# On a Nonlinear Degenerate Evolution Equation with Nonlinear Boundary Damping

Authors: A. T. Lourêdo; G. Siracusa; C. A. Silva Filho
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

This paper deals essentially with a nonlinear degenerate evolution equation of the formKu″-Δu+∑j=1nbj∂u′/∂xj+uσu=0 supplemented with nonlinear boundary conditions of Neumann type given by ∂u/∂ν+h·,u′=0. Under suitable conditions the existence and uniqueness of solutions are shown and that the boundary damping produces a uniform global stability of the corresponding solutions.


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## Body

## 1. Introduction

LetΩ be a smooth bounded open set of Rn, with n≥1, and its boundary ∂Ω=Γ of class C2. Assume that Γ is constituted by two disjoint closed parts Γ0 and Γ1 both with positive Lebesgue measure.The main goal of this paper is to prove the existence and uniqueness as well as the uniform decay rates for the energy of the following nonlinear initial boundary value problem:PKu′′-Δu+∑j=1nbj∂u′∂xj+uσu=0inΩ×0,∞,u=0on⁡Γ0×0,∞,∂u∂ν+h·,u′=0on⁡Γ1×0,∞,u0,x=u0x,u′0,x=u1xinΩ,
where K=K(x,t), bj=bj(x,t), and h=h(x,s) are real functions, ν=ν(x) denotes the unit outward normal at x∈Γ1, and σ>1 is a constant.The parabolic-hyperbolic equationKu′′-Δ(u-u′)+f(u)=0 when n=1 or n=2; this equation governs the motion of a nonlinear Kelvin solid. That is, a bar for n=1 and a plate for n=2, subject to no nonlinear elastic constraints. The function K represents the mass density of the solid.The existence of solutions of the linear problem associated withP  (K=1, bj=0, and without the function f(s)=|s|σs and with h(x,s)=δ(x)s) was established by Komornik and Zuazua in [1], via semigroup theory and by Milla Miranda and Medeiros in [2], applying the Galerkin’s method, with a special basis. The advantage of this second method is to define the Sobolev space where ∂u/∂ν is lying. In the same context, applying this second method for a wave equation with a nonlinear term, Araruna and Maciel [3], derive similar results. In Cavalcanti et al. [4] the existence of solution and an exponential decay rate is established supposing f=0 and h being a particular function considered in our work; see also Cavalcanti et al. [5].For the wave equation withK=1 and b=0 there is a vast literature on this problem. We cite the papers Cavalcanti et al. [6], Lasiecka and Tataru [7], and references contained therein for the reader.Following the ideas delivered in Milla Miranda and Medeiros [2], but bringing more technical difficulties, Milla Miranda and San Gil Jutuca [8] applied the Galerkin’s method with a special basis to show the existence of solutions for Kirchhoff’s equation with a linear dissipation on the boundary. Applying a similar approach, Lourêdo and Miranda [9] obtained the existence of solutions for a coupled system of Kirchhoff’s equations with nonlinear boundary dissipation. For other models, but in the same context, we cite to the reader Lourêdo and Miranda [10], and Lourêdo et al. [11].Park and Kang studied the existence, uniqueness, and uniform decay for the nonlinear degenerate equation with memory condition on the boundary in [12]. For the asymptotic behavior they also used the Nakao’s method. de Lima Santos and Junior [13] studied the equation with a boundary condition with memory for Kirchhoff plates equations. An abstract formulation with the coefficient K satisfying the same conditions as in our paper was studied by Pereira in [14] and was established the existence, uniqueness, and asymptotic behavior for the solutions associated with a nonlinear beam equation.In this paper we are interested in showing the global existence of solutions for ProblemP under very general conditions to be fixed in the next section.In our approach, we apply the Galerkin’s method for aperturbed problem and a special basis; an appropriate Strauss’ Lipschitz-continuous approximation hl of h; the compactness method; and results on trace mapping of nonsmooth functions. Finally, the uniform stabilization of solutions is accomplished by using the Nakao’s method.


## 2. Notations and Main Results

In order to establish the main results of this paper we assume the following assumptions on the objects of problemP:(H1)
(A1)
bj∈W1,∞(0,∞;C1(Ω¯)) and there exists a positive constant b0>0 such that(A2)
div⁡b(x,t)≤-b0  ∀x∈Γ1, t∈[0,∞), where b(x,t)=(b1(x,t),…,bn(x,t));(H2)
(1)1n<σ≤2n-2ifn≥3,σ>1nifn=1,2.(H3)
h∈C0(R;L∞(Γ1)) with h:Γ1×R→R strongly monotone; that is,
(2)hx,s-hx,rs-r≥h0s-r2a.e.x∈Γ1,∀s,r∈R,where2h0>bΩ¯×(0,∞). We use the notation d0=2h0-b∞;(H4)
K∈C1([0,T];L∞(Ω)) with K(x,t)≥0, ∀t≥0, a.e. x∈Ω and there exists γ>0 such that K(x,0)≥γ>0, a.e. x∈Ω;(H5)
∂K(x,t)/∂tR≤δ+C(δ)K(x,t), for all δ>0;(H6)
(u0,u1)∈D(-Δ)×H01(Ω).The scalar product and norm ofL2(Ω) are denoted, respectively, by (·,·) and ·. By V we represent the Hilbert space V={v∈H1(Ω);v=0on⁡Γ0} which is equipped with the scalar product and norm
(3)u,v=∑i=1n∂u∂xi,∂v∂xi,u2=∑i=1n∂u∂xi2.The operator-Δ is defined by the triplet {V,L2(Ω),((·,·))}. Then its domain is given by
(4)D-Δ=u∈V∩H2Ω;∂u∂ν=0on⁡Γ1.
From spectral theory it follows that D(-Δ) is dense in V; see [15]. Moreover, it will be denoted
(5)u,vL2Γ1=∫Γ1uxvxdΓ,uL2Γ12=∫Γ1u2xdΓ,u∞=esssup⁡t≥0⁡utL∞Ω.Theorem 1.
Assume hypotheses(H1)–(H6); there exists at least a function u:Ω×(0,∞)→R in the class
(6)u∈Lloc⁡∞0,∞;V,u′∈Lloc⁡∞0,∞;V,u′′∈Lloc⁡∞0,∞;L2Ω,Δu∈Lloc⁡∞0,∞;L2Ω,∂u∂ν∈Lloc⁡10,∞;L1Γ1,
satisfying
(7)Ku′′-Δu+∑j=1nbj∂u′∂xj+uσu=00000000000inLloc⁡∞0,∞;L2Ω,∂u∂ν+h·,u′=0inLloc⁡10,∞;L1Γ1,u0=u0,u′0=u1inΩ.In addition, if(H7)
|h(x,s)|≤h1|s|  ∀s∈R and a.e. x in Γ1 (h1 a positive constant), holds true, we have the following result.Corollary 2.
Under the hypothesis of Theorem1 and(H7), the solution u of Problem P (obtained in Theorem 1) is unique and has the following regularity:
(8)u∈Lloc⁡∞0,∞;V∩Lloc⁡20,∞;H3/2Ω,u′∈Lloc⁡∞0,∞;V,u′′∈Lloc⁡∞0,∞;L2Ω,Ku′′-Δu+∑j=1nbj∂u′∂xj+uσu=00000000000inLloc⁡∞0,∞;L2Ω,∂u∂ν+h·,u′=0inLloc⁡20,∞;L2Γ1,u0=u0,u′0=u1inΩ.Remark 3.
If we replace the functionf(s)=|s|σs in Theorem 1 by a continuous function g such that
(9)gs-grs-r≥g0s-r2,∀s,r∈R,g0apositiveconstant,
and further
(10)gs≤g1s,∀s∈R,
Theorem 1 remains valid. Indeed, from (10), we obtain
(11)gls≤g1s,∀s∈R,
where (gl) is the Strauss’ approximations (see [16]) of the function g.Remark 4.
Analogously if|h(x,s)|≤h1|s|  ∀s∈R and a.e. x in Γ1 (h1 a positive constant), we obtain |hl(x,s)|≤3/2h1|s| for all s∈R (see [10]).For use later, note that(12)∀w∈Vw2≤1λ1w2,∀w∈VwL2Γ1≤k1w,∀w∈V,k1positiveconstants;
where λ1 is the first eigenvalue of the spectral problem ((u,v))=λ(u,v) for all v∈V (see [15]).In order to establish the uniform decay rate for energy, we assume(H8)
K′∞<b0/2;(H9)
K∞=esssup⁡t≥0⁡KtL∞(Ω);(H10)
div⁡b∞≤δ0;(H11)
b∞<21/32λ1, b0>32(1/λ1)/MK∞, K∞>b∞/2+8(1/λ1)/M, where M=min⁡⁡1,1-(9/32+(1/λ1)b∞).Remark 5.
There are functions that satisfy hypothesis (H5) and (H8). In fact, the function K(x,t)=α(x)/α∞β(t) with α∈C1(Ω¯), α(x)≥0, and β(t)=b0/4e-t satisfies such hypothesis, since
(13)K′x,t≤αxα∞b04e-t≤b04<b02,
and so
(14)K′x,t≤δ+Kx,t,∀δ≥0.Theorem 6.
Under the hypothesis of Theorem1 and(H7)–(H11), with h1(s) satisfying (9) and (10), the energy
(15)Et=12K1/2tu′t2+ut22σ+2∫Ωutσ+2dxiiiiiiiiiiiiii+2σ+2∫Ωutσ+2dxfort≥1,
associated with the solution, u, obtained in Corollary 2 is uniformly stable; that is, there exists a positive constant such that
(16)Et≤Cexp⁡-ϖt,∀t≥1,
where C and ϖ are positive constants.For use later, we observe that hypothesis in(H3)1 on σ implies q*=2n/(n-2)≥2σ+2 and q*≥σn. Thus, the Sobolev’s embedding gives
(17)V↪Lq*Ω↪L2σ+2Ωforn≥3,V↪L2σ+2Ωforn=1,2,(18)V↪Lq*Ω↪LσnΩforn≥3,V↪LσnΩforn=1,2.
Here, X↪Y indicates that the subspace X is continuously embedded in the space Y.Next, following the ideas contained in Strauss [16], we approximate the function h by Lipschitz-continuous ones hl.


