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Energy decay of dissipative plate equations with memory-type boundary conditions


In this paper we consider a plate equation with internal feedback and viscoelastic damping localized on a part of the boundary. Without imposing restrictive assumptions on the time-dependent frictional damping, we establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalizes previous results existing in the literature.


In this paper we are concerned with the following problem

(1.1)utt+Δ2u+θ(t)h(ut)=0in Ω×(0,),(1.2)u=uν=0on Γ0×(0,),(1.3)u+0tg1(ts)β2u(s)ds=0on Γ1×(0,),(1.4)uν+0tg2(ts)β1u(s)ds=0on Γ1×(0,),(1.5)u(x,y,0)=u0(x,y),ut(x,y,0)=u1(x,y)in Ω,
which is a Kirchhoff plates equation with internal frictional damping and memory conditions at a part of the boundary. Here Ω is a bounded domain of R2 with a smooth boundary Ω=Γ0Γ1,ν=(ν1,ν2) is the unit outward normal to Ω,η=(ν2,ν1) is the unit tangent positively oriented on Ω, the integral terms in (1.3) and (1.4) are the memories responsible for the viscoelastic damping where g1, g2 are positive functions called the relaxation functions, θ is a time dependent coefficient of the frictional damping, and h is a specific function. We are denoting by β1, β2 the following differential operators:
and μ(0,12) represents the Poisson coefficient. This system describes the transversal displacement u=u(x,y,t) of a thin vibrating plate subjected to internal time-dependent frictional damping and boundary viscoelastic damping.

The uniform stabilization of Kirchhoff plates with linear or nonlinear internal feedback, with θ1, was investigated by several authors. In Ammari and Tucsnak [4], Cavalcanti et al. [7], Guzman and Tucsnak [11], Komornik [18], Pazoto et al. [39], and Vasconcellos and Teixeira [41], it was proved that if h satisfies

where c1, c2 are positive constants, then for q=1 the energy decay rate is exponential while for q>1 we obtain a polynomial decay rate. Similar results were also obtained for boundary frictional damping (see Horn [14], Komornik [19], Lagnese [21], Lasiecka [22], and Ji and Lasiecka [15]). Decay results for arbitrary growth of the frictional damping term have been given by Amroun and Benaissa [5] motivated by the works done by Lasiecka and Tataru [22], Liu and Zuazua [24], and Martinez [26,27] for damped wave equations. They established an explicit formula for the energy decay rates that need not to be of exponential or polynomial types. Similarly, Han and Wang [12] studied a coupled system of plate and wave equations and used internal frictional damping terms without imposing growth conditions near zero to achieve the stability and controllability of the system. In the presence of the time dependent coefficient θ(t), Mustafa [35] and Mustafa and Messaoudi [36] established for the wave equation a general energy decay result depending on both h and θ.

On the other hand, when the unique damping mechanism is given by memory conditions, we refer to Lagnese [20] and Rivera et al. [32] who considered internal viscoelastic damping and proved that the energy decays exponentially if the relaxation function g decays exponentially and polynomially if g decays polynomially. The same results were obtained by Alabau-Boussouira et al. [3] for a more general abstract equation. For boundary viscoelastic damping, if ki is the resolvent kernel of gigi(0) for i=1,2, Santos and Junior [40] showed that the energy decays exponentially (polynomially), provided the resolvent kernels also decay exponentially (polynomially). In Rivera et al. [33,34] investigated a class of abstract viscoelastic systems of the form

where A is a strictly positive, self-adjoint operator with D(A) a subset of a Hilbert space and ∗ denotes the convolution product in the variable t. The authors showed that solutions for (1.6), when 0<β<1, decay polynomially even if the kernel g decays exponentially. While, in the case β=1, the solution energy decays at the same decay rate as the relaxation function.

