1.Introduction
In this paper we are concerned with the following problem
(1.1)utt+Δ2u+θ(t)h(ut)=0in Ω×(0,∞),(1.2)u=∂u∂ν=0on Γ0×(0,∞),(1.3)−u+∫0tg1(t−s)β2u(s)ds=0on Γ1×(0,∞),(1.4)∂u∂ν+∫0tg2(t−s)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=Γ0∪Γ1,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:
β1u=Δu+(1−μ)B1u,β2u=∂Δu∂ν+(1−μ)∂B2u∂η,
where
B1u=2ν1ν2uxy−ν12uyy−ν22uxx,B2u=(ν12−ν22)uxy+ν1ν2(uyy−uxx)
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 θ≡1, 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
c1min{|s|,|s|q}⩽|h(s)|⩽c2max{|s|,|s|1q},
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 −gi′gi(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
(1.6)utt(t)+Au(t)−(g∗Aβu)(t)=0,
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
(1.7)g′(t)⩽−ξ(t)g(t),∀t⩾0,
where
ξ:R+→R+ is a nonincreasing differentiable function with
(1.8)|ξ′(t)ξ(t)|⩽k,∀t⩾0
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
g′⩽−χ(g(t)),
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
∫0k0dxχ(x)=+∞,∫0k0xdxχ(x)<1,lims→0+infχ(s)/sχ′(s)>12
and proved an energy decay result. In addition to these assumptions, if
lims→0+supχ(s)/sχ′(s)<1andg′=−χ(g(t))
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
utt−Δu+∫0tdiv[a(x)g(t−s)∇u(s)]ds+b(x)h(ut)+f(u)=0,x∈Ω,t>0
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.
2.Preliminaries
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
(A1).
Ω is a bounded domain of R2 with a smooth boundary ∂Ω=Γ0∪Γ1, 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 x0∈R2 such that, for m(x)=x−x0, m·ν⩽0 on Γ0 and m·ν>0 on Γ1.
Remark.
Hypothesis (A1) implies that there exist constants δ0 and R such that
m·ν⩾δ0>0on Γ1and|m(x)|⩽Rfor all x∈Ω.
We denote by
ki the resolvent kernel of
(−gi′/gi(0)) which satisfies
ki(t)+1gi(0)(gi′∗ki)(t)=−1gi(0)gi′(t),∀i=1,2,
where ∗ points to the convolution product
(u∗v)(t)=∫0tu(t−s)v(s)ds.
By differentiating Eqs (
1.3) and (
1.4), we arrive at the following Volterra equations:
β2u+1g1(0)g1′∗β2u=1g1(0)ut,β1u+1g2(0)g2′∗β1u=−1g2(0)∂ut∂ν.
Using the Volterra’s inverse operator and taking
τi=1gi(0), for
i=1,2, we get
β2u=τ1{ut+k1∗ut},on Γ1×(0,∞),β1u=−τ2{∂ut∂ν+k2∗∂ut∂ν},on Γ1×(0,∞),
which gives, assuming throughout the paper that
u0≡0,
(2.1)β2u=τ1{ut+k1(0)u+k1′∗u},on Γ1×(0,∞),(2.2)β1u=−τ2{∂ut∂ν+k2(0)∂u∂ν+k2′∗∂u∂ν},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.
(A2).
ki:R+→R+, for i=1,2, are C2 functions such that
ki(0)>0,limt→∞ki(t)=0,ki′(t)⩽0
and there exists a positive function
H∈C1(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
(2.3)ki″(t)⩾H(−ki′(t)),(i=1,2)∀t>0.
(A3).
h:R→R 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)⩽H−1(sh(s))if |s|⩽r.
(A4).
θ:R+→R+ is a nonincreasing C1 function.
In the sequel we assume that system (1.1)–(1.5) has a unique solution
u∈L∞(R+;H4(Ω)∩W)∩W1,∞(R+;W)∩W2,∞(R+;L2(Ω)),
where
W={w∈H2(Ω):w=∂w∂ν=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
(2.4)a(u,v)=∫Ω{uxxvxx+uyyvyy+μ(uxxvyy+uyyvxx)+2(1−μ)uxyvxy}dxdy
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,
α‖u‖H2(Ω)2⩽a(u,u)⩽β‖u‖H2(Ω)2.