## 3. Proof of Theorem1

For our purposes we need the following previous results, whose proof can be seen in [10].Lemma 7.
Leth be a function satisfying hypothesis(H3). Then there exists a sequence (hl) of functions in C0(R;L∞(Γ1)) such that(i)
hl(x,0)=0 a.e. x in Γ1;(ii)
[hl(x,s)-hl(x,r)](s-r)≥h0(s-r)2, ∀s,r∈R, and a.e. x in Γ1; (h0 positive constant);(iii)
for anyl there exists a function cl in L∞(Γ1) satisfying
(19)hlx,s-hlx,r≤cls-r,∀s,r∈R,a.e.xinΓ1;(iv)
(hl) converges to g uniformly on bounded sets on R and a.e. x in Γ1.Lemma 8.
LetT>0 be a real number. Consider the sequence (wl) of vectors in L2(0,T;H-1/2(Γ1))∩L1(0,T;L1(Γ1)) and vectors w∈L2(0,T;H-1/2(Γ1)) and χ∈L1(0,T;L1(Γ1)) such that(i)
wl→w weak in L2(0,T;H-1/2(Γ1));(ii)
wl→χ in L1(0,T;L1(Γ1)).
Then,w=χ.Lemma 9.
Letp:R→R be a globally Lipschitz-continuous function with p(0)=0 and let
(20)γ0:V⟶H1/2Γ1
be the continuous trace of order zero. Consider u∈L2(0,T;V) then p(u)∈L2(0,T;V), p(γ0u)∈L2(0,T;H1/2(Γ1)), and γ0p(u(t))=p(γ0u(t)) a.e. t∈(0,T).Proof.
We see that(21)p:V⟶V,p:H1/2Γ1⟶H1/2Γ1
are continuous maps (see Brezis and Cazenave [17] and Marcus and Mizel [18]). Let v∈V. Consider a sequence (vk) of functions of C1(Ω¯) such that
(22)vk⟶vinV.
Then by (21) and (22), we have p(vk)→p(v) in V, and by (20)(23)γ0pvk⟶γ0pvinH1/2Γ1.
Also by (20) and (22), we deduce
(24)pγ0vk⟶pγ0vinH1/2Γ1.
As γ0p(vk)=p(γ0vk), it follows from (23) and (24) that γ0p(v)=p(γ0v). This implies
(25)γ0put=pγ0uta.e.in0,T.
Now, we consider the setO={s∈R:pisnotdifferentiableins}. Then
(26)∂∂xipux,t=p′ux,t∂u∂xix,t,ux,t∉O,0,ux,t∈O,
with i=1,2,…,n (see Brezis and Cazenave, loc. cit). As p′∈L∞(R) then
(27)pu∈L20,T;V.
From this and since γ0:L2(0,T;V)→L2(0,T;H1/2(Γ1)) then γ0p(u)∈L2(0,T;H1/2(Γ1)). This and (25) furnish
(28)pγ0u∈L20,T;H1/2Γ1.
From (25) to (28) we have the results of this Lemma.Proof of Theorem1.
We will use the Faedo-Galerkin’s method with a special basis ofV∩H2(Ω). Thus, let us consider the Strauss’ approximation (hl) of h given by Lemma 7. Let us consider (ul1) a sequence of vectors in D(Ω) such that
(29)ul1⟶u1inH01Ω.
Note that hl(x,ul1)=0 and ∂u0/∂ν=0 on Γ1 since u0∈D(-Δ). Thus,
(30)∂u0∂ν+hl·,ul1=0on⁡Γ1,∀l.
Now, we fix l and construct the basis {w1l,w2l,…,} of V∩H2(Ω) such that u0, ul1 belong to the subspace [w1l,w2l] spanned by w1l and w2l. Let Vml=[w1l,w2l,…,wml] be the subspace of V∩H2(Ω) spanned by w1l,…,wml. With this basis we determine the approximate solutions uɛlm(t) of Problem AP, where 0<ɛ<1 fixed.
Approximated Perturbed Problem. This consists to find the functions uɛlm(t)=∑j=1mgɛjlm(t)wjl, solutions of the problem
APKt+ɛuɛlm′′t,wjl+uɛlmt,wjl+∑j=1nbjt∂uɛlm′t∂xj,wjl+uɛlmtσuɛlmt,wjl+hl·,uɛlm′t,wjlL2Γ1=0,j=1,2,…,m,uɛlm0=u0,uɛlm′0=ul1.
The above finite-dimensional system has a solution,uɛlm(t), defined on [0,tɛlm[. The following estimate allows us to extend this solution to the whole interval [0,∞).


### 3.1. Estimates

#### 3.1.1. First Estimate

Consideringw=2uɛlm′(t) in AP1, integrating from 0 to t  (0≤t<tɛlm), using the fact that hl(x,s)s≥h0s2 (see Part (ii) of Lemma 7), assumptions (H4) and (H5), and since 0<ɛ<1, we obtain
(31)Kt,uɛlm′2t+ɛuɛlm′t2+uɛlmt2iiiiiiii+2σ+2∫Ωuɛlmtσ+2dxiiiiiiii+2h0∫0t∫Γ1uɛlm′x,s2dΓdsiiiiiiii+2∫0t∑j=1n∫Ωbjs∂uɛlm′s∂xjuɛlm′sdxds≤ul12+u02+K0ul12+2σ+2∫Ωu0σ+2dxiiiiiiii+∫0tδuɛlm′s2+CδKs,uɛlm′2sds.
Note that
(32)∑j=1n∫Ωbjt∂uɛlm′∂xjtuɛlm′tdx=-12∫Ωdiv⁡buɛlm′2dx+12∫Γ1b·νuɛlm′2dΓ.
In fact, by the Gauss’s formula we have
(33)∫Ωbjuɛlm′∂uɛlm′∂xjdx=-12∫Ω∂bj∂xjuɛlm′2dx+12∫Γ1bjνjuɛlm′2dΓ,
where νj is the jth entry of the normal vector ν. Hence, by (A2) we obtain
(34)∫0t∑j=1n∫Ωbjuɛlm′∂uɛlm′s∂xjdxds≥b02∫0t∫Ωuɛlm′s2dxdsiiiiiiii+12∫0t∫Γ1bs·νuɛlm′t2dΓds.
Using the hypothesis div⁡b(t)≤-b0, choosing δ=b0/4>0, and plugging (34) in (31), we find
(35)Kt,uɛlm′2t+ɛuɛlm′t2+uɛlmt2iiiiiiii+b02∫0tuɛlm′s2ds+2σ+2∫Ωuɛlmtσ+2dxiiiiiiii+d0∫0t∫Γ1uɛlm′x,s2dΓds≤c0u12+u02+c1u0σ+2iiiiiiii+Cδ∫0tKs,uɛlm′2sds,
where d0=2h0-b∞/2>0 for all l≥l0, ∀m and 0<ɛ<1. Moreover, vLσ+2(Ω)≤c1v for all v∈V. Therefore by the Gronwall’s inequality and (35)(36)K1/2uɛlm′isboundedinLloc⁡∞0,∞;L2Ω,uɛlmisboundedinLloc⁡∞0,∞;V,uɛlm′isboundedinLloc⁡20,∞;L2Γ1.