Then, a natural question was raised: how does the energy behave as the kernel function does not necessarily decay polynomially or exponentially? Han and Wang gave an answer to the above question when treating (1.6), for β=1, in [13]. They considered relaxation functions satisfying

where ξ:R+R+ is a nonincreasing differentiable function with
for some constant k and showed that the rate of the decay of the energy is exactly the rate of decay of g which is not necessarily of polynomial or exponential decay type. These conditions (1.7) and (1.8) on g where first used by Messaoudi [28,29] in studying a viscoelastic wave equation. After that, Messaoudi and Mustafa [30] and Mustafa and Messaoudi [37] eliminated condition (1.8) and used only (1.7) to establish more general stability results of viscoelastic Timoshenko beams. Similarly, condition (1.7) was used by Ferreira and Messaoudi [9] to treat a nonlinear viscoelastic plate equation with a p(x,t)-Laplacian operator. We also mention the work of Alabau-Boussouira and Cannarsa [2] who considered wave equation with memory whose relaxation function is satisfying
where χ is a non-negative function, with χ(0)=χ(0)=0, and χ is strictly increasing and strictly convex on (0,k0], for some k0>0. They also required that
and proved an energy decay result. In addition to these assumptions, if
then, in this case, an explicit rate of decay is given. Recently, the above conditions were strongly weakened by Mustafa and Messaoudi [38] and an explicit and general decay rate formula was obtained.

The interaction between viscoelastic and frictional dampings was considered by several authors. Cavalcanti and Oquendo [8] looked into wave equation of the form

and established exponential stability for g decaying exponentially and h linear and polynomial stability for g decaying polynomially and h having a polynomial growth near zero. Using (1.7), h having no restrictive growth assumption near the origin, with time dependent coefficient and a(x)=b(x)=1, Liu [25] proved a more general decay result. Similarly, Guesmia and Messaoudi [10] studied Timoshenko systems with frictional versus viscoelastic damping and Messaoudi and Mustafa [31] studied viscoelastic wave equation with boundary feedback and obtained general energy decay estimates. Once again, Kang [16,17] imposed the condition (1.7) on the relaxation functions for viscoelastic dampings in plate models which are also subject to frictional damping and they established general stability results.

Our aim in this work is to investigate (1.1)–(1.5) with both weak frictional damping and boundary viscoelastic damping. We obtain a general relation between the decay rate for the energy (when t goes to infinity) and the functions gi, θ, and h using resolvent kernels of general-type decay and without imposing any growth assumption near the origin on h. The result of this paper generalizes previous related results where it allows a larger class of functions g and h, from which the energy decay rates are not necessarily of exponential or polynomial types and takes into account the effect of a time dependent coefficient θ(t). The proof is based on the multiplier method and makes use of some properties of convex functions including the use of the general Young’s inequality and Jensen’s inequality. These convexity arguments were introduced by Lasiecka and Tataru [23] and used by Liu and Zuazua [24] and Alabau-Boussouira [1]. The paper is organized as follows. In Section 2, we present some notation and material needed for our work. Some technical lemmas and the proof of our main result will be given in Section 3.


We use the standard Lebesgue and Sobolev spaces with their usual scalar products and norms. Throughout this paper, c is used to denote a generic positive constant. We first consider the following hypothesis


Ω is a bounded domain of R2 with a smooth boundary Ω=Γ0Γ1, where Γ0 and Γ1 are closed and disjoint, with meas(Γ0)>0, ν =(ν1,ν2) is the unit outward normal to Ω,η=(ν2,ν1) is the unit tangent positively oriented on Ω, and there exists a fixed point x0R2 such that, for m(x)=xx0, m·ν0 on Γ0 and m·ν>0 on Γ1.


