# General Decay for the Degenerate Equation with a Memory Condition at the Boundary

Authors: Su-Young Shin; Jum-Ran Kang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We consider a degenerate equation with a memory condition at the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.


---

## Body

## 1. Introduction

The main purpose of this paper is to investigate the asymptotic behavior of the solutions of the degenerate equation with a memory condition at the boundary(1)K(x)u′′+Δ2u+f(u)=0inQ=Ω×(0,∞),(2)u=∂u∂ν=0onΓ0×(0,∞),(3)-u+∫0t‍g1(t-s)ℬ2u(s)ds=0onΓ1×(0,∞),(4)∂u∂ν+∫0t‍g2(t-s)ℬ1u(s)ds=0onΓ1×(0,∞),(5)u(0,x)=u0(x),u′(0,x)=u1(x)inΩ,
where Ω is a bounded domain of ℝn with a smooth boundary Γ and let us assume that Γ, can be divided into two nonnull parts Γ=Γ0∪Γ1 and Γ0-∩Γ1-=∅ and K∈C1(Ω-) and K(x)≥0 for all x∈Ω which satisfies some appropriate conditions. ν is the unit outward normal to Γ and τ=(-ν2,ν1) the corresponding unit tangent vector. Here, the relaxation functions gi (i=1,2) are positive and nondecreasing, the function f∈C1(ℝ) and
(6)ℬ1u=Δu+(1-μ)B1u,ℬ2u=∂Δu∂ν+(1-μ)∂B2u∂τ,B1u=2ν1ν2uxy-ν12uyy-ν22uxx,B2u=(ν12-ν22)uxy+ν1ν2(uyy-uxx),
and the constant μ, 0<μ<1/2, represents Poisson’s ratio.From the physical point of view, we know that the memory effect described in integral equations (3) and (4) can be caused by the interaction with another viscoelastic element. In fact, the boundary conditions (3) and (4) mean that Ω is composed of a material which is clamped in a rigid body in Γ0 and is clamped in a body with viscoelastic properties in the complementary part of its boundary named Γ1. Problems related to (1)–(5) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics.The existence of global solutions and exponential decay to the degenerate equation with∂Ω=Γ0 has been investigated by several authors. See Cavalcanti et al. [1] and Menezes et al. [2]. For instance, when K(x) is equal to 1, (1) describes the transverse deflection u(x,t) of beams. There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain or at the boundary (see [3–17]). Santos et al. [18] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Cavalcanti et al. [19] proved the uniform decay rates of solutions to a degenerate system with a memory condition at the boundary. Santos and Junior [20] investigated the stability of solutions for Kirchhoff plate equations with a boundary memory condition. Rivera et al. [21] showed the asymptotic behavior to a von Karman plate with boundary memory conditions. Park and Kang [22] studied the exponential decay for the Kirchhoff plate equations with nonlinear dissipation and boundary memory condition. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation functions. In the present work, we generalize the earlier decay results of the solution of (1)–(5). More precisely, we show that the energy decays at the rate similar to the relaxation functions, which are not necessarily decaying like polynomial or exponential functions. In fact, our result allows a larger class of relaxation functions. Recently, Messaoudi and Soufyane [23], Mustafa and Messaoudi [24], and Santos and Soufyane [25] proved the general decay for the wave equation, Timoshenko system, and von Karman plate system with viscoelastic boundary conditions, respectively.The organization of this paper is as follows. In Section2, we present some notations and material needed for our work and state the existence result to system (1)–(5). In Section 3, we prove the general decay of the solutions to the degenerate equation with a memory condition at the boundary.


