Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback

1

Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback

Abbes Benaissa

, Aissa Benguessoum

and Salim A. Messaoudi

1Laboratory of Analysis and Control of PDE’s, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, Algeria

2Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia

Received 2 September 2013, appeared 21 March 2014 Communicated by Paul Eloe

Abstract. In this paper, we consider the wave equation with a weak internal constant delay term:

u⁰⁰(x,t)−_∆xu(x,t) +µ₁(t)u⁰(x,t) +µ₂(t)u⁰(x,t−τ) =0

in a bounded domain. Under appropriate conditions on µ1 and µ2, we prove global existence of solutions by the Faedo–Galerkin method and establish a decay rate estimate for the energy using the multiplier method.

Keywords:nonlinear wave equation, delay term, decay rate, multiplier method.

2010 Mathematics Subject Classification:35B40, 35L70.

1 Introduction

In this paper we investigate the decay properties of solutions for the initial boundary value problem for the linear wave equation of the form



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

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



u⁰⁰(x,t)−_∆xu(x,t) +µ₁(t)u⁰(x,t) +µ₂(t)u⁰(x,t−τ) =0 inΩ×]0,+_∞[,

u(x,t) =0 onΓ×]0,+_∞[,

u(x, 0) =u₀(x), u_t(x, 0) =u₁(x) onΩ, u_t(x,t−τ) = f₀(x,t−τ) onΩ×]0,τ[,

(P)

where Ωis a bounded domain in IRⁿ, n ∈ IN^∗, with a smooth boundary∂Ω = _Γ, τ > 0 is a time delay and the initial data(u₀,u₁,f₀)belong to a suitable function space.

In absence of delay (µ2 = 0), the energy of problem (P) is exponentially decaying to zero provided that µ₁is constant, see, for instance, [3,4, 7,8] and [12]. On the contrary, if µ₁ = 0 and µ₂ > 0 (a constant weight), that is, there exists only the internal delay, the system (P)

1Corresponding author. Email: benaissa−abbes@yahoo.com

becomes unstable (see, for instance, [5]). In recent years, the PDEs with time delay effects have become an active area of research since they arise in many practical problems (see, for example, [1, 19]). In [5], it was shown that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one and, therefore, delay becomes a source of instability. To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary (see [13,15,20]). For instance, the authors of [13] studied the wave equation with a linear internal damping term with constant delay (τ = const in the problem (P) and determined suitable relations betweenµ₁andµ₂, for which the stability or alternatively instability takes place. More precisely, they showed that the energy is exponentially stable if µ₂ < µ₁ and they also found a sequence of delays for which the corresponding solution of (P) will be unstable ifµ2 ≥ µ₁. The main approach used in [13] is an observability inequality obtained with a Carleman estimate. The same results were obtained if both the damping and the delay are acting on the boundary. We also recall the result by Xu, Yung and Li [20], where the authors proved a result similar to the one in [13] for the one-space dimension by adopting the spectral analysis approach.

In [17], Nicaise, Pignotti and Valein extended the above result to higher space dimensions and established an exponential decay.

Our purpose in this paper is to give an energy decay estimate of the solution of problem (P) in the presence of a delay term with a weight depending on time. We use the Galerkin approximation scheme and the multiplier technique to prove our results.

2 Preliminaries and main results

First assume the following hypotheses:

(H1)µ₁: IR+ →]0,+_∞[is a non-increasing function of classC¹(IR+)satisfying

µ⁰₁(t) µ₁(t)

≤ M, (2.1)

(H2)µ₂: IR+ → IR is a function of classC¹(IR+), which is not necessarily positive or mono- tone, such that

µ₂(t)≤ βµ₁(t), (2.2)

µ⁰₂(t)≤ Mµ^˜ ₁(t), (2.3) for some 0<β<1 and ˜M >0.

We now state a Lemma needed later.

Lemma 2.1(Martinez [10]). Let E: IR+→ IR+be a non increasing function andφ: IR+ → IR+an increasing C¹function such that

φ(0) =0 and φ(t)→+_∞ as t→+_∞.

Assume that there existσ>−1andω>0such that Z _+∞

S E^1+σ(t)φ⁰(t)dt≤ ¹

ωE^σ(0)E(S), 0≤S<+_∞. (2.4)

Then

E(t) =0 ∀t≥ ^E(0)^σ

ω|σ| ^{, if} −1<σ <0, (2.5) E(t)≤ E(0)

1+σ 1+ωσφ(t)

¹_σ

∀t≥0, if σ >0, (2.6)

E(t)≤ E(₀)e^1−ωφ(t)∀t≥_0, if σ=_{0. (2.7)}

We introduce, as in [13], the new variable

z(x,ρ,t) =ut(x,t−τρ), x∈_Ω, ρ∈ (0, 1), t>0. (2.8) Then, we have

τzt(x,ρ,t) +zρ(x,ρ,t) =0, inΩ×(0, 1)×(0,+_∞). (2.9) Therefore, problem (P) takes the form:























u⁰⁰(x,t)−_∆xu(x,t) +µ₁(t)u⁰(x,t) +µ₂(t)z(x, 1,t) =0, x∈_Ω,t >0,

τz_t(x,ρ,t) +z_ρ(x,ρ,t) =0, x∈_Ω,ρ∈ (0, 1),t>0,

u(x,t) =0, x∈_∂Ω,t>0,

z(x, 0,t) =u⁰(x,t), x∈_Ω,t >0, u(x, 0) =u₀(x), u_t(x, 0) =u₁(x), x∈_Ω,

z(x,ρ, 0) = f0(x,−τρ), x∈_Ω,ρ∈ (0, 1).