Hypothesis (A1) implies that there exist constants δ0 and R such that

m·νδ0>0on Γ1and|m(x)|Rfor all xΩ.
We denote by ki the resolvent kernel of (gi/gi(0)) which satisfies
where ∗ points to the convolution product
By differentiating Eqs (1.3) and (1.4), we arrive at the following Volterra equations:
Using the Volterra’s inverse operator and taking τi=1gi(0), for i=1,2, we get
β2u=τ1{ut+k1ut},on Γ1×(0,),β1u=τ2{utν+k2utν},on Γ1×(0,),
which gives, assuming throughout the paper that u00,
(2.1)β2u=τ1{ut+k1(0)u+k1u},on Γ1×(0,),(2.2)β1u=τ2{utν+k2(0)uν+k2uν},on Γ1×(0,).
Therefore, we use (2.1) and (2.2) instead of the boundary conditions (1.3) and (1.4), and also consider the following assumptions.


ki:R+R+, for i=1,2, are C2 functions such that

and there exists a positive function HC1(R+) and H is a linear function or it is a strictly increasing and strictly convex C2 function on (0,r], r<1, with H(0)=H(0)=0, such that


h:RR is a nondecreasing C0 function and there exist constants c1,c2>0 such that

c1|s||h(s)|c2|s|if |s|r,s2+h2(s)H1(sh(s))if |s|r.


θ:R+R+ is a nonincreasing C1 function.

In the sequel we assume that system (1.1)–(1.5) has a unique solution

where W={wH2(Ω):w=wν=0 on Γ0}. This result can be proved, for initial data in suitable function spaces, using standard arguments such as the Galerkin method (see [40]).

Let us define the bilinear form a(·,·) as follows

and, as measΓ0>0, we know that a(u,u) is an equivalent norm on W; that is, for some positive constants α and β,
We state the following lemma which will be useful in what follows.

Lemma 2.1

Lemma 2.1([21]).

Let u and v be functions in H4(Ω) and μR. Then we have

Now, we introduce the energy functional

Our main stability result is the following

Theorem 2.1.

Assume that (A1)(A4) hold. Then there exist positive constants c1, c2, c3 and ε0 such that the solution of (1.1)–(1.5) satisfies

provided that D is a positive C1 function, with D(0)=0, for which H0 is a strictly increasing and strictly convex C2 function on (0,r] and
(2.8)0+ki(s)H01(ki(s))ds<+for i=1,2.
Moreover, if 01H1(t)dt<+ for some choice of D, then we have the improved estimate
(2.9)E(t)c3G1(c10tθ(s)ds+c2)where G(t)=t11sH(ε0s)ds.
In particular, this last estimate is valid for the special case H(t)=ctp, for 1p<32.


  • 1. Using the properties of H, one can show that the function H1 is strictly decreasing and convex on (0,1], with limt0H1(t)=+. Therefore, Theorem 2.1 ensures


  • 2. Hypothesis (A3) implies that sh(s)>0, for all s0.

  • 3. The condition (A3), with r=1 and θ1, was introduced and employed by Lasiecka and Tataru [23] in their study of the asymptotic behavior of solutions of nonlinear wave equations with nonlinear frictional boundary damping where they obtained decay estimates that depend on the solution of an explicit nonlinear ordinary differential equation. It was also shown there that the monotonicity and continuity of h guarantee the existence of the function H with the properties stated in (A3). In our present work, we study the plate equation with both frictional damping, modulated by a time dependent coefficient θ(t), and boundary viscoelastic damping. We investigate the influence of these simultaneous damping mechanisms on the decay rate of the energy and establish an explicit and general energy decay formula, depending on the resolvent kernels k1 and k2, h, and θ.

  • 4. The usual exponential and polynomial decay rate estimates, already proved for H(t)=ctp, 1p<3/2, are special cases of our result. We will provide a “simpler” proof for these special cases.