#### 3.1.2. Second Estimate

Differentiating with respect tot the approximate equation AP1 and taking w=2uɛlm′′(t), we obtain
(37)ddtKtuɛlm′′t,uɛlm′′t+ddtɛuɛlm′′t2+ddtuɛlm′t2+K′tuɛlm′′t,uɛlm′′t+2∫Γ1hl′·,uɛlm′tuɛlm′′t2dΓ+2∑i=1nbi∂uɛlm′′t∂xi,uɛlm′′t+2σ+1uɛlmtσuɛlm′t,uɛlm′′t+2∑i=1nbi′t∂uɛlm′t∂xit,uɛlm′′tt=0.
Note that
(38)∑j=1nbjt∂uɛlm′′t∂xjt,uɛlm′′t=-12∫Ωdiv⁡buɛlm′′t2dx+12∫Γ1bt·νuɛlm′′t2dΓ
and that
(39)∑j=1nbj′t∂uɛlm′t∂xj,uɛlm′′t≤nmax⁡1≤j≤nbj′∞∂uɛlm′∂xjtuɛlm′′t≤nmax⁡1≤j≤nbj′∞∇uɛlm′tuɛlm′′t≤n1/2max⁡1≤j≤nbj′∞2Cρ∇uɛlm′t2+ρuɛlm′′t2,
where the last inequality becomes from Young’s inequality. Therefore, if
(40)K2≔Cρn1/2max⁡1≤j≤nbj∞2,
then |∑j=1nbj′(t)∂uɛlm′(t)/∂xj(t),uɛlm′′(t)|≤K2uɛlm′(t)2+ρ|uɛlm′′(t)|2. Combining this inequality and (38) with (37), after that using that hl′(·,s)≥h0, div⁡b(t)≤-b0 and integrating from 0 to t, we obtain
(41)Kt,uɛlm′′2t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b02-ρ∫0tuɛlm′′s2dsiiiiiiii+2h0-b∞2∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+K1/20uɛlm′′0iiiiiiii-2σ+1∫0tuɛlmsσuɛlm′s,uɛlm′′sdsiiiiiiii+δ∫0tuɛlm′′s2ds+K2∫0tuɛlm′s2iiiiiiii+∫0tCδKs,uɛlm′′2sds.
We also have used the above hypotheses (H4) and (H5). Now, by Hölder’s inequality for 1/q1*+1/n+1/2=1, embedding (18) and first estimate (36), we find
(42)uεlmtσuεlm′t,uεlm′′t≤∫Ωuεlmtσn1/n∫Ωuεlm′tq1*1/q1*iiiiiiii·∫Ωuεlm′′t21/2≤Cuεlmtσuεlm′tuεlm′′t≤Cuεlm′tuεlm′′t≤Cηulm′t2+ηulm′′t2,
where the constant C is independent of l, m, ɛ and t≥0, and Cη is a positive constant that depends on η. Substituting this inequality in (41) yields
(43)Ktuɛlm′′t,uɛlm′′t+ɛuɛlm′′t2iiiiiiii+uɛlm′t2b02-ρ+η∫0tuɛlm′′s2dsiiiiiiii+2h0-b∞2∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+sup⁡x∈Ω⁡Kx,0uɛlm′′02iiiiiiii+K2∫0tuɛlm′s2+δ∫0tuɛlm′′s2dsiiiiiiii+∫0tCδKsuɛlm′′s,uɛlm′′sds.
Choosing δ=b0/4 and ρ, η small such that b1=b0/2-(ρ+η)>0, we find
(44)Kt,uɛlm′′2t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b1∫0tuɛlm′′s2ds+d0∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+sup⁡x∈Ω⁡Kx,0uɛlm′′02iiiiiiii+K2∫0tuɛlm′s2+CδKsuɛlm′′s,uɛlm′′sds.
Now we need to derive an estimate for (uɛlm′′(0)). Thus, taking t=0 in approximate Problem AP1 and choosing v=uɛlm′′(0), one has
(45)K0+ɛuɛlm′′0,uɛlm′′0+u0,uɛlm′′0+∫Γ1hl·,ul1uɛlm′′0dΓ+∑i=1nbj0∂ul1∂xi,uɛlm′′0+u0σu0,uɛlm′′0=0.
Applying Green’s formula
(46)K0+ɛuɛlm′′0,uɛlm′′0-Δu0,uɛlm′′0+∫Γ1∂u0∂ν+hl·,ul1uɛlm′′0dΓ+∑i=1nbj0∂ul1∂xi,uɛlm′′0+u0σu0,uɛlm′′0=0.
Thanks to (30), the integrals on Γ1 in (46) are null. Using convergence (29), embedding (17), and the hypothesis (H4) give
(47)uɛlm′′0≤C,∀l,m,ɛ.
From this boundedness and (46), we get
(48)Ktuɛlm′′t,uɛlm′′t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b1∫0tuɛlm′′s2ds+d0∫0t∫Γ1uɛlm′′s2dΓds≤C1+K2∫0tuɛlm′s2iiiiiiii+∫0tCδKsuɛlm′′t,uɛlm′′sds,
where C1 is constant independent of l, m, and ɛ. Applying Gronwall’s inequality in (48) and using estimate (47), we have
(49)K1/2uɛlm′′isboundedinLloc⁡∞0,∞;L2Ω;uɛlm′′isboundedinLloc⁡20,∞;L2Ω;uɛlm′isboundedinLloc⁡∞0,∞;V;uɛlm′′isboundedinLloc⁡20,∞;L2Γ1.
From estimates (36), (49), induction, and diagonal process, we obtain a subsequence of (uɛlm), which is still denoted by (uɛlm), and a function uɛl:Ω×]0,∞[→R, such that
(50)uɛlm⟶uϵlweakstarinLloc⁡∞0,∞;V;uɛlm′⟶uϵl′weakstarinLloc⁡∞0,∞;V;Kuɛlm′′⟶Kuϵl′′weakstarinLloc⁡∞0,∞;L2Ω;uɛlm′⟶uϵl′weakinLloc⁡20,∞;L2Γ1;uɛlm′′⟶uϵl′′weakinLloc⁡20,∞;L2Γ1.
From the convergence (49)2 we obtain
(51)∑i=1nbj·∂uɛlm′∂xj⟶∑i=1nbj·∂uɛl′∂xjweakstarinLloc⁡∞0,∞;L2Ω.


#### 3.1.3. Analysis of the Nonlinear Terms

By estimates (36)1, (36)2, compactness method (cf. Lions [19] or Simon [20]), embedding (17), induction, and diagonal process, we obtain a subsequence of (uɛlm), which also is denoted by (uɛlm), such that
(52)uɛlmσulm⟶uɛlσuɛlweakstarinLloc⁡∞0,∞;L2Ω.
From (48)1, we have that (uɛlm′) is bounded in Lloc⁡∞(0,∞;H1/2(Γ1)). Thus, estimate (49)3 and the compactness embedding of H1/2(Γ1) in L2(Γ1), give uɛlm′→uɛl′ in Lloc⁡2(0,∞;L2(Γ1)). From this, property (iii) and Lemma 7 yield
(53)hl·,uɛlm′⟶hl·,uɛl′inLloc⁡20,∞;L2Γ1.


#### 3.1.4. Passage to the Limit asm

Convergences (49), (51), and (53) permit us to pass to the limits in approximate equations AP, as m→∞. Thus, this fact and the density of Vml in V, give
(54)∫0TK+ɛuɛl′′t,vψtdt+∫0Tuɛlt,vψtdt+∫0T∫Γ1hl·,uɛl′tvψtdΓdt+∑i=1n∫0Tbi∂uɛl′∂xit,vψtdt+∫0Tuɛltσuɛt,vψt=0,00000000000∀ψ∈D0,T,∀v∈V.
Now, if ψ∈D(0,T) and v∈D(Ω) we obtain using the regularity of uɛl (given by (49)), that
(55)K+ɛuɛl′′-Δuɛl+∑i=1nbi∂uɛl′∂xi+uɛlσuɛl=0inLloc⁡∞0,∞;L2Ω.
From (49)1 and (55) we have uɛl∈L∞(0,T;V) and Δuɛl∈L∞(0,T;L2(Ω)), respectively. Thus, ∂uɛl/∂ν∈L∞(0,T;H-1/2(Γ1)). (compare to Lions [19] and Medeiros and Milla Miranda [21]). Multiplying both sides of (55) by vψ, with v∈V and ψ∈D(0,T), and integrating over Ω×[0,T[, then the preceding regularity, ∂uɛl/∂ν, gives
(56)∫0TK+ɛuɛl′′,vψdt+∫0Tuɛl,vψdt-∫0T∂uɛl∂ν,vψdt+∫0Tuɛlσuɛl,vψdt+∑i=1n∫0Tbi∂uɛl′∂xi,vψdt=0,
where ·,· denotes the duality paring between H-1/2(Γ1) and H1/2(Γ1).Comparing (54) and (56) and using the Lipschitz property of hl, we obtain
(57)∂uɛl∂ν+hl·,uɛl′=0inLloc⁡20,∞;L2Γ1.


