt ) . On the other hand, by integration by parts and using Lemma 2.3 (for − g (cid:48) and ψ x ) and the factthat E is non-increasing, we get (cid:16) ρ k k − ρ (cid:17)(cid:82) t g ( s ) ds (cid:90) L ϕ t (cid:90) t g ( t − s ) ψ xt ( s ) dsdx = (cid:16) ρ k k − ρ (cid:17)(cid:82) t g ( s ) ds (cid:90) L ϕ t (cid:18) g (0) ψ x − g ( t ) ψ x + (cid:90) t g (cid:48) ( t − s ) ψ x ( s ) ds (cid:19) dx = (cid:16) ρ k k − ρ (cid:17)(cid:82) t g ( s ) ds (cid:90) L ϕ t (cid:18) g ( t )( ψ x − ψ x ) − (cid:90) t g (cid:48) ( t − s )( ψ x ( t ) − ψ x ( s )) ds (cid:19) dx ε ρ (cid:90) L ϕ t dx + cε g ( t ) (cid:90) L ( ψ x + ψ x ) dx − cε g (cid:48) ◦ ψ x ≤ ε ρ (cid:90) L ϕ t dx + cε E (0) g ( t ) − cε g (cid:48) ◦ ψ x . Inserting the last two inequalities in (5.7), we get (5.6).
Theorem 5.1.
Let ( ϕ , ϕ ) ∈ (cid:0) H (0 , L ) ∩ H (0 , L ) (cid:1) × H (0 , L ) and ( ψ , ψ ) , ( w , w ) ∈ ( H ∗ (0 , L ) ∩ H ∗ (0 , L )) × H ∗ (0 , L ) . Assume that conditions ( A , ( A hold and that ρ k (cid:54) = ρ k and k = k . Then for l small enough and for any t > , there exists a positive constant C that may dependon the initial data but independent of t , for which the strong solution of ( P ) satisfies, for t > t , E ( t ) ≤ C (cid:32) (cid:82) tt ξ p − ( s ) ds (cid:33) p − , for 1 ≤ p < . (5.8) Proof.
Repeating the steps of the proof of Theorem 4.1 up to inequality (4.6), then inserting(5.6) into (4.6) we obtain L (cid:48) ( t ) ≤ − ( k − ε ) E ( t ) + (cid:20) N − c (cid:18) ε (cid:19)(cid:21) E (cid:48) ( t ) + cg ◦ ψ x + cε g ◦ ψ xt + cε g ( t ) , ∀ t ≥ t . Now we choose ε so small that k − ε > , and then pick N > c (cid:0) ε (cid:1) to get L (cid:48) ( t ) ≤ − k E ( t ) + c ( g ◦ ψ x + g ◦ ψ xt ) + cg ( t ) , ∀ t ≥ t , for some k > . We then multiply both sides of the above inequality by ξ ( t ) and use Corollaries2.1 and 5.1 to get ξ ( t ) L (cid:48) ( t ) ≤ − k ξ ( t ) E ( t ) + cξ ( t )( g ◦ ψ x + g ◦ ψ xt ) + cξ ( t ) g ( t ) ≤ − k ξ ( t ) E ( t ) + c (cid:104)(cid:0) − E (cid:48) ( t ) (cid:1) p − + (cid:0) − E (cid:48)∗ ( t ) + c g ( t ) (cid:1) p − (cid:105) + cξ ( t ) g ( t ) . Next, we set α = 2 p − , then multiply both sides of the above inequality by ( ξE ) α ( t ) andexploit Young’s inequality, with q = α + 1 α and q (cid:48) = α + 1 , to obtain ξ α +1 ( t ) E α ( t ) L (cid:48) ( t ) ≤ − ( k − cγ )( ξE ) α +1 ( t ) − cE (cid:48) ( t ) − cE (cid:48)∗ ( t )+ c g ( t )+ cξ α +1 ( t ) E α ( t ) g ( t ) , ∀ γ > . We choose γ > so small such that λ := k − cγ > and use the non-increasing property of ξ and g to get ( ξ α +1 E α L + cE + cE ∗ ) (cid:48) ( t ) ≤ − λ ( ξE ) α +1 ( t ) + cξ α +1 ( t ) E α ( t ) g ( t ) + c g ( t ) , λ ( ξE ) α +1 ( t ) ≤ − ( ξ α +1 E α L + cE + cE ∗ ) (cid:48) ( t ) + cξ α +1 ( t ) E α ( t ) g ( t ) + c g ( t ) . Then integration over ( t , t ) together with the non-increasing property of E and ξ , and thehypothesis ( A yield, for t ≥ t , λ E α +1 ( t ) (cid:90) tt ξ α +1 ( s ) ds ≤ λ (cid:90) tt ( ξE ) α +1 ( s ) ds ≤ − ( ξ α +1 E α L + cE + cE ∗ )( t )+( ξ α +1 E α L + cE + cE ∗ )(0) + (cid:90) L ψ xx dx +( cξ α +1 (0) E α (0) + c ) (cid:90) tt g ( s ) ds ≤ ( ξ α +1 E α L + cE + cE ∗ )(0) + (cid:90) L ψ xx dx +( cξ α +1 (0) E α (0) + c ) (cid:90) ∞ g ( s ) ds. Therefore, we get E ( t ) ≤ C (cid:32) (cid:82) tt ξ p − ( s ) ds (cid:33) p − , ∀ t > t . This completes the proof of the Theorem 5.1.