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 μ∈R. Then we have
(2.5)∫Ω(Δ2u)vdx=a(u,v)+∫Γ{(β2u)v−(β1u)∂v∂ν}dΓ
and
∫Ω(m·∇v)Δ2vdx=a(v,v)+12∫Γm·ν[vxx2+vyy2+2μvxxvyy+2(1−μ)vxy2]dΓ(2.6)+∫Γ[(β2v)m·∇v−(β1v)∂∂ν(m·∇v)]dΓ.
Now, we introduce the energy functional
E(t):=12[∫Ω|ut|2dx+a(u,u)+τ1∫Γ1(k1(t)|u|2−(k1′∘u))dΓ+τ2∫Γ1(k2(t)|∂u∂ν|2−(k2′∘∂u∂ν))dΓ],
where
(f∘w)(t)=∫0tf(t−s)|w(t)−w(s)|2ds.
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
(2.7)E(t)⩽c3H1−1(c1∫0tθ(s)ds+c2)∀t⩾0,
where
H1(t)=∫t11sH0′(ε0s)dsandH0(t)=H(D(t))
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)H0−1(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)⩽c3G−1(c1∫0tθ(s)ds+c2)where G(t)=∫t11sH′(ε0s)ds.
In particular, this last estimate is valid for the special case H(t)=ctp, for 1⩽p<32.Remarks.
1. Using the properties of H, one can show that the function H1 is strictly decreasing and convex on (0,1], with limt→0H1(t)=+∞. Therefore, Theorem 2.1 ensures
limt→+∞E(t)=0.
2. Hypothesis (A3) implies that sh(s)>0, for all s≠0.
3. The condition (A3), with r=1 and θ≡1, 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, 1⩽p<3/2, are special cases of our result. We will provide a “simpler” proof for these special cases.
5. The condition ki″⩾d(−ki′)p, 1⩽p<3/2, assumes (−ki′(t))⩽ωe−dt when p=1 and (−ki′(t))⩽ωt1p−1 when 1<p<3/2. Our result allows resolvent kernels whose derivatives are not necessarily of exponential or polynomial decay. For instance, if
ki′(t)=−exp(−tq),i=1,2
for 0<q<1, then ki″(t)=H(−ki′(t)) where, for t∈(0,r], r<1,
H(t)=qt[ln(1/t)]1q−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
E(t)⩽cexp(−ω[∫0tθ(s)ds]q).
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
F[1C∫Ωf(x)j(x)dx]⩽1C∫ΩF[f(x)]j(x)dx.
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
(2.10)max{ki(t),−ki′(t),ki″(t)}<min{r,H(r),H0(r)},(i=1,2)∀t⩾t1.
As ki′ is nondecreasing, ki′(0)<0 and ki′(t1)<0, then ki′(t)<0 for any t∈[0,t1] and
0<−ki′(t1)⩽−ki′(t)⩽−ki′(0),(i=1,2)∀t∈[0,t1].
Therefore, since H is a positive continuous function, then
a⩽H(−ki′(t))⩽b,(i=1,2)∀t∈[0,t1]
for some positive constants a and b. Consequently, for all t∈[0,t1],
ki″(t)⩾H(−ki′(t))⩾a=aki′(0)ki′(0)⩾aki′(0)ki′(t)(i=1,2)
which gives, for some positive constant d,
(2.11)ki″(t)⩾−dki′(t),(i=1,2)∀t∈[0,t1].
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)⩽H3−1(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), ∀t⩾t1.
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
E′(t)=−θ(t)∫Ωuth(ut)dx−τ12∫Γ1(2|ut|2−k1′(t)|u|2+k1″∘u)dΓ−τ22∫Γ1(2|∂ut∂ν|2−k2′(t)|∂u∂ν|2+k2″∘∂u∂ν)dΓ(3.1)⩽0.
Proof.