#### 3.1.5. Passage to the Limit inε→0 and l→∞

As estimates (36) and (49) are independent of l, m, and ɛ we obtain a subsequence of (uɛl), which still denoted by (uɛl), and a function ul such that all convergences (49) and (52) are valid. These convergences will be denoted by (49)ɛ, (51)ɛ, and (52)ɛ, respectively. These results imply that there exists a function ul belonging to class (49) and it is a solution of equation
(58)Kul′′-Δul+∑i=1nbi∂ul′∂xi+ulσul=0inLloc⁡∞0,∞;L2Ω,(59)∂ul∂ν+hl·,ul′=0inLloc⁡20,∞;L2Γ1.
Denoting these convergence in l by (49)l, (51)l, and (52)l, respectively, then the convergence (49)l gives us ul→u weak star in L∞(0,∞;V). From this and (55), Δul→Δu weak star in Lloc⁡∞(0,∞;L2(Ω)). Then
(60)∂ul∂ν⟶∂u∂νweakstarinLloc⁡∞0,∞;H-1/2Γ1.
Moreover, convergence (49)l furnishes ul′→u′inLloc⁡2(0,∞;L2(Γ1)). Now, we fix T>0. The preceding convergence implies
(61)ul′x,t⟶u′x,ta.einΣ1=Γ1×0,T.
Fixing (x,t)∈Σ1, then by (61) the set {ul′(x,t):l∈N} is bounded. Part (iv) of Lemma 7 says that (hl) converges to h uniformly in bounded sets of R, a.e. x in Γ1. These two results and (61) give
(62)hlx,ul′x,t⟶hx,u′x,ta.einΣ1=Γ1×0,T.On the other hand, by (58) and (59), we obtain
(63)∫Γ1hl·,ul′tul′tdΓ=-Ktul′′t,ul′t-12ddtult2iiiiiiii-∑i=1nbi∂ul′t∂xi,ul′t-ultσult,ul′t.
By familiar inequalities,
(64)∑i=1nbi∂ul′t∂xi,ul′t≤Cul′t2+ul′t2,
and from embedding (17),
(65)ultσult,ul′t≤Cult2σ+2+ul′t2,
where C>0 is a constant independent of l and t∈[0,T].As(66)Ktul′′t,ul′t=12ddtKtul′t,ul′t-12K′tul′t,ul′t,
integrating the inequality above from 0 to T, and using the hypothesis (H5) and (H6), we find
(67)∫0TKtul′′t,ul′tdt=12KTul′T,ul′T-K0ul1,ul1iiiiiiii-12∫0TK′tul′t,ul′tdt≤K∞ul′T2+ul12iiiiiiii+∫0Tδul′s2+CδKs,ul′2sds≤K∞ul′T2+ul12iiiiiiii+Cδ,K∞∫0Tul′t2dt.Note thatul∈C0([0,T];V), ul′∈C0([0,T];L2(Ω)) and that (ul(T)) and (ul′(T)) are bounded in V and in L2(Ω), respectively. Taking into account the preceding considerations, estimates (49)l and convergence (36)l in (63), we obtain
(68)∫0T∫Γ1hl·,ul′tul′tdΓdt≤C,0000000000∀l≥l0,∀t∈0,T,
where C>0 is a constant independent of l≥l0 and t∈[0,T]. Note that hl(·,ul′(t))ul′(t)≥0.From (62), (68), Strauss’ approximations, Lemma 9 and from a diagonal process, we get
(69)hl·,ul′⟶h·,u′inLloc⁡10,∞;L1Γ1.Convergences (60) and (69) imply ∂ul/∂ν→∂u/∂ν in D′(Γ1×]0,∞[) and hl(·,ul′)→h(·,u′) in D′(Γ1×]0,∞[). Now we take the limit in (59). Moreover the last two convergences and the regularity of h(·,u′) imply ∂u/∂ν+h(·,u′)=0 in Lloc⁡1(0,∞;L1(Γ1)), which shows that u satisfies
(70)Ku′′-Δu+∑i=1nbi∂u′∂xi+uσu=00000000000inLloc⁡∞0,∞;L2Ω.
Hence, the result is done as in (58).The verification of the initial conditions follows by convergence (49)l.Remark 10.
If|h(x,s)|≤h1|s|a.e.x∈Γ, then the sequence hl, which converges to h, satisfies |hl(x,s)|≤3/2h1|s| (see Lourêdo and Miranda [10]). In these conditions, the solution is unique and ∂u/∂ν+h(·,u′)=0inLloc⁡2(0,∞;L2(Γ1)). Consequently, u∈L∞(0,∞,V∩H3/2(Ω)).
The proof of Corollary2 follows from Remark 10 and from regularity of elliptic problems (see Lions and Magenes [22]).


## 3.1. Estimates

### 3.1.1. First Estimate

Consideringw=2uɛlm′(t) in AP1, integrating from 0 to t  (0≤t<tɛlm), using the fact that hl(x,s)s≥h0s2 (see Part (ii) of Lemma 7), assumptions (H4) and (H5), and since 0<ɛ<1, we obtain
(31)Kt,uɛlm′2t+ɛuɛlm′t2+uɛlmt2iiiiiiii+2σ+2∫Ωuɛlmtσ+2dxiiiiiiii+2h0∫0t∫Γ1uɛlm′x,s2dΓdsiiiiiiii+2∫0t∑j=1n∫Ωbjs∂uɛlm′s∂xjuɛlm′sdxds≤ul12+u02+K0ul12+2σ+2∫Ωu0σ+2dxiiiiiiii+∫0tδuɛlm′s2+CδKs,uɛlm′2sds.
Note that
(32)∑j=1n∫Ωbjt∂uɛlm′∂xjtuɛlm′tdx=-12∫Ωdiv⁡buɛlm′2dx+12∫Γ1b·νuɛlm′2dΓ.
In fact, by the Gauss’s formula we have
(33)∫Ωbjuɛlm′∂uɛlm′∂xjdx=-12∫Ω∂bj∂xjuɛlm′2dx+12∫Γ1bjνjuɛlm′2dΓ,
where νj is the jth entry of the normal vector ν. Hence, by (A2) we obtain
(34)∫0t∑j=1n∫Ωbjuɛlm′∂uɛlm′s∂xjdxds≥b02∫0t∫Ωuɛlm′s2dxdsiiiiiiii+12∫0t∫Γ1bs·νuɛlm′t2dΓds.
Using the hypothesis div⁡b(t)≤-b0, choosing δ=b0/4>0, and plugging (34) in (31), we find
(35)Kt,uɛlm′2t+ɛuɛlm′t2+uɛlmt2iiiiiiii+b02∫0tuɛlm′s2ds+2σ+2∫Ωuɛlmtσ+2dxiiiiiiii+d0∫0t∫Γ1uɛlm′x,s2dΓds≤c0u12+u02+c1u0σ+2iiiiiiii+Cδ∫0tKs,uɛlm′2sds,
where d0=2h0-b∞/2>0 for all l≥l0, ∀m and 0<ɛ<1. Moreover, vLσ+2(Ω)≤c1v for all v∈V. Therefore by the Gronwall’s inequality and (35)(36)K1/2uɛlm′isboundedinLloc⁡∞0,∞;L2Ω,uɛlmisboundedinLloc⁡∞0,∞;V,uɛlm′isboundedinLloc⁡20,∞;L2Γ1.


### 3.1.2. Second Estimate

Differentiating with respect tot the approximate equation AP1 and taking w=2uɛlm′′(t), we obtain
(37)ddtKtuɛlm′′t,uɛlm′′t+ddtɛuɛlm′′t2+ddtuɛlm′t2+K′tuɛlm′′t,uɛlm′′t+2∫Γ1hl′·,uɛlm′tuɛlm′′t2dΓ+2∑i=1nbi∂uɛlm′′t∂xi,uɛlm′′t+2σ+1uɛlmtσuɛlm′t,uɛlm′′t+2∑i=1nbi′t∂uɛlm′t∂xit,uɛlm′′tt=0.
Note that
(38)∑j=1nbjt∂uɛlm′′t∂xjt,uɛlm′′t=-12∫Ωdiv⁡buɛlm′′t2dx+12∫Γ1bt·νuɛlm′′t2dΓ
and that
(39)∑j=1nbj′t∂uɛlm′t∂xj,uɛlm′′t≤nmax⁡1≤j≤nbj′∞∂uɛlm′∂xjtuɛlm′′t≤nmax⁡1≤j≤nbj′∞∇uɛlm′tuɛlm′′t≤n1/2max⁡1≤j≤nbj′∞2Cρ∇uɛlm′t2+ρuɛlm′′t2,
where the last inequality becomes from Young’s inequality. Therefore, if
(40)K2≔Cρn1/2max⁡1≤j≤nbj∞2,
then |∑j=1nbj′(t)∂uɛlm′(t)/∂xj(t),uɛlm′′(t)|≤K2uɛlm′(t)2+ρ|uɛlm′′(t)|2. Combining this inequality and (38) with (37), after that using that hl′(·,s)≥h0, div⁡b(t)≤-b0 and integrating from 0 to t, we obtain
(41)Kt,uɛlm′′2t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b02-ρ∫0tuɛlm′′s2dsiiiiiiii+2h0-b∞2∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+K1/20uɛlm′′0iiiiiiii-2σ+1∫0tuɛlmsσuɛlm′s,uɛlm′′sdsiiiiiiii+δ∫0tuɛlm′′s2ds+K2∫0tuɛlm′s2iiiiiiii+∫0tCδKs,uɛlm′′2sds.
We also have used the above hypotheses (H4) and (H5). Now, by Hölder’s inequality for 1/q1*+1/n+1/2=1, embedding (18) and first estimate (36), we find
(42)uεlmtσuεlm′t,uεlm′′t≤∫Ωuεlmtσn1/n∫Ωuεlm′tq1*1/q1*iiiiiiii·∫Ωuεlm′′t21/2≤Cuεlmtσuεlm′tuεlm′′t≤Cuεlm′tuεlm′′t≤Cηulm′t2+ηulm′′t2,
where the constant C is independent of l, m, ɛ and t≥0, and Cη is a positive constant that depends on η. Substituting this inequality in (41) yields
(43)Ktuɛlm′′t,uɛlm′′t+ɛuɛlm′′t2iiiiiiii+uɛlm′t2b02-ρ+η∫0tuɛlm′′s2dsiiiiiiii+2h0-b∞2∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+sup⁡x∈Ω⁡Kx,0uɛlm′′02iiiiiiii+K2∫0tuɛlm′s2+δ∫0tuɛlm′′s2dsiiiiiiii+∫0tCδKsuɛlm′′s,uɛlm′′sds.
Choosing δ=b0/4 and ρ, η small such that b1=b0/2-(ρ+η)>0, we find
(44)Kt,uɛlm′′2t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b1∫0tuɛlm′′s2ds+d0∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+sup⁡x∈Ω⁡Kx,0uɛlm′′02iiiiiiii+K2∫0tuɛlm′s2+CδKsuɛlm′′s,uɛlm′′sds.
Now we need to derive an estimate for (uɛlm′′(0)). Thus, taking t=0 in approximate Problem AP1 and choosing v=uɛlm′′(0), one has
(45)K0+ɛuɛlm′′0,uɛlm′′0+u0,uɛlm′′0+∫Γ1hl·,ul1uɛlm′′0dΓ+∑i=1nbj0∂ul1∂xi,uɛlm′′0+u0σu0,uɛlm′′0=0.
Applying Green’s formula
(46)K0+ɛuɛlm′′0,uɛlm′′0-Δu0,uɛlm′′0+∫Γ1∂u0∂ν+hl·,ul1uɛlm′′0dΓ+∑i=1nbj0∂ul1∂xi,uɛlm′′0+u0σu0,uɛlm′′0=0.
Thanks to (30), the integrals on Γ1 in (46) are null. Using convergence (29), embedding (17), and the hypothesis (H4) give
(47)uɛlm′′0≤C,∀l,m,ɛ.
From this boundedness and (46), we get
(48)Ktuɛlm′′t,uɛlm′′t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b1∫0tuɛlm′′s2ds+d0∫0t∫Γ1uɛlm′′s2dΓds≤C1+K2∫0tuɛlm′s2iiiiiiii+∫0tCδKsuɛlm′′t,uɛlm′′sds,
where C1 is constant independent of l, m, and ɛ. Applying Gronwall’s inequality in (48) and using estimate (47), we have
(49)K1/2uɛlm′′isboundedinLloc⁡∞0,∞;L2Ω;uɛlm′′isboundedinLloc⁡20,∞;L2Ω;uɛlm′isboundedinLloc⁡∞0,∞;V;uɛlm′′isboundedinLloc⁡20,∞;L2Γ1.
From estimates (36), (49), induction, and diagonal process, we obtain a subsequence of (uɛlm), which is still denoted by (uɛlm), and a function uɛl:Ω×]0,∞[→R, such that
(50)uɛlm⟶uϵlweakstarinLloc⁡∞0,∞;V;uɛlm′⟶uϵl′weakstarinLloc⁡∞0,∞;V;Kuɛlm′′⟶Kuϵl′′weakstarinLloc⁡∞0,∞;L2Ω;uɛlm′⟶uϵl′weakinLloc⁡20,∞;L2Γ1;uɛlm′′⟶uϵl′′weakinLloc⁡20,∞;L2Γ1.
From the convergence (49)2 we obtain
(51)∑i=1nbj·∂uɛlm′∂xj⟶∑i=1nbj·∂uɛl′∂xjweakstarinLloc⁡∞0,∞;L2Ω.