Example 5.1.
Let g ( t ) = e − at , where a > . Then g (cid:48) ( t ) = − ξ ( t ) g ( t ) with ξ ( t ) = a . It followsfrom (5.8) that for any fixed t > , there exists C > such that E ( t ) ≤ Ct − t , ∀ t > t . Example 5.2.
Consider the same function g as in Example 4.1 and write g (cid:48) as in Example 4.1.Then it follows from (5.8) that for any fixed t > , there exists C > such that E ( t ) ≤ C (cid:32) (cid:82) tt ξ p − ( s ) ds (cid:33) p − = c (1 + t ) qq +2 ∀ t > t . For more examples, see [14].
In this section, we introduce a scheme for the problem based on P -finite element method inspace and implicit Euler scheme for time discretization. Then we draw graphs for the discreditedenergy showing it’s decay in both cases, polynomial and exponential. Finally, we implementthe approximation of the solutions ϕ, ψ and w in D and their cross section at x = 0 . .23 .1 Finite element setup We denote by (Γ h ) h a partition of Ω which fulfills the following conditions:1. Γ h = { R ⊂ ¯Ω; R is closed in Ω } ;2. ∀ ( R, R (cid:48) ) ∈ Γ h × Γ h ; | R | = | R (cid:48) | , where their intersection is either empty or an end point;3. ¯Ω = (cid:83) R ∈ Γ h R .We define the uniform partition of Ω as x < x < · · · < x s and denote the length of ( x j , x j +1 ) as h = Ls . Now for time discretization, denote by ∆ t = TN the step time, where T isthe total time and N is a positive integer. Finally we define the discrete finite element spaceby s h = { u h ∈ H (0 , L ) : ∀ R ∈ Γ h ; u h | R ∈ P ( R ) } , where P k ( R ) denotes the space of restrictions of R of polynomials with one variable and oforder less than or equal to k .Now we introduce the scheme and the discrete energy by using implicit Euler scheme ρ ∆ t (Φ nh − Φ n − h , ¯ ϕ h ) + k ( ϕ nh,x + ψ nh + lw nh , ¯ ϕ h,x ) − lk ( w nh,x − lϕ nh , ¯ ϕ h ) = 0 ρ ∆ t (Ψ nh − Ψ n − h , ¯ ψ h ) + k ( ψ nh,x , ¯ ψ x ) + k ( ϕ nh,x + ψ nh + + lw nh , ¯ ψ h ) − ∆ t n (cid:80) m =1 g ( t n − m )( ψ mh,x , ¯ ψ h,x ) = 0 ρ ∆ t ( W nh − W n − h , ¯ w h ) + k ( w nh,x − lϕ nh , ¯ w h,x ) + k l ( ϕ nh,x + ψ nh + + lw nh , ¯ w h ) = 0 where t j = j ∆ t and E n = ρ || Φ nh || + ρ || W nh || + ρ || Ψ nh || + k (cid:12)(cid:12)(cid:12)(cid:12) ϕ nh,x + ψ nh + lw nh (cid:12)(cid:12)(cid:12)(cid:12) + k (cid:12)(cid:12)(cid:12)(cid:12) w nh,x − lϕ nh (cid:12)(cid:12)(cid:12)(cid:12) + k (cid:12)(cid:12)(cid:12)(cid:12) ψ nh,x (cid:12)(cid:12)(cid:12)(cid:12) − t n (cid:90) g ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ψ nh,x (cid:12)(cid:12)(cid:12)(cid:12) + 12 ∆ t L (cid:90) n (cid:88) m =1 g ( t n − m )( ψ nh,x − ψ mh,x ) dx By using the following data k = k = k = 1 , ρ = ρ = 0 . , ∆ t = 0 . , h = 0 . , T = 7 . g ( x ) = e − x ; we draw the solutions ϕ, ψ, and w in D (see Figures 1, 2 and 