(2.10)

Letξbe a positive constant such that

τβ<ξ <τ(2−β). (2.11)

We define the energy of the solution by:

E(t) = ¹

2ku⁰(t)k²₂+¹

2k∇_xu(t)k²₂+^ξ(t) 2

Z

Ω

Z ₁

0 z²(x,ρ,t)dρdx, (2.12) where

ξ(t) =ξµ₁(t). We have the following theorem.

Theorem 2.2. Let(u₀,u₁,f₀)∈(H²(_Ω)∩H₀¹(_Ω))×H₀¹(_Ω)×H₀¹(_Ω;H¹(0, 1))satisfy the compat- ibility condition

f₀(·_{, 0}) =u₁.

Assume that(H1)and(H2)hold. Then problem(P)admits a unique global weak solution

u∈ L^∞_loc((−τ,∞); H²(_Ω)∩H₀¹(_Ω)), u⁰ ∈ L^∞_loc((−τ,∞); H₀¹(_Ω)), u⁰⁰ ∈ L^∞_loc((−τ,∞); L²(_Ω)). Moreover, for some positive constants c, ω, we obtain the following decay property:

E(t)≤ cE(0)e^−ω

R_t

0µ₁(s)ds, ∀t ≥0. (2.13)

Lemma 2.3. Let(u,z)be a solution to the problem(2.10). Then, the energy functional defined by(2.12) satisfies

E⁰(t)≤ −

µ₁(t)− ^ξ(t)

2τ − |µ₂(t)|

2

kut(x,t)k²₂− ξ(t)

2τ − |µ₂(t)|

2

kz(x, 1,t)k²₂

≤0. (2.14)

Proof. Multiplying the first equation in (2.10) byu_t(x,t), integrating overΩand using Green’s identity, we obtain:

1 2

∂

∂tku_t(x,t)k²₂+ ¹ 2

∂

∂tk∇u(x,t)k²₂ +µ₁(t)ku_t(x,t)k²₂+µ₂(t)

Z

Ωu_t(x,t−τ)u_t(x,t)dx=0.

(2.15)

We multiply the second equation in (2.10) byξ(t)zand integrate overΩ×(0, 1)to obtain:

ξ(t)τ Z

Ω

Z ₁

0 z_t(x,ρ,t)z(x,ρ,t)dρdx+ξ(t)

Z

Ω

Z ₁

0 z_ρ(x,ρ,t)z(x,ρ,t))dρdx=0. (2.16) This yields

ξ(t)τ 2

d dt

Z

Ω

Z ₁

0 z²(x,ρ,t)dρdx+^ξ(t) 2

Z

Ω

Z ₁

0

∂

∂ρz²(x,ρ,t))dρdx=0, which gives

τ 2

d dt

ξ(t)

Z

Ω

Z ₁

0 z²(x,ρ,t)dρdx

−ξ⁰(t)

Z ₁

0

Z

Ωz²(x,ρ,t)dρdx

+ ^ξ(t) 2

Z

Ωz²(x, 1,t)dx− ^ξ(t) 2

Z

Ωu²_t(x,t)dx=0.

Consequently, τ

2 d dt

ξ(_t)

Z

Ω

Z ₁

0 z²(_x,ρ,t)_dρdx

= ^τ 2ξ⁰(t)

Z

Ω

Z ₁

0 z²(x,ρ,t)dρdx− ^ξ(t) 2

Z

Ωz²(x, 1,t)dx+ ^ξ(t) 2

Z

Ωu²_t(x,t)dx.

(2.17)

Combination of (2.15) and (2.17) leads to 1

2

∂

∂t

ku_t(x,t)k²₂+k∇u(x,t)k²₂+ξ(t)

Z

Ω

Z ₁

0 z²(x,ρ,t)dρdx

=−µ₁(t)ku_t(x,t)k²₂−µ₂(t)

Z

Ωz(x, 1,t)u_t(x,t)dx +¹

2ξ⁰(t)

Z

Ω

Z ₁

0 z²(x,ρ,t)dρdx−^ξ(t) 2τ

Z

Ωz²(x, 1,t)dx+ ^ξ(t)

2τ ku_t(x,t)k²₂.