  • 5. The condition kid(ki)p, 1p<3/2, assumes (ki(t))ωedt when p=1 and (ki(t))ωt1p1 when 1<p<3/2. Our result allows resolvent kernels whose derivatives are not necessarily of exponential or polynomial decay. For instance, if

    for 0<q<1, then ki(t)=H(ki(t)) where, for t(0,r], r<1,
    which satisfies hypothesis (A2). Also, by taking D(t)=tα, (2.8) is satisfied for any α>1. Therefore, if h satisfies (A3) with this function H, then we can use Theorem 2.1 and do some calculations (see [38]) to deduce that the energy decays at the rate

  • 6. The well-known Jensen’s inequality will be of essential use in establishing our main result. If F is a convex function on [a,b], f:Ω[a,b] and j are integrable functions on Ω, j(x)0, and Ωj(x)dx=C>0, then Jensen’s inequality states that


  • 7. Since limt+ki(t)=0 and (ki(t)) is nonnegative and nonincreasing, then we can easily deduce that limt+(ki(t))=0. Similarly, assuming the existence of the limit, we find that limt+ki(t)=0. Hence, there is t1>0 large enough such that ki(t1)<0 and

    As ki is nondecreasing, ki(0)<0 and ki(t1)<0, then ki(t)<0 for any t[0,t1] and
    Therefore, since H is a positive continuous function, then
    for some positive constants a and b. Consequently, for all t[0,t1],
    which gives, for some positive constant d,

  • 8. If different functions H1, H2, and H3 have the properties mentioned in (A2) and (A3) such that k1(t)H1(k1(t)), k2(t)H2(k2(t)), and s2+h2(s)H31(sh(s)), then there is r<min{r1,r2,r3} small enough so that, say, H1(t)min{H2(t),H3(t)} on the interval (0,r]. Thus, the function H(t)=H1(t) satisfies both (A2) and (A3), tt1.

3.Proof of the main result

In this section we prove Theorem 2.1. For this purpose, we establish several lemmas.

Lemma 3.1.

Under the assumptions (A1)(A4), the energy functional satisfies, along the solution of (1.1), the estimate



Multiplying the equation (1.1) by ut, integrating by parts over Ω, and using (2.5) and the boundary conditions (2.1) and (2.2), we get

Then, making use of the identity
our conclusion follows. □

Now we are going to construct a Lyapunov functional L equivalent to E, with which we can show the desired result.

Lemma 3.2.

Under the assumptions (A1)(A4), the functional

satisfies, along the solution,the estimate


Direct computations, taking v=u in (2.6), we get

Let us examine the integrals over Γ0 in (3.3). Since u=uν=0 on Γ0, we have B1u=B2u=0 on Γ0 and
ν(m·u)=(m·ν)Δu,uxx2+uyy2+2μuxxuyy+2(1μ)uxy2=(Δu)2on Γ0
uxxuyy(uxy)2=0on Γ0.
Therefore, from (3.3), we have
Using the Young inequality, we have
where ϵ is a positive constant. Using the trace theory, we obtain
Substituting the inequalities (3.5)–(3.8) into (3.4) and taking into account the fact that m·ν0 on Γ0, we have
Since, by Hölder inequality,
Consequently, our conclusion easily follows. □

Proof of Theorem 2.1.

For N>0, we define

Combining (3.1) and (3.2) and using the facts that ki<0, ki>0 and |m·ν|R, we obtain
Then choosing 0<ϵ<1c, and N large enough so that
so, we arrive at
which, using Trace theory and the fact that limtki(t)=0, for i=1,2, yields, for large t1,
On the other hand, we can choose N even larger (if needed) so that
which means that, for some constants α1,α2>0,
Now, we consider the following partition of Ω
and use (A3), (A4), (2.11), and (3.1) to conclude that, for any tt1,
Next, we take F(t)=θ(t)L(t)+2cE(t), which is clearly equivalent to E(t) as θ is nonincreasing, and use (3.10) and (3.12)–(3.13), to get, for all tt1,

  • (I) H(t)=ctp and 1p<32: This means, using Holder’s inequality, that


    • Case 1. p=1: Estimate (3.14) yields

      which gives
      Hence, using the fact that F+cEE, we easily obtain

    • Case 2. 1<p<32: One can easily show that 0+[ki(s)]1δ0ds<+ for any δ0<2p and i=1,2. Using this fact (3.1), and the trace theory and choosing t1 even larger if needed, we deduce that, for all tt1,