3, respectively) and their crosssection at x = 0 . (see Figures 4, 5 and 6, respectively).For the energy we have two cases, taking the conditions of equal and non-equl speeds ofwave propagation.If k k = ρ ρ and k = k we obtain an exponential decay by using the same data taken for thesolutions as shown in the following Figures 7 – 10.24igure 1: The evolution in time and space of ϕ Figure 2: The evolution in time and space of ψ Figure 3: The evolution in time and space of w ϕ at x = 0 . Figure 5: The evolution in time of ψ at x = 0 . w at x = 0 . Figure 7: The evolution in time of E n ln ( E n ) that shows the exponential decayFigure 9: The evolution in time of ln ( E n ) with it’s regression line28igure 10: The evolution in time of ln ( E n ) /t Figure 11: The evolution in time of E n − ln ( E n ) with respect to ln ( t ) Figure 13: The variation of − ln ( E n ) with respect to ln ( t ) with it’s regression line30f k k (cid:54) = ρ ρ and k (cid:54) = k we obtain a polynomial decay by taking the following data k = 5 , k = k = 1 , ρ = 0 . , ρ = 0 . , ∆ t = 0 . , h = 0 . , and total time T = 16 . . with g ( x ) = 1 / ( x + 1) as show in the following Figures 11 – 13. Acknowledgement
The authors would like to express their gratitude to King Fahd University of Petroleum andMinerals (KFUPM) for its continuous support. This work is partially funded by KFUPM underProject SB181018.
References [1]
Afilal, M., Merabtene, T., Rhofir, K., and Soufyane, A.
Decay rates of thesolution of the Cauchy thermoelastic Bresse system.
Zeitschrift für Angew. Math. undPhys. 67 , 5 (2016), 119.[2]
Alabau-Boussouira, F., Muñoz Rivera, J. E., and Almeida Júnior, D. d. S.
Stability to weak dissipative Bresse system.
J. Math. Anal. Appl. 374 , 2 (2011), 481–498.[3]
Alves, M., Fatori, L., Jorge Silva, M., and Monteiro, R.
Stability and optimalityof decay rate for a weakly dissipative Bresse system.
Math. Methods Appl. Sci. 38 , 5 (mar2015), 898–908.[4]
Alves, M. S., V., O. V., Rambaud, A., and Muñoz-Rivera, J.
Exponential stabilityto the Bresse system with boundary dissipation conditions. arXiv Prepr. arXiv1506.01657 (2015).[5]
Borichev, A., and Tomilov, Y.
Optimal polynomial decay of functions and operatorsemigroups.
Math. Ann. 347 , 2 (2010), 455–478.[6]
Bresse, J. A. C.
Cours de mecanique appliquee par M. Bresse: Résistance des matériauxet stabilité des constructions . Mallet-Bachelier, Paris, 1859.[7]
Dell’Oro, F.
Asymptotic stability of thermoelastic systems of Bresse type.
J. Differ.Equ. 258 , 11 (2015), 3902–3927.[8]
Fatori, L. H., and Monteiro, R. N.
The optimal decay rate for a weak dissipativeBresse system.
Appl. Math. Lett. 25 , 3 (2012), 600–604.[9]
Fatori, L. H., and Muñoz Rivera, J. E.
Rates of decay to weak thermoelastic Bressesystem.