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
12ddt{∫Ω|ut|2dx+a(u,u)}=−∫Γ1(β2u)utdΓ+∫Γ1(β1u)∂ut∂νdΓ−θ(t)∫Ωuth(ut)dx=−τ1∫Γ1ut2−k1(0)τ12ddt∫Γ1u2−τ1∫Γ1{k1′∗u}utdΓ−τ2∫Γ1|∂ut∂ν|2−k2(0)τ22ddt∫Γ1|∂u∂ν|2−τ2∫Γ1{k2′∗∂u∂ν}∂ut∂νdΓ−θ(t)∫Ωuth(ut)dx.
Then, making use of the identity
(f∗w)wt=−12f(t)|w(t)|2+12f′∘w−12ddt[f∘w−(∫0tf(s)ds)|w|2],
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
ψ(t):=∫Ω(m·∇u)utdx
satisfies, along the solution,the estimate
ddtψ(t)⩽12∫Γ1m·ν|ut|2dΓ−∫Ω|ut|2dx−(1−ϵc2)a(u,u)+2τ12ϵ∫Γ1[|ut|2+k12(t)|u|2−k1(0)(k1′∘u)]dΓ+c2ϵ∫Ωh2(ut)dx+2τ22ϵ∫Γ1[|∂ut∂ν|2+k22(t)|∂u∂ν|2−k2(0)(k2′∘∂u∂ν)]dΓ(3.2)−(12−ϵc2)∫Γ1m·ν[uxx2+uyy2+2μuxxuyy+2(1−μ)uxy2]dΓ.
Proof.
Direct computations, taking v=u in (2.6), we get
ddtψ(t)=∫Ω(m·∇ut)utdx+∫Ω(m·∇u)uttdx=12∫Γ1m·ν|ut|2dΓ−∫Ω|ut|2dx−θ(t)∫Ω(m·∇u)h(ut)dx−a(u,u)−∫Γ[(β2u)(m·∇u)−(β1u)∂∂ν(m·∇u)]dΓ(3.3)−12∫Γm·ν[uxx2+uyy2+2μuxxuyy+2(1−μ)uxy2]dΓ.
Let us examine the integrals over
Γ0 in (
3.3). Since
u=∂u∂ν=0 on
Γ0, we have
B1u=B2u=0 on
Γ0 and
∂∂ν(m·∇u)=(m·ν)Δu,uxx2+uyy2+2μuxxuyy+2(1−μ)uxy2=(Δu)2on Γ0
since
uxxuyy−(uxy)2=0on Γ0.
Therefore, from (
3.3), we have
ddtψ(t)=12∫Γ1m·ν|ut|2dΓ−∫Ω|ut|2dx−θ(t)∫Ω(m·∇u)h(ut)dx−a(u,u)+12∫Γ0m·ν(Δu)2dΓ−12∫Γ1m·ν[uxx2+uyy2+2μuxxuyy+2(1−μ)uxy2]dΓ(3.4)−∫Γ1(β2u)(m·∇u)dΓ+∫Γ1(β1u)∂∂ν(m·∇u)dΓ.
Using the Young inequality, we have
(3.5)|∫Γ1(β2u)(m·∇u)dΓ|⩽12ϵ∫Γ1|β2u|2dΓ+ϵ2∫Γ1|m·∇u|2dΓ,(3.6)|∫Γ1(β1u)∂∂υ(m·∇u)dΓ|⩽12ϵ∫Γ1|β1u|2dΓ+ϵ2∫Γ1|∂∂υ(m·∇u)|2dΓ,(3.7)−θ(t)∫Ω(m·∇u)h(ut)dx⩽c2ϵ∫Ωh2(ut)dx+ϵ2∫Ω|m·∇u|2dx,
where
ϵ is a positive constant. Using the trace theory, we obtain
∫Γ1|m·∇u|2dΓ+∫Γ1|∂∂ν(m·∇u)|2dΓ+∫Ω|m·∇u|2dx(3.8)⩽ca(u,u)+c∫Γ1m·ν[uxx2+uyy2+2μuxxuyy+2(1−μ)uxy2]dΓ.