### 3.1.3. Analysis of the Nonlinear Terms

By estimates (36)1, (36)2, compactness method (cf. Lions [19] or Simon [20]), embedding (17), induction, and diagonal process, we obtain a subsequence of (uɛlm), which also is denoted by (uɛlm), such that
(52)uɛlmσulm⟶uɛlσuɛlweakstarinLloc⁡∞0,∞;L2Ω.
From (48)1, we have that (uɛlm′) is bounded in Lloc⁡∞(0,∞;H1/2(Γ1)). Thus, estimate (49)3 and the compactness embedding of H1/2(Γ1) in L2(Γ1), give uɛlm′→uɛl′ in Lloc⁡2(0,∞;L2(Γ1)). From this, property (iii) and Lemma 7 yield
(53)hl·,uɛlm′⟶hl·,uɛl′inLloc⁡20,∞;L2Γ1.


### 3.1.4. Passage to the Limit asm

Convergences (49), (51), and (53) permit us to pass to the limits in approximate equations AP, as m→∞. Thus, this fact and the density of Vml in V, give
(54)∫0TK+ɛuɛl′′t,vψtdt+∫0Tuɛlt,vψtdt+∫0T∫Γ1hl·,uɛl′tvψtdΓdt+∑i=1n∫0Tbi∂uɛl′∂xit,vψtdt+∫0Tuɛltσuɛt,vψt=0,00000000000∀ψ∈D0,T,∀v∈V.
Now, if ψ∈D(0,T) and v∈D(Ω) we obtain using the regularity of uɛl (given by (49)), that
(55)K+ɛuɛl′′-Δuɛl+∑i=1nbi∂uɛl′∂xi+uɛlσuɛl=0inLloc⁡∞0,∞;L2Ω.
From (49)1 and (55) we have uɛl∈L∞(0,T;V) and Δuɛl∈L∞(0,T;L2(Ω)), respectively. Thus, ∂uɛl/∂ν∈L∞(0,T;H-1/2(Γ1)). (compare to Lions [19] and Medeiros and Milla Miranda [21]). Multiplying both sides of (55) by vψ, with v∈V and ψ∈D(0,T), and integrating over Ω×[0,T[, then the preceding regularity, ∂uɛl/∂ν, gives
(56)∫0TK+ɛuɛl′′,vψdt+∫0Tuɛl,vψdt-∫0T∂uɛl∂ν,vψdt+∫0Tuɛlσuɛl,vψdt+∑i=1n∫0Tbi∂uɛl′∂xi,vψdt=0,
where ·,· denotes the duality paring between H-1/2(Γ1) and H1/2(Γ1).Comparing (54) and (56) and using the Lipschitz property of hl, we obtain
(57)∂uɛl∂ν+hl·,uɛl′=0inLloc⁡20,∞;L2Γ1.


### 3.1.5. Passage to the Limit inε→0 and l→∞

As estimates (36) and (49) are independent of l, m, and ɛ we obtain a subsequence of (uɛl), which still denoted by (uɛl), and a function ul such that all convergences (49) and (52) are valid. These convergences will be denoted by (49)ɛ, (51)ɛ, and (52)ɛ, respectively. These results imply that there exists a function ul belonging to class (49) and it is a solution of equation
(58)Kul′′-Δul+∑i=1nbi∂ul′∂xi+ulσul=0inLloc⁡∞0,∞;L2Ω,(59)∂ul∂ν+hl·,ul′=0inLloc⁡20,∞;L2Γ1.
Denoting these convergence in l by (49)l, (51)l, and (52)l, respectively, then the convergence (49)l gives us ul→u weak star in L∞(0,∞;V). From this and (55), Δul→Δu weak star in Lloc⁡∞(0,∞;L2(Ω)). Then
(60)∂ul∂ν⟶∂u∂νweakstarinLloc⁡∞0,∞;H-1/2Γ1.
Moreover, convergence (49)l furnishes ul′→u′inLloc⁡2(0,∞;L2(Γ1)). Now, we fix T>0. The preceding convergence implies
(61)ul′x,t⟶u′x,ta.einΣ1=Γ1×0,T.
Fixing (x,t)∈Σ1, then by (61) the set {ul′(x,t):l∈N} is bounded. Part (iv) of Lemma 7 says that (hl) converges to h uniformly in bounded sets of R, a.e. x in Γ1. These two results and (61) give
(62)hlx,ul′x,t⟶hx,u′x,ta.einΣ1=Γ1×0,T.On the other hand, by (58) and (59), we obtain
(63)∫Γ1hl·,ul′tul′tdΓ=-Ktul′′t,ul′t-12ddtult2iiiiiiii-∑i=1nbi∂ul′t∂xi,ul′t-ultσult,ul′t.
By familiar inequalities,
(64)∑i=1nbi∂ul′t∂xi,ul′t≤Cul′t2+ul′t2,
and from embedding (17),
(65)ultσult,ul′t≤Cult2σ+2+ul′t2,
where C>0 is a constant independent of l and t∈[0,T].As(66)Ktul′′t,ul′t=12ddtKtul′t,ul′t-12K′tul′t,ul′t,
integrating the inequality above from 0 to T, and using the hypothesis (H5) and (H6), we find
(67)∫0TKtul′′t,ul′tdt=12KTul′T,ul′T-K0ul1,ul1iiiiiiii-12∫0TK′tul′t,ul′tdt≤K∞ul′T2+ul12iiiiiiii+∫0Tδul′s2+CδKs,ul′2sds≤K∞ul′T2+ul12iiiiiiii+Cδ,K∞∫0Tul′t2dt.Note thatul∈C0([0,T];V), ul′∈C0([0,T];L2(Ω)) and that (ul(T)) and (ul′(T)) are bounded in V and in L2(Ω), respectively. Taking into account the preceding considerations, estimates (49)l and convergence (36)l in (63), we obtain
(68)∫0T∫Γ1hl·,ul′tul′tdΓdt≤C,0000000000∀l≥l0,∀t∈0,T,
where C>0 is a constant independent of l≥l0 and t∈[0,T]. Note that hl(·,ul′(t))ul′(t)≥0.From (62), (68), Strauss’ approximations, Lemma 9 and from a diagonal process, we get
(69)hl·,ul′⟶h·,u′inLloc⁡10,∞;L1Γ1.Convergences (60) and (69) imply ∂ul/∂ν→∂u/∂ν in D′(Γ1×]0,∞[) and hl(·,ul′)→h(·,u′) in D′(Γ1×]0,∞[). Now we take the limit in (59). Moreover the last two convergences and the regularity of h(·,u′) imply ∂u/∂ν+h(·,u′)=0 in Lloc⁡1(0,∞;L1(Γ1)), which shows that u satisfies
(70)Ku′′-Δu+∑i=1nbi∂u′∂xi+uσu=00000000000inLloc⁡∞0,∞;L2Ω.
Hence, the result is done as in (58).The verification of the initial conditions follows by convergence (49)l.Remark 10.
If|h(x,s)|≤h1|s|a.e.x∈Γ, then the sequence hl, which converges to h, satisfies |hl(x,s)|≤3/2h1|s| (see Lourêdo and Miranda [10]). In these conditions, the solution is unique and ∂u/∂ν+h(·,u′)=0inLloc⁡2(0,∞;L2(Γ1)). Consequently, u∈L∞(0,∞,V∩H3/2(Ω)).
The proof of Corollary2 follows from Remark 10 and from regularity of elliptic problems (see Lions and Magenes [22]).