## 2. Preliminaries

In this section, we introduce some notations and establish the existence of solutions of the problem (1)–(5).Note that, because of condition (2), the solution of system (1)–(5) must belong to the following space:
(7)W={v∈H2(Ω):v=∂v∂ν=0onΓ0}.
Let us define the bilinear form a(·,·) as follows:
(8)a(u,v)=∫Ω‍{uxxvxx+uyyvyy+μ(uxxvyy+uyyvxx)+2(1-μ)uxyvxy}dxdy.
Since Γ0≠∅, we know that a(u,u) is equivalent to the H2(Ω) norm on W; that is,
(9)c0∥u∥H2(Ω)2≤a(u,u)≤C0∥u∥H2(Ω)2,
and here and in the sequel, we denote by c0 and C0 generic positive constants. Simple calculation, based on the integration by parts formula, yields
(10)(Δ2u,v)=a(u,v)+(ℬ2u,v)Γ-(ℬ1u,∂v∂ν)Γ.
We assume that there exists x0∈ℝn such that
(11)Γ0={x∈Γ:ν(x)·(x-x0)≤0},(12)Γ1={x∈Γ:ν(x)·(x-x0)>0}.
If we denote the compactness of Γ1 by m(x)=x-x0, condition (12) implies that there exists a small positive constant δ0 such that 0<δ0≤m(x)·ν(x), for all x∈Γ1.Next, we will use (3) and (4) to estimate the values ℬ1 and ℬ2 on Γ1. Denoting by
(13)(g*φ)(t)=∫0t‍g(t-s)φ(s)ds
the convolution product operator and differentiating (3) and (4), we arrive at the following Volterra equations:
(14)ℬ2u+1g1(0)g1′*ℬ2u=1g1(0)u′,ℬ1u+1g2(0)g2′*ℬ1u=-1g2(0)∂u′∂ν.
Applying the Volterra inverse operator, we get
(15)ℬ2u=1g1(0){u′+k1*u′},ℬ1u=-1g2(0){∂u′∂ν+k2*∂u′∂ν},
where the resolvent kernels satisfy
(16)ki+1gi(0)gi′*ki=-1gi(0)gi′,∀i=1,2.
Denoting that τ1=1/g1(0) and τ2=1/g2(0), we have
(17)ℬ2u=τ1{u′+k1(0)u-k1(t)u0+k1′*u},ℬ1u=-τ2{∂u′∂ν+k2(0)∂u∂ν-k2(t)∂u0∂ν+k2′*∂u∂ν}.
Therefore, we use (17) instead of the boundary conditions (3) and (4).Let us denote that(18)(g⋄v)(t):=∫0t‍g(t-s)|v(t)-v(s)|2ds.
The following lemma states an important property of the convolution operator.Lemma 1.
Forg,v∈C1([0,∞):ℝ), one has
(19)(g*v)v′=-12g(t)|v(t)|2+12g′⋄v-12ddt×[g⋄v-(∫0t‍g(s)ds)|v|2].The proof of this lemma follows by differentiating the termg⋄v.Lemma 2 (see [26]).
Suppose thatf∈L2(Ω),  g∈H1/2(Γ1) and h∈H3/2(Γ1); then, any solution of
(20)a(v,w)=∫Ω‍fwdx+∫Γ1‍gwdΓ+∫Γ1‍h∂w∂νdΓ,∀w∈W,
satisfies v∈H4(Ω) and also
(21)Δ2v=f,v=∂v∂ν=0onΓ0,ℬ1v=h,ℬ2v=gonΓ1.We formulate the following assumptions.(A1)
Letf∈C1(ℝ) satisfy
(22)f(s)s≥0,∀s∈ℝ.Additionally, we suppose that f is superlinear; that is,
(23)f(s)s≥(2+η)F(s),F(z)=∫0z‍f(s)ds,∀s∈ℝ,
for some η>0 with the following growth condition:
(24)|f(x)-f(y)|≤c(1+|x|ρ-1+|y|ρ-1)|x-y|,∀x,y∈ℝ
for some c>0 and ρ≥1 such that (n-2)ρ≤n.(A2)
K∈C1(Ω-);  H02(Ω)∩L∞(Ω) with K(x)≥0, for all x∈Ω, and satisfy the following condition
(25)∇K·m≥0inΩ.The well-posedness of system (1)–(5) is given by the following theorem.Theorem 3 (see [27]).
Consider assumptions (A1)-(A2) and letki∈C2(ℝ+) be such that
(26)ki,-ki′,ki′′≥0(i=1,2).
If u0∈W∩H4(Ω), u1∈W, satisfying the compatibility condition
(27)ℬ1u0=-τ2∂u1∂ν,ℬ2u0=τ1u1onΓ1,
then there is only one solution u of the system (1)–(5) satisfying
(28)u∈L∞(0,T:W∩H4(Ω)),u′∈L∞(0,T:W),u′′∈L∞(0,T:L2(Ω)).