Substituting the inequalities (
3.5)–(
3.8) into (
3.4) and taking into account the fact that
m·ν⩽0 on
Γ0, we have
ddtψ(t)⩽12∫Γ1m·ν|ut|2dΓ−∫Ω|ut|2dx−(1−ϵc2)a(u,u)+12ϵ∫Γ1|β1u|2dΓ+12ϵ∫Γ1|β2u|2dΓ+c2ϵ∫Ωh2(ut)dx−(12−ϵc2)∫Γ1m·ν[uxx2+uyy2+2μuxxuyy+2(1−μ)uxy2]dΓ.
Since, by Hölder inequality,
(k1′∗u)(t)=−∫0tk1′(t−s)(u(t)−u(s))ds+k1(t)u(t)−k1(0)u(t)⩽[−∫0tk1′(s)ds]12[(k1′∘u)(t)]12+k1(t)u(t)−k1(0)u(t)⩽[−k1(0)(k1′∘u)(t)]12+k1(t)u(t)−k1(0)u(t)
then
β2u⩽τ1{ut+k1(t)u+[−k1(0)(k1′∘u)(t)]12},
similarly
β1u⩽−τ2{∂ut∂ν+k2(t)∂u∂ν+[−k2(0)(k2′∘∂u∂ν)(t)]12}.
Consequently, our conclusion easily follows. □
Proof of Theorem 2.1.
For N>0, we define
L(t):=NE(t)+ψ(t).
Combining (
3.1) and (
3.2) and using the facts that
ki′<0,
ki″>0 and
|m·ν|⩽R, we obtain
L′(t)⩽−(τ1N−R2−2τ12ϵ)∫Γ1|ut|2dΓ−(τ2N−2τ22ϵ)∫Γ1|∂ut∂ν|2dΓ−(1−ϵc2)a(u,u)+2τ12ϵ∫Γ1k12(t)|u|2dΓ+2τ22ϵ∫Γ1k22(t)|∂u∂υ|2dΓ−∫Ω|ut|2dx−2τ12k1(0)ϵ∫Γ1k1′∘udΓ−2τ22k2(0)ϵ∫Γ1k2′∘∂u∂νdΓ−(12−ϵc2)∫Γ1m·ν[uxx2+uyy2+2μuxxuyy+2(1−μ)uxy2]dΓ(3.9)+c2ϵ∫Ωh2(ut)dx.
Then choosing
0<ϵ<1c, and
N large enough so that
τ1N−R2−2τ12ϵ>0,τ2N−2τ22ϵ>0,
so, we arrive at
L′(t)⩽−∫Ω|ut|2dx−12a(u,u)+c∫Ωh2(ut)dx+2ι12ϵ∫Γ1k12(t)|u|2dΓ+2ι22ϵ∫Γ1k22(t)|∂u∂ν|2dΓ−c∫Γ1k1′∘udΓ−c∫Γ1k2′∘∂u∂νdΓ,
which, using Trace theory and the fact that
limt→∞ki(t)=0, for
i=1,2, yields, for large
t1,
(3.10)L′(t)⩽−mE(t)−c∫Γ1k1′∘udΓ−c∫Γ1k2′∘∂u∂νdΓ+c∫Ωh2(ut)dx,∀t⩾t1.
On the other hand, we can choose
N even larger (if needed) so that
(3.11)L(t)∼E(t),
which means that, for some constants
α1,α2>0,
α1E(t)⩽L(t)⩽α2E(t).
Now, we consider the following partition of Ω
Ω1={x∈Ω:|ut|⩽r},Ω2={x∈Ω:|ut|>r}
and use
(A3),
(A4), (
2.11), and (
3.1) to conclude that, for any
t⩾t1,
−θ(t)∫0t1k1′(s)∫Γ1|u(t)−u(t−s)|2dΓds+cθ(t)∫Ω2h2(ut)dx⩽cd∫0t1k1″(s)∫Γ1|u(t)−u(t−s)|2dΓds+cθ(t)∫Ω2uth(ut)dx(3.12)⩽−cE′(t),
−θ(t)∫0t1k2′(s)∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds(3.13)⩽cd∫0t1k2″(s)∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2⩽−cE′(t).
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
t⩾t1,
F′(t)⩽−mθE(t)−cθ∫t1tk1′(s)∫Γ1|u(t)−u(t−s)|2dΓds(3.14)−cθ∫t1tk2′(s)∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds+cθ∫Ω1h2(ut)dx.