IMA J. Appl. Math. (Institute Math. Its Appl. 75 , 6 (2010), 881–904.[10]
Gallego, F. A., and Noz Rivera, J. E. M.
Decay rates for solutions to thermoelasticBresse systems of types I and III.
Electron. J. Differ. Equations 2017 , 73 (2017), 1–26.[11]
Guesmia, A.
Asymptotic stability of Bresse system with one infinite memory in thelongitudinal displacements.
Mediterr. J. Math. 14 , 2 (2017), 49.3112]
Guesmia, A., and Kafini, M.
Bresse system with infinite memories.
Math. MethodsAppl. Sci. 38 , 11 (2015), 2389–2402.[13]
Guesmia, A., and Kirane, M.
Uniform and weak stability of Bresse system with twoinfinite memories.
Zeitschrift für Angew. Math. und Phys. 67 , 5 (2016), 124.[14]
Guesmia, A., and Messaoudi, S. A.
On the stabilization of Timoshenko systems withmemory and different speeds of wave propagation.
Appl. Math. Comput. 219 , 17 (2013),9424–9437.[15]
Keddi, A., Apalara, T., and Messaoudi, S.
Exponential and polynomial decay in athermoelastic-Bresse system with second sound.
Appl. Math. Optim. (2016).[16]
Lasiecka, I., and Tataru, D.
Uniform boundary stabilization of semilinear wave equa-tions with nonlinear boundary damping.
Differ. Integr. Equations 6 , 3 (1993), 507–533.[17]
Liu, Z., and Rao, B.
Energy decay rate of the thermoelastic Bresse system.
Zeitschriftfür Angew. Math. und Phys. 60 , 1 (2009), 54–69.[18]
Liu, Z., and Zheng, S.
Semigroups Associated with Dissipative Systems . CRC Press,1999.[19]
Messaoudi, S. A.
On the control of solutions of a viscoelastic equation.
J. Franklin Inst.344 , 5 (2007), 765–776.[20]
Messaoudi, S. A., and Al-Khulaifi, W.
General and optimal decay for a quasilinearviscoelastic equation.
Appl. Math. Lett. 66 (2017), 16–22.[21]
Messaoudi, S. A., and Edwin Mukiawa, S.
Existence and decay of solutions to aviscoelastic plate equation.
Electron. J. Differ. Equations 2016 , 22 (2016), 1–14.[22]
Najdi, N., and Wehbe, A.
Weakly locally thermal stabilization of Bresse systems.
Electron. J. Differ. Equations 2014 (2014), 1–19.[23]
Qin, Y., Yang, X., and Ma, Z.
Global existence of solutions for the thermoelasticBresse system.
Commun. Pure Appl. Anal. 13 , 4 (2014), 1395–1406.[24]
Said-Houari, B., and Hamadouche, T.
The asymptotic behavior of the Bresse-Cattaneo system.
Commun. Contemp. Math. 18 , 4 (2015), 18 pages.[25]
Said-Houari, B., and Hamadouche, T.
The Cauchy problem of the Bresse system inthermoelasticity of type III.
Appl. Anal. 95 , 11 (nov 2016), 2323–2338.[26]
Said-Houari, B., and Soufyane, A.
The Bresse system in thermoelasticity.
Math.Methods Appl. Sci. 38 , 17 (nov 2015), 3642–3652.[27]
Santos, M. d. L., Soufyane, A., and Júnior, D. A.
Asymptotic behavior to Bressesystem with past history.
Q. Appl. Math. 73 , 2014 (2015), 23–54.[28]
Santos, M. L., and Júnior, D. d. S. A.
Numerical exponential decay to dissipativeBresse system.
J. Appl. Math. 2010 (2010), 1–17.3229]
Soriano, J., Charles, W., and Schulz, R.
Asymptotic stability for Bresse systems.
J. Math. Anal. Appl. 412 , 1 (2014), 369–380.[30]
Soriano, J. A., Muñoz Rivera, J. E., and Fatori, L. H.
Bresse system withindefinite damping.
J. Math. Anal. Appl. 387 , 1 (2012), 284–290.[31]
Wehbe, A., and Youssef, W.
Exponential and polynomial stability of an elastic Bressesystem with two locally distributed feedbacks.