## 3. General Decay

In this section, we show that the solution of system (1)–(5) may have a general decay not necessarily of exponential or polynomial type. For this we consider that the resolvent kernels satisfy the following hypothesis.(H)ki:ℝ+→ℝ+ is twice differentiable function such that
(29)ki(0)>0,limt→∞ki(t)=0,ki′(t)≤0,
and there exists a nonincreasing continuous function ξi:ℝ+→ℝ+ satisfying
(30)ki′′(t)≥-ξi(t)ki′(t),i=1,2,∀t≥0.The following identity will be used later.Lemma 4 (see [26]).
For everyv∈H4(Ω) and for every μ∈ℝ, one has
(31)∫Ω‍(m·∇v)Δ2vdx=a(v,v)+12∫Γ‍m·ν[vxx2+vyy2+2μvxxvyy+2(1-μ)vxy2]dΓ+∫Γ‍[(ℬ2v)m·∇v-(ℬ1v)∂∂ν(m·∇v)]dΓ.Let us introduce the energy function(32)E(t)=12∫Ω‍K(x)|u′|2dx+12a(u,u)+∫Ω‍F(u)dx+τ12∫Γ1‍(k1(t)|u|2-k1′⋄u)dΓ+τ22∫Γ1‍(k2(t)|∂u∂ν|2-k2′⋄∂u∂ν)dΓ.
Now, we establish some inequalities for the strong solution of system (1)–(5).Lemma 5.
The energy functionalE satisfies, along the solution of (1)–(5), the estimate
(33)E′(t)≤-τ12∫Γ1‍(|u′|2-k12(t)|u0|2-k1′(t)|u|2+k1′′⋄u)dΓ-τ22∫Γ1‍(|∂u′∂ν|2-k22(t)|∂u0∂ν|2-k2′(t)|∂u∂ν|2+k2′′⋄∂u∂ν|∂u′∂ν|2)dΓ.Proof.
Multiplying (1) by u′, integrating over Ω, and using (10), we get
(34)12ddt{∫Ω‍K|u′|2dx+a(u,u)+2∫Ω‍F(u)dx}=-∫Γ1‍(ℬ2u)u′dΓ+∫Γ1‍(ℬ1u)∂u′∂νdΓ.
Substituting the boundary terms by (17) and using Lemma 1 and the Young inequality, our conclusion follows.Let us consider the following binary operator:(35)(k∘u)(t):=∫0t‍k(t-s)(u(t)-u(s))ds.
Then applying the Holder inequality for 0≤α≤1 we have
(36)|(k∘u)(t)|2≤[∫0t‍|k(s)|2(1-α)ds](|k|2α⋄u)(t).
Let us define the functional
(37)ψ(t)=∫Ω‍[m·∇u+(n2-θ)u]Ku′dx.
The following lemma plays an important role in the construction of the desired functional.Lemma 6.
Suppose that the initial data(u0,u1)∈(H4(Ω)∩W)×W, satisfying the compatibility condition (27). Then, the solution of system (1)–(5) satisfies
(38)ψ′(t)≤12∫Γ1‍m·νK|u′|2dΓ-θ∫Ω‍K|u′|2dx-(1+n2-θ-ϵλ0)a(u,u)-(nη2-2θ-ηθ)∫Ω‍F(u)dx+2τ12ϵ×∫Γ1‍{|u′|2+k12(t)|u|2+k12(t)|u0|2+|k1′∘u|2}dΓ+2τ22ϵ∫Γ1‍{|∂u′∂ν|2+k22(t)|∂u∂ν|2+k22(t)|∂u0∂ν|2+|k2′∘∂u∂ν|2|∂u′∂ν|2|∂u0∂ν|2}dΓ-(12-ϵλ0δ0)∫Γ1‍m·ν[uxx2+uyy2+2μuxxuyy+2(1-μ)uxy2]dΓ.Proof.