(I) H(t)=ctp and 1⩽p<32: This means, using Holder’s inequality, that
cθ∫Ω1h2(ut)dx⩽cθ∫Ω1[uth(ut)]1pdx⩽cθ[∫Ω1uth(ut)dx]1p⩽cθp−1p[−E′(t)]1p.
Case 1. p=1: Estimate (3.14) yields
F′(t)⩽−mθ(t)E(t)+cθ(t)∫Γ1(k1″∘u)(t)dΓ+cθ(t)∫Γ1k2″∘∂u∂ν−cE′(t)⩽−mθ(t)E(t)−cE′(t),∀t⩾t1,
which gives
(F+cE)′(t)⩽−mθ(t)E(t),∀t⩾t1.
Hence, using the fact that F+cE∼E, we easily obtain
E(t)⩽c′e−c∫0tθ(s)ds=c′G−1(c∫0tθ(s)ds).
Case 2. 1<p<32: One can easily show that ∫0+∞[−ki′(s)]1−δ0ds<+∞ for any δ0<2−p 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 t⩾t1,
η(t):=∫t1t[−k1′(s)]1−δ0∫Γ1|u(t)−u(t−s)|2dΓds⩽2∫t1t[−k1′(s)]1−δ0∫Γ1(|u(t)|2+|u(t−s)|2)dΓds(3.15)⩽cE(0)∫t1t[−k1′(s)]1−δ0ds<1
and
γ(t):=∫t1t[−k2′(s)]1−δ0∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds⩽2∫t1t[−k2′(s)]1−δ0∫Γ1|∂u(t)∂ν|2+|∂u(t−s)∂ν|2dΓds(3.16)⩽cE(0)∫t1t[−k2′(s)]1−δ0ds<1.
Then, Jensen’s inequality (3.1), hypothesis (A2), and (3.15) lead to
−∫t1tk1′(s)∫Γ1|u(t)−u(t−s)|2dΓds=∫t1t[−k1′(s)]δ0[−k1′(s)]1−δ0∫Γ1|u(t)−u(t−s)|2dΓds=∫t1t[−k1′(s)](p−1+δ0)(δ0p−1+δ0)[−k1′(s)]1−δ0∫Γ1|u(t)−u(t−s)|2dΓds⩽η(t)[1η(t)∫t1t[−k1′(s)](p−1+δ0)[−k1′(s)]1−δ0∫Γ1|u(t)−u(t−s)|2dΓds]δ0p−1+δ0⩽[∫t1t[−k1′(s)]p∫Γ1|u(t)−u(t−s)|2dΓds]δ0p−1+δ0⩽c[∫t1tk1″(s)∫Γ1|u(t)−u(t−s)|2dΓds]δ0p−1+δ0⩽c[−E′(t)]δ0p−1+δ0.
Similarly
−∫t1tk2′(s)∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds⩽c[−E′(t)]δ0p−1+δ0.
Then, particularly for δ0=12, we find that (3.14) becomes
F′(t)⩽−mθE(t)+cθ[−E′(t)]12p−1+cθp−1p[−E′(t)]1p.
Now, we multiply by E2p−2(t) to get, using (3.1),
(FE2p−2)′⩽F′(t)E2p−2⩽−mθE2p−1+cθE2p−2[−E′]12p−1+cθp−1pE2p−2[−E′]1p.
Then, Young’s inequality gives
(FE2p−2)′⩽−mθE2p−1(t)+εθE2p−1(t)+Cεθ(−E′(t))+δθE2p+Cδ(−E′(t)).
Consequently, as E2p(t)⩽E(0)E2p−1(t), picking ε+δE(0)<m, we obtain
F0′(t)⩽−m′θ(t)E2p−1(t),
where F0=FE2p−2+CE∼E. Hence we have, for some a0>0,
F0′(t)⩽−a0θ(t)F02p−1(t)
from which we easily deduce that
(3.17)E(t)⩽a(a′∫0tθ(s)ds+a″)12p−2.