## 3.1.1. First Estimate

Consideringw=2uɛlm′(t) in AP1, integrating from 0 to t  (0≤t<tɛlm), using the fact that hl(x,s)s≥h0s2 (see Part (ii) of Lemma 7), assumptions (H4) and (H5), and since 0<ɛ<1, we obtain
(31)Kt,uɛlm′2t+ɛuɛlm′t2+uɛlmt2iiiiiiii+2σ+2∫Ωuɛlmtσ+2dxiiiiiiii+2h0∫0t∫Γ1uɛlm′x,s2dΓdsiiiiiiii+2∫0t∑j=1n∫Ωbjs∂uɛlm′s∂xjuɛlm′sdxds≤ul12+u02+K0ul12+2σ+2∫Ωu0σ+2dxiiiiiiii+∫0tδuɛlm′s2+CδKs,uɛlm′2sds.
Note that
(32)∑j=1n∫Ωbjt∂uɛlm′∂xjtuɛlm′tdx=-12∫Ωdiv⁡buɛlm′2dx+12∫Γ1b·νuɛlm′2dΓ.
In fact, by the Gauss’s formula we have
(33)∫Ωbjuɛlm′∂uɛlm′∂xjdx=-12∫Ω∂bj∂xjuɛlm′2dx+12∫Γ1bjνjuɛlm′2dΓ,
where νj is the jth entry of the normal vector ν. Hence, by (A2) we obtain
(34)∫0t∑j=1n∫Ωbjuɛlm′∂uɛlm′s∂xjdxds≥b02∫0t∫Ωuɛlm′s2dxdsiiiiiiii+12∫0t∫Γ1bs·νuɛlm′t2dΓds.
Using the hypothesis div⁡b(t)≤-b0, choosing δ=b0/4>0, and plugging (34) in (31), we find
(35)Kt,uɛlm′2t+ɛuɛlm′t2+uɛlmt2iiiiiiii+b02∫0tuɛlm′s2ds+2σ+2∫Ωuɛlmtσ+2dxiiiiiiii+d0∫0t∫Γ1uɛlm′x,s2dΓds≤c0u12+u02+c1u0σ+2iiiiiiii+Cδ∫0tKs,uɛlm′2sds,
where d0=2h0-b∞/2>0 for all l≥l0, ∀m and 0<ɛ<1. Moreover, vLσ+2(Ω)≤c1v for all v∈V. Therefore by the Gronwall’s inequality and (35)(36)K1/2uɛlm′isboundedinLloc⁡∞0,∞;L2Ω,uɛlmisboundedinLloc⁡∞0,∞;V,uɛlm′isboundedinLloc⁡20,∞;L2Γ1.


## 3.1.2. Second Estimate

Differentiating with respect tot the approximate equation AP1 and taking w=2uɛlm′′(t), we obtain
(37)ddtKtuɛlm′′t,uɛlm′′t+ddtɛuɛlm′′t2+ddtuɛlm′t2+K′tuɛlm′′t,uɛlm′′t+2∫Γ1hl′·,uɛlm′tuɛlm′′t2dΓ+2∑i=1nbi∂uɛlm′′t∂xi,uɛlm′′t+2σ+1uɛlmtσuɛlm′t,uɛlm′′t+2∑i=1nbi′t∂uɛlm′t∂xit,uɛlm′′tt=0.
Note that
(38)∑j=1nbjt∂uɛlm′′t∂xjt,uɛlm′′t=-12∫Ωdiv⁡buɛlm′′t2dx+12∫Γ1bt·νuɛlm′′t2dΓ
and that
(39)∑j=1nbj′t∂uɛlm′t∂xj,uɛlm′′t≤nmax⁡1≤j≤nbj′∞∂uɛlm′∂xjtuɛlm′′t≤nmax⁡1≤j≤nbj′∞∇uɛlm′tuɛlm′′t≤n1/2max⁡1≤j≤nbj′∞2Cρ∇uɛlm′t2+ρuɛlm′′t2,
where the last inequality becomes from Young’s inequality. Therefore, if
(40)K2≔Cρn1/2max⁡1≤j≤nbj∞2,
then |∑j=1nbj′(t)∂uɛlm′(t)/∂xj(t),uɛlm′′(t)|≤K2uɛlm′(t)2+ρ|uɛlm′′(t)|2. Combining this inequality and (38) with (37), after that using that hl′(·,s)≥h0, div⁡b(t)≤-b0 and integrating from 0 to t, we obtain
(41)Kt,uɛlm′′2t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b02-ρ∫0tuɛlm′′s2dsiiiiiiii+2h0-b∞2∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+K1/20uɛlm′′0iiiiiiii-2σ+1∫0tuɛlmsσuɛlm′s,uɛlm′′sdsiiiiiiii+δ∫0tuɛlm′′s2ds+K2∫0tuɛlm′s2iiiiiiii+∫0tCδKs,uɛlm′′2sds.
We also have used the above hypotheses (H4) and (H5). Now, by Hölder’s inequality for 1/q1*+1/n+1/2=1, embedding (18) and first estimate (36), we find
(42)uεlmtσuεlm′t,uεlm′′t≤∫Ωuεlmtσn1/n∫Ωuεlm′tq1*1/q1*iiiiiiii·∫Ωuεlm′′t21/2≤Cuεlmtσuεlm′tuεlm′′t≤Cuεlm′tuεlm′′t≤Cηulm′t2+ηulm′′t2,
where the constant C is independent of l, m, ɛ and t≥0, and Cη is a positive constant that depends on η. Substituting this inequality in (41) yields
(43)Ktuɛlm′′t,uɛlm′′t+ɛuɛlm′′t2iiiiiiii+uɛlm′t2b02-ρ+η∫0tuɛlm′′s2dsiiiiiiii+2h0-b∞2∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+sup⁡x∈Ω⁡Kx,0uɛlm′′02iiiiiiii+K2∫0tuɛlm′s2+δ∫0tuɛlm′′s2dsiiiiiiii+∫0tCδKsuɛlm′′s,uɛlm′′sds.
Choosing δ=b0/4 and ρ, η small such that b1=b0/2-(ρ+η)>0, we find
(44)Kt,uɛlm′′2t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b1∫0tuɛlm′′s2ds+d0∫0t∫Γ1uɛlm′′s2dΓds≤uɛlm′′02+ul12+sup⁡x∈Ω⁡Kx,0uɛlm′′02iiiiiiii+K2∫0tuɛlm′s2+CδKsuɛlm′′s,uɛlm′′sds.
Now we need to derive an estimate for (uɛlm′′(0)). Thus, taking t=0 in approximate Problem AP1 and choosing v=uɛlm′′(0), one has
(45)K0+ɛuɛlm′′0,uɛlm′′0+u0,uɛlm′′0+∫Γ1hl·,ul1uɛlm′′0dΓ+∑i=1nbj0∂ul1∂xi,uɛlm′′0+u0σu0,uɛlm′′0=0.
Applying Green’s formula
(46)K0+ɛuɛlm′′0,uɛlm′′0-Δu0,uɛlm′′0+∫Γ1∂u0∂ν+hl·,ul1uɛlm′′0dΓ+∑i=1nbj0∂ul1∂xi,uɛlm′′0+u0σu0,uɛlm′′0=0.
Thanks to (30), the integrals on Γ1 in (46) are null. Using convergence (29), embedding (17), and the hypothesis (H4) give
(47)uɛlm′′0≤C,∀l,m,ɛ.
From this boundedness and (46), we get
(48)Ktuɛlm′′t,uɛlm′′t+ɛuɛlm′′t2+uɛlm′t2iiiiiiii+b1∫0tuɛlm′′s2ds+d0∫0t∫Γ1uɛlm′′s2dΓds≤C1+K2∫0tuɛlm′s2iiiiiiii+∫0tCδKsuɛlm′′t,uɛlm′′sds,
where C1 is constant independent of l, m, and ɛ. Applying Gronwall’s inequality in (48) and using estimate (47), we have
(49)K1/2uɛlm′′isboundedinLloc⁡∞0,∞;L2Ω;uɛlm′′isboundedinLloc⁡20,∞;L2Ω;uɛlm′isboundedinLloc⁡∞0,∞;V;uɛlm′′isboundedinLloc⁡20,∞;L2Γ1.
From estimates (36), (49), induction, and diagonal process, we obtain a subsequence of (uɛlm), which is still denoted by (uɛlm), and a function uɛl:Ω×]0,∞[→R, such that
(50)uɛlm⟶uϵlweakstarinLloc⁡∞0,∞;V;uɛlm′⟶uϵl′weakstarinLloc⁡∞0,∞;V;Kuɛlm′′⟶Kuϵl′′weakstarinLloc⁡∞0,∞;L2Ω;uɛlm′⟶uϵl′weakinLloc⁡20,∞;L2Γ1;uɛlm′′⟶uϵl′′weakinLloc⁡20,∞;L2Γ1.
From the convergence (49)2 we obtain
(51)∑i=1nbj·∂uɛlm′∂xj⟶∑i=1nbj·∂uɛl′∂xjweakstarinLloc⁡∞0,∞;L2Ω.