Differentiatingψ using (1) and Lemma 4, we get
(39)ψ′(t)=∫Ω‍[m·∇u′+(n2-θ)u′]Ku′dx+∫Ω‍[m·∇u+(n2-θ)u]Ku′′dx=12∫Γ1‍m·νK|u′|2dΓ-θ∫Ω‍K|u′|2dx-12∫Ω‍∇K·m|u′|2dx-(1+n2-θ)a(u,u)+n∫Ω‍F(u)dx-(n2-θ)∫Ω‍f(u)udx-12∫Γ‍m·ν[uxx2+uyy2+2μuxxuyy+2(1-μ)uxy2]dΓ-∫Γ‍(ℬ2u)[(m·∇u)+(n2-θ)u]dΓ+∫Γ‍(ℬ1u)[∂∂ν(m·∇u)+(n2-θ)∂u∂ν]dΓ.
Let us next examine the integrals over Γ0 in (39). Since u=∂u/∂ν=0 on Γ0, we have B1u=B2u=0 on Γ0 and
(40)∂∂ν(m·∇u)=(m·ν)Δu,uxx2+uyy2+2μuxxuyy+2(1-μ)uxy2=(Δu)2onΓ0,
since
(41)uxxuyy-uxy2=0onΓ0.
Therefore, from (39) and (40), we have
(42)ψ′(t)≤12∫Γ1‍m·νK|u′|2dΓ-θ∫Ω‍K|u′|2dx-12∫Ω‍∇K·m|u′|2dx-(1+n2-θ)a(u,u)+n∫Ω‍F(u)dx-(n2-θ)∫Ω‍f(u)udx+12∫Γ0‍m·ν(Δu)2dΓ-12∫Γ1‍m·ν[uxx2+uyy2+2μuxxuyy+2(1-μ)uxy2]dΓ-∫Γ1‍(ℬ2u)[(m·∇u)+(n2-θ)u]dΓ+∫Γ1‍(ℬ1u)[∂∂ν(m·∇u)+(n2-θ)∂u∂ν]dΓ.
Using the Young inequality, we get
(43)|∫Γ1‍(ℬ2u)[(m·∇u)+(n2-θ)u]dΓ|≤12ϵ∫Γ1‍|ℬ2u|2dΓ+ϵ∫Γ1‍(|m·∇u|2+(n2-θ)2|u|2)dΓ,(44)|∫Γ1‍(ℬ1u)[∂∂ν(m·∇u)+(n2-θ)∂u∂ν]dΓ|≤12ϵ∫Γ1‍|ℬ1u|2dΓ+ϵ∫Γ1‍(|∂∂ν(m·∇u)|2+(n2-θ)2|∂u∂ν|2)dΓ,
where ϵ is a positive constant. Since the bilinear form a(u,u) is strictly coercive on W, using the trace theory, we obtain
(45)∫Γ1‍(|m·∇u|2+(n2-θ)2|u|2)dΓ+∫Γ1‍(|∂∂ν(m·∇u)|2+(n2-θ)2|∂u∂ν|2)dΓ≤λ0a(u,u)+λ0δ0×∫Γ1‍m·ν[uxx2+uyy2+2μuxxuyy+2(1-μ)uxy2]dΓ,
where λ0 is a constant depending on Ω,  μ,  θ and n. Substituting inequalities (43)–(45) into (42) and taking into account that m·ν≤0 on Γ0, as well as (23) and (25), we have
(46)ψ′(t)≤12∫Γ1‍m·νK|u′|2dΓ-θ∫Ω‍K|u′|2dx-(1+n2-θ-ϵλ0)a(u,u)-(nη2-2θ-ηθ)∫Ω‍F(u)dx+12ϵ∫Γ1‍(|ℬ1u|2+|ℬ2u|2)dΓ-(12-ϵλ0δ0)∫Γ1‍m·ν[uxx2+uyy2+2μuxxuyy+2(1-μ)uxy2]dΓ.
Since the boundary conditions (17) can be written as
(47)ℬ2u=τ1{u′+k1(t)u-k1(t)u0-k1′∘u},ℬ1u=-τ2{∂u′∂ν+k2(t)∂u∂ν-k2(t)∂u0∂ν-k2′∘∂u∂ν},
our conclusion follows.Let us introduce the Lyapunov functional(48)ℒ(t)=NE(t)+ψ(t),
with N>0. Now, we are in a position to show the main result of this paper.Theorem 7.