By recalling that p<3/2 and using (3.17), we find that ∫0+∞θ(s)E(s)ds<+∞. Hence, by noting that
θ(t)∫0t∫Γ1|u(t)−u(t−s)|2dΓds⩽c∫0tθ(s)E(s)ds,θ(t)∫0t∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds⩽c∫0tθ(s)E(s)ds,
estimate (3.14) gives
F′(t)⩽−mθE(t)+cθ∫Γ1([−k1′]p·1p∘u)(t)dΓ+cθ∫Γ1([−k2′]p·1p∘∂u∂ν)(t)dΓ+cθp−1p[−E′(t)]1p⩽−mθE(t)+cθ1p[∫Γ1([−k1′]p∘u)(t)dΓ]1p+cθ1p[∫Γ1([−k2′]p∘∂u∂ν)(t)dΓ]1p+cθp−1p[−E′(t)]1p⩽−mθE(t)+cθp−1p[∫Γ1(k1″∘u)(t)dΓ]1p+cθp−1p[∫Γ1(k2″∘∂u∂ν)(t)dΓ]1p+cθp−1p[−E′(t)]1p⩽−mθE(t)+cθp−1p[−E′(t)]1p.
Therefore, repeating the above steps, with multiplying by Ep−1(t), we arrive at
E(t)⩽a(a′∫0tθ(s)ds+a″)1p−1=cG−1(c′∫0tθ(s)ds+c″).
(II) The general case: We define I(t) by
I(t):=∫t1t−k1′(s)H0−1(k1″(s))∫Γ1|u(t)−u(t−s)|2dΓds,
where H0 is such that (2.8) is satisfied. As in (3.15), we find that I(t) satisfies, for all t⩾t1,
(3.18)I(t)<1.
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
ξ(t):=∫t1tk1″(s)−k1′(s)H0−1(k1″(s))∫Γ1|u(t)−u(t−s)|2dΓds
and infer from (A2) and the properties of H0 and D that
−ki′(s)H0−1(ki″(s))⩽−ki′(s)H0−1(H(−ki′(s)))=−ki′(s)D−1(−ki′(s))⩽k0∀i=1,2,
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,
ξ(t)⩽k0∫t1tk1″(s)∫Γ1|u(t)−u(t−s)|2dΓds⩽cE(0)∫t1tk1″(s)⩽−ck1′(t1)E(0)(3.19)<13min{r,H(r),H0(r)}.
Since H0 is strictly convex on (0,r] and H0(0)=0, then
H0(μx)⩽μH0(x)
provided 0⩽μ⩽1 and x ∈(0,r]. The use of this fact, hypothesis (A2), (2.10), (3.18), (3.19), and Jensen’s inequality leads to
ξ(t)=1I(t)∫t1tI(t)H0[H0−1(k1″(s))]−k1′(s)H0−1(k1″(s))∫Γ1|u(t)−u(t−s)|2dΓds⩾1I(t)∫t1tH0[I(t)H0−1(k1″(s))]−k1′(s)H0−1(k1″(s))∫Γ1|u(t)−u(t−s)|2dΓds⩾H0(1I(t)∫t1tI(t)H0−1(k1″(s))−k1′(s)H0−1(k1″(s))∫Γ1|u(t)−u(t−s)|2dΓds)=H0(−∫t1tk1′(s)∫Γ1|u(t)−u(t−s)|2dΓds).
This implies that
(3.20)−∫t1tk1′(s)∫Γ1|u(t)−u(t−s)|2dΓds⩽H0−1(ξ(t)).
We also define
ϕ(t):=∫t1t−k2′(s)H0−1(k2″(s))∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds,χ(t):=∫t1tk2″(s)−k2′(s)H0−1(k2″(s))∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds.
We similarly deduce, for all t>t1, that
ϕ(t)<1
and
(3.21)χ(t)<13min{r,H(r),H0(r)}.
Repeating the above steps, we arrive at
(3.22)−∫t1tk2′(s)∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds⩽H0−1(χ(t)).
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
S(t):=1|Ω1|∫Ω1uth(ut)dx.
(A3) and Jensen’s inequality give
(3.24)H−1(S(t))⩾c∫Ω1H−1(uth(ut))dx⩾c∫Ω1h2(ut)dx.