## 3.1.3. Analysis of the Nonlinear Terms

By estimates (36)1, (36)2, compactness method (cf. Lions [19] or Simon [20]), embedding (17), induction, and diagonal process, we obtain a subsequence of (uɛlm), which also is denoted by (uɛlm), such that
(52)uɛlmσulm⟶uɛlσuɛlweakstarinLloc⁡∞0,∞;L2Ω.
From (48)1, we have that (uɛlm′) is bounded in Lloc⁡∞(0,∞;H1/2(Γ1)). Thus, estimate (49)3 and the compactness embedding of H1/2(Γ1) in L2(Γ1), give uɛlm′→uɛl′ in Lloc⁡2(0,∞;L2(Γ1)). From this, property (iii) and Lemma 7 yield
(53)hl·,uɛlm′⟶hl·,uɛl′inLloc⁡20,∞;L2Γ1.


## 3.1.4. Passage to the Limit asm

Convergences (49), (51), and (53) permit us to pass to the limits in approximate equations AP, as m→∞. Thus, this fact and the density of Vml in V, give
(54)∫0TK+ɛuɛl′′t,vψtdt+∫0Tuɛlt,vψtdt+∫0T∫Γ1hl·,uɛl′tvψtdΓdt+∑i=1n∫0Tbi∂uɛl′∂xit,vψtdt+∫0Tuɛltσuɛt,vψt=0,00000000000∀ψ∈D0,T,∀v∈V.
Now, if ψ∈D(0,T) and v∈D(Ω) we obtain using the regularity of uɛl (given by (49)), that
(55)K+ɛuɛl′′-Δuɛl+∑i=1nbi∂uɛl′∂xi+uɛlσuɛl=0inLloc⁡∞0,∞;L2Ω.
From (49)1 and (55) we have uɛl∈L∞(0,T;V) and Δuɛl∈L∞(0,T;L2(Ω)), respectively. Thus, ∂uɛl/∂ν∈L∞(0,T;H-1/2(Γ1)). (compare to Lions [19] and Medeiros and Milla Miranda [21]). Multiplying both sides of (55) by vψ, with v∈V and ψ∈D(0,T), and integrating over Ω×[0,T[, then the preceding regularity, ∂uɛl/∂ν, gives
(56)∫0TK+ɛuɛl′′,vψdt+∫0Tuɛl,vψdt-∫0T∂uɛl∂ν,vψdt+∫0Tuɛlσuɛl,vψdt+∑i=1n∫0Tbi∂uɛl′∂xi,vψdt=0,
where ·,· denotes the duality paring between H-1/2(Γ1) and H1/2(Γ1).Comparing (54) and (56) and using the Lipschitz property of hl, we obtain
(57)∂uɛl∂ν+hl·,uɛl′=0inLloc⁡20,∞;L2Γ1.


## 3.1.5. Passage to the Limit inε→0 and l→∞

As estimates (36) and (49) are independent of l, m, and ɛ we obtain a subsequence of (uɛl), which still denoted by (uɛl), and a function ul such that all convergences (49) and (52) are valid. These convergences will be denoted by (49)ɛ, (51)ɛ, and (52)ɛ, respectively. These results imply that there exists a function ul belonging to class (49) and it is a solution of equation
(58)Kul′′-Δul+∑i=1nbi∂ul′∂xi+ulσul=0inLloc⁡∞0,∞;L2Ω,(59)∂ul∂ν+hl·,ul′=0inLloc⁡20,∞;L2Γ1.
Denoting these convergence in l by (49)l, (51)l, and (52)l, respectively, then the convergence (49)l gives us ul→u weak star in L∞(0,∞;V). From this and (55), Δul→Δu weak star in Lloc⁡∞(0,∞;L2(Ω)). Then
(60)∂ul∂ν⟶∂u∂νweakstarinLloc⁡∞0,∞;H-1/2Γ1.
Moreover, convergence (49)l furnishes ul′→u′inLloc⁡2(0,∞;L2(Γ1)). Now, we fix T>0. The preceding convergence implies
(61)ul′x,t⟶u′x,ta.einΣ1=Γ1×0,T.
Fixing (x,t)∈Σ1, then by (61) the set {ul′(x,t):l∈N} is bounded. Part (iv) of Lemma 7 says that (hl) converges to h uniformly in bounded sets of R, a.e. x in Γ1. These two results and (61) give
(62)hlx,ul′x,t⟶hx,u′x,ta.einΣ1=Γ1×0,T.On the other hand, by (58) and (59), we obtain
(63)∫Γ1hl·,ul′tul′tdΓ=-Ktul′′t,ul′t-12ddtult2iiiiiiii-∑i=1nbi∂ul′t∂xi,ul′t-ultσult,ul′t.
By familiar inequalities,
(64)∑i=1nbi∂ul′t∂xi,ul′t≤Cul′t2+ul′t2,
and from embedding (17),
(65)ultσult,ul′t≤Cult2σ+2+ul′t2,
where C>0 is a constant independent of l and t∈[0,T].As(66)Ktul′′t,ul′t=12ddtKtul′t,ul′t-12K′tul′t,ul′t,
integrating the inequality above from 0 to T, and using the hypothesis (H5) and (H6), we find
(67)∫0TKtul′′t,ul′tdt=12KTul′T,ul′T-K0ul1,ul1iiiiiiii-12∫0TK′tul′t,ul′tdt≤K∞ul′T2+ul12iiiiiiii+∫0Tδul′s2+CδKs,ul′2sds≤K∞ul′T2+ul12iiiiiiii+Cδ,K∞∫0Tul′t2dt.Note thatul∈C0([0,T];V), ul′∈C0([0,T];L2(Ω)) and that (ul(T)) and (ul′(T)) are bounded in V and in L2(Ω), respectively. Taking into account the preceding considerations, estimates (49)l and convergence (36)l in (63), we obtain
(68)∫0T∫Γ1hl·,ul′tul′tdΓdt≤C,0000000000∀l≥l0,∀t∈0,T,
where C>0 is a constant independent of l≥l0 and t∈[0,T]. Note that hl(·,ul′(t))ul′(t)≥0.From (62), (68), Strauss’ approximations, Lemma 9 and from a diagonal process, we get
(69)hl·,ul′⟶h·,u′inLloc⁡10,∞;L1Γ1.Convergences (60) and (69) imply ∂ul/∂ν→∂u/∂ν in D′(Γ1×]0,∞[) and hl(·,ul′)→h(·,u′) in D′(Γ1×]0,∞[). Now we take the limit in (59). Moreover the last two convergences and the regularity of h(·,u′) imply ∂u/∂ν+h(·,u′)=0 in Lloc⁡1(0,∞;L1(Γ1)), which shows that u satisfies
(70)Ku′′-Δu+∑i=1nbi∂u′∂xi+uσu=00000000000inLloc⁡∞0,∞;L2Ω.
Hence, the result is done as in (58).The verification of the initial conditions follows by convergence (49)l.Remark 10.
If|h(x,s)|≤h1|s|a.e.x∈Γ, then the sequence hl, which converges to h, satisfies |hl(x,s)|≤3/2h1|s| (see Lourêdo and Miranda [10]). In these conditions, the solution is unique and ∂u/∂ν+h(·,u′)=0inLloc⁡2(0,∞;L2(Γ1)). Consequently, u∈L∞(0,∞,V∩H3/2(Ω)).
The proof of Corollary2 follows from Remark 10 and from regularity of elliptic problems (see Lions and Magenes [22]).