Let(u0,u1)∈W×L2(Ω). Suppose that the resolvent kernels k1,  k2 satisfy the condition (H). Then, there exist constants ω,C>0 such that, for some t0 large enough, the solution of (1)–(5) satisfies
(49)E(t)≤CE(0)e-ω∫0t‍ξ(s)ds,∀t≥t0,ifu0=∂u0∂ν=0onΓ1.
Otherwise,
(50)E(t)≤C(E(0)+∫0t‍k0(s)eω∫t0s‍ξ(τ)dτds)e-ω∫0t‍ξ(s)ds,
for all t≥t0, where
(51)ξ(t)=min{ξ1(t),ξ2(t)},k0(t)=∫Γ1‍k12(t)|u0|2dΓ+∫Γ1‍k22(t)|∂u0∂ν|2dΓ.Proof.
Applying inequality (36) with α=1/2 in Lemma 6 and from Lemma 5, we obtain
(52)ℒ′(t)≤-∫Ω‍θK|u′|2dx-τ1N2∫Γ1‍{|u′|2-k12(t)|u0|2-k1′(t)|u|2+k1′′⋄u}dΓ-τ2N2∫Γ1‍{|∂u′∂ν|2-k22(t)|∂u0∂ν|2-k2′(t)|∂u∂ν|2+k2′′⋄∂u∂ν|∂u′∂ν|2|∂u′∂ν|2}dΓ-(1+n2-θ-ϵλ0)a(u,u)-(nη2-2θ-ηθ)×∫Ω‍F(u)dx+2τ12ϵ∫Γ1‍{|u′|2+k12(t)|u|2+k12(t)|u0|2-k1(0)k1′⋄u|u′|2}dΓ+2τ22ϵ∫Γ1‍{|∂u′∂ν|2+k22(t)|∂u∂ν|2+k22(t)|∂u0∂ν|2-k2(0)k2′⋄∂u∂ν|∂u0∂ν|2|∂u0∂ν|2|∂u′∂ν|2}dΓ+12∫Γ1‍m·νK|u′|2dΓ-(12-ϵλ0δ0)×∫Γ1‍m·ν[uxx2+uyy2+2μuxxuyy+2(1-μ)uxy2]dΓ.
We take θ and ϵ so small such that
(53)nη2-2θ-ηθ>0,1+n2-θ-ϵλ0>0,12-ϵλ0δ0>0.
Since K∈L∞(Ω) and then choosing N large enough, we obtain
(54)ℒ′(t)≤-c0E(t)+c∫Γ1‍k12(t)|u0|2dΓ+c∫Γ1‍k22(t)|∂u0∂ν|2dΓ-c∫Γ1‍k1′⋄udΓ-c∫Γ1‍k2′⋄∂u∂νdΓ,∀t≥t0.
On the other hand, we can choose N even larger so that
(55)ℒ(t)~E(t).
If ξ(t)=min{ξ1(t),ξ2(t)},  t≥t0, then, using (30) and (33), we have
(56)ξ(t)ℒ′(t)≤-c0ξ(t)E(t)+cξ(t)∫Γ1‍k12(t)|u0|2dΓ+cξ(t)∫Γ1‍k22(t)|∂u0∂ν|2dΓ-cξ1(t)∫Γ1‍k1′⋄udΓ-cξ2(t)∫Γ1‍k2′⋄∂u∂νdΓ≤-c0ξ(t)E(t)+cξ(t)∫Γ1‍k12(t)|u0|2dΓ+cξ(t)∫Γ1‍k22(t)|∂u0∂ν|2dΓ+c∫Γ1‍k1′′⋄udΓ+c∫Γ1‍k2′′⋄∂u∂νdΓ≤-c0ξ(t)E(t)+cξ(t)∫Γ1‍k12(t)|u0|2dΓ+cξ(t)∫Γ1‍k22(t)|∂u0∂ν|2dΓ-cE′(t),∀t≥t0,
which gives
(57)ξ(t)ℒ′(t)+cE′(t)≤-c0ξ(t)E(t)+cξ(t)∫Γ1‍k12(t)|u0|2dΓ+cξ(t)∫Γ1‍k22(t)|∂u0∂ν|2dΓ,∀t≥t0.
Using the fact that ξ is a nonincreasing continuous function as ξ1 and ξ2 are nonincreasing, and so ξ is differentiable, with ξ′(t)≤0, for a.e. t, then we infer that
(58)(ξℒ+cE)′(t)≤ξ(t)ℒ′(t)+cE′(t)≤-c0ξ(t)E(t)+cξ(t)∫Γ1‍k12(t)|u0|2dΓ+cξ(t)∫Γ1‍k22(t)|∂u0∂ν|2dΓ,∀t≥t0.
Since using (55),
(59)F=ξℒ+cE~E,
we obtain, for some positive constant ω,
(60)F′(t)≤-ωξ(t)F(t)+c∫Γ1‍k12(t)|u0|2dΓ+c∫Γ1‍k22(t)|∂u0∂ν|2dΓ,∀t≥t0.