Inserting the estimates (3.20), (3.22), and (3.24) into (3.14), we obtain
F′(t)⩽−mθ(t)E(t)+cθ(t)[H0−1(ξ(t))+H0−1(χ(t))+H−1(S(t))],∀t⩾t1.
One can easily make use of the properties of H,D,H0 and the fact that H0−1(S(t))=D−1(H−1(S(t))), D−1(0)=0, and H−1(S(t))⩽r to deduce, for some positive constant c, that H−1(S(t))⩽cH0−1(S(t)). Therefore
F′(t)⩽−mθ(t)E(t)+cθ(t)[H0−1(ξ(t))+H0−1(χ(t))+H0−1(S(t))](3.25)⩽−mθ(t)E(t)+cθ(t)H0−1(ξ(t)+χ(t)+S(t)).
Now, for ε0<r and c0>0, using (3.25), and the fact that E′⩽0, H0′>0,H0″>0 on (0,r], we find that the functional F1, defined by
F1(t):=H0′(ε0E(t)E(0))F(t)+c0E(t)
satisfies, for some α1,α2>0,
(3.26)α1F1(t)⩽E(t)⩽α2F1(t)
and, for all t⩾t1,
F1′(t)=ε0E′(t)E(0)H0″(ε0E(t)E(0))F(t)+H0′(ε0E(t)E(0))F′(t)+c0E′(t)(3.27)⩽−mθE(t)H0′(ε0E(t)E(0))+cθH0′(ε0E(t)E(0))H0−1(ξ(t)+χ(t)+S(t))+c0E′(t).
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)AB⩽H0∗(A)+H0(B),if A∈(0,H0′(r)],B∈(0,r].
With A=H0′(ε0E(t)E(0)) and B=H0−1(ξ(t)+χ(t)+S(t)), using (3.1), (3.19), (3.21), (3.23), and (3.27)–(3.29), we arrive at
F1′(t)⩽−mθ(t)E(t)H0′(ε0E(t)E(0))+cθ(t)H1∗(H0′(ε0E(t)E(0)))+cθ(t)(ξ(t)+χ(t)+S(t))+c0E′(t)⩽−mθ(t)E(t)H0′(ε0E(t)E(0))+cε0θ(t)E(t)E(0)H0′(ε0E(t)E(0))−cE′(t)+c0E′(t).
Consequently, with a suitable choice of ε0 and c0, we obtain, for all t>t1,
(3.30)F1′(t)⩽−τθ(t)(E(t)E(0))H0′(ε0E(t)E(0))=−τθ(t)H2(E(t)E(0)),
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
R(t)=α1F1(t)E(0),
taking in account (3.26) and (3.30), we have
(3.31)R(t)∼E(t)
and, for some c1>0,
R′(t)⩽−c1θ(t)H2(R(t)),∀t>t1.
Considering H1(t)=∫t11H2(s)ds, we deduce that (H1(R))′(t)>0, ∀t⩾t1, which implies that H1(R(t)), t⩾t1, is increasing. Thus,
k1∫t1tθ(s)ds⩽∫t1t(H1(R))′(s)ds⩽H1(R(t))−H1(R(t1)),
and so, for some c2>0,
(3.32)R(t)⩽H1−1(c1∫t1tθ(s)ds+c2),∀t⩾t1.
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+∞H1−1(t)dt<+∞, and so, by (2.7), ∫0+∞E(t)dt<+∞. Then, we have
∫0t∫Γ1|u(t)−u(t−s)|2dΓds⩽c∫0tE(s)ds<+∞,∫0t∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds⩽c∫0tE(s)ds<+∞.
Therefore, we can repeat the same procedures with
I(t):=∫t1t∫Γ1|u(t)−u(t−s)|2dΓds,ϕ(t):=∫t1t∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds
and
ξ(t):=∫t1tk1″(s)∫Γ1|u(t)−u(t−s)|2dΓds,χ(t):=∫t1tk2″(s)∫Γ1|∂u(t)∂ν−∂u(t−s)∂ν|2dΓds
to establish (2.9).
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