## 4. Asymptotic Behavior

In this section, by applying Nakao’s method (see [23]), we will prove the uniform stabilization of the energy associated with the solution of the Problem P.Proof of Theorem6.
First, we prove the inequality (15) for the approximate energy El(t) given by
(71)Elt=12K1/2tul′t2+ult22σ+2∫Ωultσ+2dxiiiiiiiiiiiiiiii+2σ+2∫Ωultσ+2dxfort≥1,
and Theorem 6 will follow by taking the liminf⁡⁡ in l.
Taking the scalar product ofL2(Ω) in both sides of
(72)Kul′′-Δul+∑j=1nbj∂ul′∂xj+ulσul=0000000000000inLloc⁡∞0,∞;L2Ω
with ul′, we find
(73)ddtElt=12∂Kt∂tul′t,ul′t-hl·,ul′t,ul′tΓ1+12∫Ωdiv⁡btul′t2dx-12∫Γ1bt·νult2dx.
Using (73) and the hypotheses (A2), (H3), (H8), and (H2), we obtain
(74)ddtElt≤-b04ul′t2+b∞2-h0ul′tΓ12≤0.
Note that El(t) is decreasing. From (73), the hypotheses (H1), (H2), and Remark 10, we have that
(75)ddtElt≤b02ul′t2+d1ul′tΓ12+δ02ul′t2+b∞2ul′tΓ12.
Thus,
(76)ddtElt≤N1ul′t2+N2ul′tΓ12,
where N1=b0/2+δ0/2>0 and N2=d1+b∞/2>0. Integrating (76) from t to t+1, we obtain
(77)Elt+1-Elt≤N1∫tt+1ul′s2ds+N2∫tt+1ul′sΓ12ds.
Furthermore, taking the scalar product in both sides of (72) with ul, we find
(78)Ktul′′t,ult+ult2+b·∇ul′t,ult+hl·,ul′t,ultΓ1+ultσult,ult=0.
As
(79)Ktul′′t,ult=ddtKtul′t,ult-K′tul′t,ult-Ktul′t,ul′t,
then
(80)ult2+ultLσ+2Ωσ+2=-b·∇ul′t,ult-hl·,ul′t,ultΓ1iiiiiiii+K′tul′t,ult+Ktul′t,ul′tiiiiiiii-ddtKtul′t,ult.
Now note that
(81)b·∇ul′t,ult=∫Ω∑j=1nbjt∂ul′∂xjtultdx
and by Gauss’ formula, we have
(82)∂∂xjbjtultul′t=bjtult∂ul′∂xjt+∂bjt∂xjultul′t+bjt∂ul∂xjtul′t.
Therefore,
(83)∑j=1n∫Ωbjtult∂ul′t∂xjdx=∫Γbt·νultul′tdΓ-∫Ωdiv⁡btultul′tdx-∫Ωbt·∇ultul′tdx.
Hence,
(84)b·∇ul′t,ult=∫Γ1btνultul′tdΓ-∫Ωdiv⁡btultul′tdx-∫Ωbt·∇ultul′tdx.
Substituting (84) in (80), we obtain
(85)ult2+ultLσ+2Ωσ+2=-∫Γ1bt·νultul′tdΓ+∫Ωdiv⁡btultul′tdx+∫Ωbt·∇ultul′tdx-hl·,ul′t,ultL2Γ1+K′tul′t,ult+Ktul′t,ul′t-ddtKtul′t,ult.
Integrating (85) from t1 to t2 yields
(86)∫t1t2ult2+ultLσ+2Ωσ+2dt≤b∞∫t1t2ultΓ1ul′tΓ1dt+δ0∫t1t2ultul′tdt+b∞∫t1t2∇ultul′tdt+∫t1t2hl·,ul′tΓ1ultΓ1dt+∫t1t2b02ul′tultdt+K∞∫t1t2ul′t2dt-Kt2ul′t2,ult2-Kt1ul′t1,ult1.
Using the embedding of V in L2(Ω) and V in L2(Γ1) (see (12)), it follows that(i)
b∞∫t1t2|ul(t)|Γ1|ul′(t)|Γ1dt≤32b∞2k12∫t1t2|ul′(t)|Γ12dt+1/32∫t1t2ul(t)2dt;(ii)
δ0∫t1t2|ul(t)||ul′(t)|dt≤32(1/λ1)∫t1t2|ul′(t)|2dt+1/32∫t1t2ul(t)2dt;(iii)
b∞∫t1t2|∇ul(t)||ul′(t)|dt≤b∞2/2∫t1t2ul(t)2dt+1/2∫t1t2|ul′(t)|2dt.(iv)
∫t1t2|hl(·,ul′(t))|Γ1|ul(t)|Γ1dt≤3/2h1k1∫t1t2|ul′(t)|Γ1|ul(t)|Γ1dt≤3/2h1k12(λ11/2)216∫t1t2|ul′(t)|Γ12dt+ 1/32∫t1t2ul(t)2dt;(v)
∫t1t2b0/2|ul′(t)||ul(t)|dt≤32b0/22(1/λ1)∫t1t2|ul′(t)|2+1/32∫t1t2ul(t)2dt;(vi)
|(K(t2)ul′(t2),ul(t2))-(K(t1)ul′(t1),ul(t1)|≤K∞[|ul′(t2)||ul(t2)|+|ul′(t1)||ul(t1)|].
Note that(87)ult2≤1λ1ult≤1λ1El1/2t2,
since El(t) is decreasing and t≤t1, t2≤t+1. Analogously, we obtain
(88)ult1≤1λ1El1/2t1.
Using (87) and (88) in (vi) we find
(89)Kt2ul′t2,ult2-Kt1ul′t1,ult1≤K∞1λ11/2sup⁡t1≤s≤t2⁡El1/2sul′t2+ul′t1.
Substituting (i)–(vi) and (89) in (86), we obtain
(90)∫t1t2ult2+ultLσ+2Ωσ+2dt≤32+b∞1λ1∫t1t2ult2dtiiiiiiii+321λ12+8b021λ12∫t1t2ul′t2dtiiiiiiii+32b∞2k12+18h1k12λ1+12∫t1t2ul′tΓ12dtiiiiiiii+K∞1λ1sup⁡t1≤s≤t2⁡El1/2sul′t2+ul′t1.
It follows from (74) that
(91)Elt+1-Elt≤-N3∫tt+1ul′s2ds-N4∫tt+1ul′sΓ1ds,
where N3=b0>0 and N4=h0-b∞/2>0. Hence
(92)Dl2t=Elt-Elt+1≥N3∫tt+1ul′s2ds+N4∫tt+1ul′tΓ12ds≥d1∫tt+1ul′s2ds+∫tt+1ul′tΓ12ds,
where d1=min⁡⁡N3,N4>0.
From (92), we obtain
(93)Dl2td1≥∫tt+1ul′s2ds,Dl2td1≥∫tt+1ul′sΓ12ds,
and from (93), we get
(94)∫tt+1/4ul′sΓ12ds≤∫tt+1ul′sΓ12ds≤Dl2td1,∫t+3/4t+1ul′sΓ12ds≤∫tt+1ul′sΓ12ds≤Dl2td1.
By the Mean Value Theorem, there are t1∈(t,t+1/4) and t2∈(t+3/4,t+1), such that
(95)14ul′t12=∫tt+1/4ul′s2ds≤Dl2td1,14ul′t22=∫t+3/4t+1ul′s2ds≤Dl2td1.
From (95), we can write
(96)ul′t1+ul′t2≤2Dltd1.
Now from (93) we have
(97)N5∫t1t2ul′sΓ12ds≤N5∫tt+1ul′sΓ12ds≤N5d1Dl2t,
where N5=32b∞+18(h1k1)2.
Analogously we obtain(98)N6∫t1t2ul′s2ds≤N6∫tt+1ul′s2ds≤N6d1Dl2t,
where N6=32(1/λ1)+8b02(1/λ1).
Substituting (96), (97), and (98) in (90) and considering
(99)N7=932+1λ1b∞,
we get
(100)∫t1t2ult2+ultLσ+2Ωσ+2dt≤N7∫t1t2ult2dt+N5∫t1t2ul′t2dtiiiiiiii+N6∫t1t2ul′tΓ12dtiiiiiiii+K∞1λ1sup⁡t1≤s≤t2⁡El1/2ul′t2+ul′t1≤N7∫t1t2ult2dt+N5d1Dl2t+N6d1Dl2tiiiiiiii+2K∞1/λ1d1sup⁡t1≤s≤t2⁡El1/2Dlt≤N7∫t1t2ult2dt+N5d1+N6d1Dl2tiiiiiiii+2K∞1/λ1d1DltEl1/2t.
As Dl2(t)=El(t)-El(t+1)≤El(t) then Dl(t)≤El1/2(t), and substituting this inequality in (100), we obtain
(101)∫t1t2ult2+ultLσ+2Ωσ+2dtM≤N7∫t1t2ult2dt+N5+N6d1Dl2tMM+2K∞1/λ1d1DltElt.
Replacing t=t1 and t+1=t2 in (93), we have
(102)∫t1t2Ktul′t2≤K∞∫t1t2ul′t2dt≤K∞d1Dl2t.
Adding (101) and (102), we obtain
(103)∫t1t2Ktul′t2+ult2+ultLσ+2Ωσ+2dt≤N7∫t1t2ult2dt+K∞+N5+N6d1Dl2tM+21/λ1K∞d1Elt,
and this implies
(104)∫t1t2Ktul′t2+1-N7ult2+ultLσ+2Ωσ+2dt≤N8Dl2t+N9Elt,
where
(105)N8=K∞+N5+N6d1,N9=21/λ1K∞d1.
The hypothesis (H11) yields b∞<21λ1/32 and as 1-N7>0 then
(106)∫t1t212Ktul′t2+12ult2+1σ+2ultLσ+2Ωσ+2dt≤N10Dl2t+N11Elt.
Since 1/σ+2<1, N10=N8/M and N11=N9/M, where M=min⁡⁡{1,1-N7}>0.
From (106), we obtain
(107)∫t1t2Eltdt≤N10Dl2t+N11Elt.
Note that by hypothesis (H11), N11<1/4, thus from (107), we find
(108)∫t1t2Eltdt≤N10Dl2t+14Elt.
Again, by the Mean Value Theorem there exists t*∈(t1,t2), such that
(109)∫t1t2Elsds=Elt*t2-t1≥12Elt*.
Integrating (76) from t* to t and using (77) and (93), it follows that
(110)Elt≤Elt*+N1∫t*tul′s2ds+N2∫t*tul′sΓ12ds≤Elt*+N1d1Dl2t+N2d1Dl2t.
Substituting (109) in (110), we get
(111)Elt≤2∫t1t2Elsds+N1d1+N2d1Dl2t.
Now, substituting (108) in (111), we obtain
(112)Elt≤12Elt+2N10+N1d1+N2d1Dl2t,(113)Elt≤22N10+N1d1+N2d1Dl2t.
As El(t) is decreasing, the inequality (113) provides that
(114)sup⁡t≤s≤t+1⁡Els≤N12Elt-Elt+1,
where
(115)N12=22N10+N1d1+N2d1>0.
Thus, it follows using (114) and from Nakao’s Lemma, (see [23]), that
(116)Elt≤ce-wt,t≥1.
Taking the liminf⁡⁡ as l→∞ in (116), we obtain
(117)Et≤ce-wt,∀t≥1,
where c is positive constant.



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*Source: 281032-2015-03-03.xml*