Case  1. If u0=∂u0/∂ν=0 on Γ1, then (60) reduces to
(61)F′(t)≤-ωξ(t)F(t),∀t≥t0.
A simple integration over (t0,t) yields
(62)F(t)≤F(t0)e-ω∫t0t‍ξ(s)ds,∀t≥t0.
By using (33) and (59), we then obtain for some positive constant C(63)E(t)≤CE(0)e-ω∫0t‍ξ(s)ds,∀t≥t0.
Thus, estimate (49) is proved.
Case  2. If (u0,(∂u0/∂ν))≠(0,0) on Γ1, then (60) gives
(64)F′(t)≤-ωξ(t)F(t)+ck0(t),∀t≥t0,
where
(65)k0(t)=∫Γ1‍k12(t)|u0|2dΓ+∫Γ1‍k22(t)|∂u0∂ν|2dΓ.
In this case, we introduce
(66)G(t):=F(t)-ce-ω∫t0t‍ξ(s)ds∫t0t‍k0(s)eω∫t0s‍ξ(τ)dτds.
A simple differentiation of G, using (64), leads to
(67)G′(t)=F′(t)+ωξ(t)ce-ω∫t0t‍ξ(s)ds×∫t0tk0(s)eω∫t0s‍ξ(τ)dτds-ck0(t)≤-ωξ(t)G(t),∀t≥t0.
Again, a simple integration over (t0,t) yields
(68)G(t)≤G(t0)e-ω∫t0t‍ξ(s)ds,∀t≥t0,
which implies, for all t≥t0,
(69)F(t)≤(F(t0)+c∫t0t‍k0(s)eω∫t0s‍ξ(τ)dτds)e-ω∫t0t‍ξ(s)ds.
By using (59), we deduce that
(70)E(t)≤C(E(0)+∫0t‍k0(s)eω∫t0s‍ξ(τ)dτds)e-ω∫t0t‍ξ(s)ds,∀t≥t0.
Consequently, by the boundedness of ξ, (50) is established.Remark 8.
Note that the exponential and polynomial decay estimates are only particular cases of (49) and (50). More precisely, we have exponential decay for ξ1(t)≡c1 and ξ2(t)≡c2 and polynomial decay for ξ1(t)=c1(1+t)-1 and ξ2(t)≡c2, where c1 and c2 are positive constants.Example 9.
As in [24], we give some examples to illustrate the energy decay rates given by (49).(1)
Ifk1(t)=k2(t)=ae-b(1+t)p,  0<p≤1, then, for i=1,2,ki′′(t)≥-ξ(t)ki′(t), where ξ(t)=bp(1+t)p-1. For suitably chosen positive constants a and b, ki satisfies (H) and (49) gives
(71)E(t)≤ce-ωb(1+t)p.(2)
Ifk1(t)=a1/(1+t)q,  q>0, and k2(t)=a2e-b(1+t)p,  0<p≤1, then, for i=1,2,ki′′(t)≥-ξ(t)ki′(t), where ξ(t)=q(1+t)-1. Then
(72)E(t)≤c(1+t)ωq.
The aforementioned two examples are included in the following more general one.(3)
For any nonincreasing functionski(t),  i=1,2, which satisfy (H), ξi=-k′/k are also nonincreasing differentiable functions, and cξ1(t)≤ξ2(t), for some 0<c≤1, and (49) gives
(73)E(t)≤c[k1(t)]ω.



---

*Source: 682061-2013-03-27.xml*