      Then, Jensen’s inequality (3.1), hypothesis (A2), and (3.15) lead to
      Then, particularly for δ0=12, we find that (3.14) becomes
      Now, we multiply by E2p2(t) to get, using (3.1),
      Then, Young’s inequality gives
      Consequently, as E2p(t)E(0)E2p1(t), picking ε+δE(0)<m, we obtain
      where F0=FE2p2+CEE. Hence we have, for some a0>0,
      from which we easily deduce that
      By recalling that p<3/2 and using (3.17), we find that 0+θ(s)E(s)ds<+. Hence, by noting that
      estimate (3.14) gives
      Therefore, repeating the above steps, with multiplying by Ep1(t), we arrive at

  • (II) The general case: We define I(t) by

    where H0 is such that (2.8) is satisfied. As in (3.15), we find that I(t) satisfies, for all tt1,
    We also assume, without loss of generality that I(t)>0, for all t>t1; otherwise (3.14) yields an exponential decay. In addition, we define ξ(t) by
    and infer from (A2) and the properties of H0 and D that
    for some positive constant k0. Then, using (3.1) and choosing t1 even larger (if needed), one can easily see that ξ(t) satisfies, for all t>t1,
    Since H0 is strictly convex on (0,r] and H0(0)=0, then
    provided 0μ1 and x (0,r]. The use of this fact, hypothesis (A2), (2.10), (3.18), (3.19), and Jensen’s inequality leads to
    This implies that
    We also define
    We similarly deduce, for all t>t1, that
    Repeating the above steps, we arrive at
    Now we estimate the last integral in (3.14). First, we can assume that r is small enough such that
    (3.23)sh(s)13min{r,H(r),H0(r)}for all |s|r.
    Then, with S(t) defined by
    (A3) and Jensen’s inequality give
    Inserting the estimates (3.20), (3.22), and (3.24) into (3.14), we obtain
    One can easily make use of the properties of H,D,H0 and the fact that H01(S(t))=D1(H1(S(t))), D1(0)=0, and H1(S(t))r to deduce, for some positive constant c, that H1(S(t))cH01(S(t)). Therefore
    Now, for ε0<r and c0>0, using (3.25), and the fact that E0, H0>0,H0>0 on (0,r], we find that the functional F1, defined by
    satisfies, for some α1,α2>0,
    and, for all tt1,
    Let H0 be the convex conjugate of H0 in the sense of Young (see [6] pp. 61–64), then
    (3.28)H0(s)=s(H0)1(s)H0[(H0)1(s)],if s(0,H0(r)]
    and H0 satisfies the following Young’s inequality
    (3.29)ABH0(A)+H0(B),if A(0,H0(r)],B(0,r].
    With A=H0(ε0E(t)E(0)) and B=H01(ξ(t)+χ(t)+S(t)), using (3.1), (3.19), (3.21), (3.23), and (3.27)–(3.29), we arrive at
    Consequently, with a suitable choice of ε0 and c0, we obtain, for all t>t1,
    where H2(t)=tH0(ε0t).

    Since H2(t)=H0(ε0t)+ε0tH0(ε0t), then, using the strict convexity of H0 on (0,r], we find that H2(t), H2(t)>0 on (0,1]. Thus, with

    taking in account (3.26) and (3.30), we have
    and, for some c1>0,
    Considering H1(t)=t11H2(s)ds, we deduce that (H1(R))(t)>0, tt1, which implies that H1(R(t)), tt1, is increasing. Thus,
    and so, for some c2>0,
    Here, we used, based on the properties of H2, the fact that H1 is strictly decreasing on (0,1]. Using (3.31)–(3.32) and by virtue of continuity and boundedness of E and θ, we obtain (2.7).

    Moreover, if 01H1(t)dt<+, then 0+H11(t)dt<+, and so, by (2.7), 0+E(t)dt<+. Then, we have

    Therefore, we can repeat the same procedures with
    to establish (2.9).


The author thanks University of Sharjah for its continuous support.



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