Recalling the definition ofE(t)in (2.12), we arrive at E⁰(t) = −

µ₁(t) − ^ξ(t) 2τ

kut(x,t)k²₂−µ₂(t)

Z

Ωz(x, 1,t)ut(x,t)dx +¹

2ξ⁰(t)

Z

Ω

Z ₁

0 z²(x,ρ,t)dρdx− ^ξ(t) 2τ

Z

Ωz²(x, 1,t)dx.

≤ −

µ₁(t)−^ξ(t) 2τ

ku_t(x,t)k²₂−µ₂(t)

Z

Ωz(x, 1,t)u_t(x,t)dx

−^ξ(t) 2τ

Z

Ωz²(x, 1,t)dx. (2.18)

Due to Young’s inequality, we have Z

Ωz(x, 1,t)u_t(x,t)dx≤ ¹

2ku_t(x,t)k²₂+ ¹

2kz(x, 1,t)k²₂. (2.19) Inserting (2.19) into (2.18), we obtain

E⁰(t)≤ −

µ₁(t) − ^ξ(t)

2τ − |µ₂(t)|

2

kut(x,t)k²₂− ξ(t)

2τ − |µ₂(t)|

2

kz(x, 1,t)k²₂

≤ −µ₁(t) 1− ^ξ 2τ− ^β

2

!

kut(x,t)k²₂−µ₁(t) ^ξ 2τ− ^β

2

!

kz(x, 1,t)k²₂ ≤0. (2.20) This completes the proof of the lemma.

3 Global existence

Throughout this section we assumeu0 ∈ H²(_Ω)∩H₀¹(_Ω)andu₁∈ H₀¹(_Ω),f0∈ L²(_Ω;H¹(0, 1)). We employ the Galerkin method to construct a global solution. Let T > 0 be fixed and denote byV_k the space generated by{w₁,w₂, . . . ,w_k}where the set{w_k,k ∈ IN}is a basis of H²(_Ω)∩H₀¹(_Ω).

Now, we define for 1≤ j≤ kthe sequenceφ_j(x,ρ)as follows:

φ_j(x, 0) =w_j.

Then, we may extend φ_j(x, 0) by φ_j(x,ρ) over L²(_Ω×(0, 1))such that (φ_j)_j form a basis of L²(_Ω;H¹(0, 1))and denote byZ_kthe space generated by{φ₁,φ₂, . . . ,φ_k}.

We construct approximate solutions(u_k,z_k),k =1, 2, 3, . . . , in the form u_k(t) =

∑

g_jk(t)w_j, z_k(t) =

∑

h_jk(t)φ_j,

whereg_jkandh_jk(j=1, 2, . . . ,k)are determined by the following system of ordinary differen- tial equations:

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

(u⁰⁰_k(t),w_j) + (∇_xu_k(t),∇_xw_j) +µ₁(t)(u⁰_k,w_j) +µ₂(t)(z_k(., 1),w_j) =0, 1≤ j≤ k,

z_k(x, 0,t) =u⁰_k(x,t),

(3.1)

associated with the initial conditions u_k(₀) =u_0k =

∑

(u₀,w_j)w_j →u₀in H²(_Ω)∩H₀¹(_Ω)_ask →+_{∞, (3.2)}

u⁰_k(0) =u_1k =

∑

(u₁,w_j)w_j →u₁inH¹₀(_Ω)ask→+_∞, (3.3) and

((τz_kt+z_kρ,φ_j) =0,

1≤j≤k, (3.4)

z_k(ρ, 0) =z_0k =

∑

(f₀,φ_j)φ_j → f₀inL²(_Ω;H¹(0, 1))ask→+_∞. (3.5) By virtue of the theory of ordinary differential equations, the system (3.1)–(3.5) has a unique local solution which is extended to a maximal interval [0,T_k[(with 0 < T_k ≤ +_{∞) by Zorn} lemma. Note thatu_k(t)is of classC².

In the next step, we obtain a priori estimates for the solution of the system (3.1)–(3.5), so that it can be extended beyond[0,T_k[to obtain a solution defined for allt >0. Then, we utilize a standard compactness argument for the limiting procedure.

The first estimate.Since the sequencesu_0k, u_1kandz_0k converge, then from (2.14) we can find a positive constantCindependent ofksuch that

E_k(t) +

Z _t

0 a₁(s)ku⁰_k(s)k²₂ds+

Z _t

0 a₂(s)kz_k(x, 1,s)k²₂ds≤ E_k(0)≤C, (3.6) where

E_k(t) = ¹

2ku⁰_k(t)k²₂+¹

2k∇_xu_k(t)k²₂+ ^ξ(t) 2

Z

Ω

Z ₁

0 z²_k(x,ρ,t)dρdx, a₁(t) =µ₁(t) 1− ^ξ

2τ− ^β 2

!

anda₂(t) =µ₁(t) ^ξ 2τ− ^β

2

! . These estimates imply that the solution(u_k,z_k)exists globally in[0,+_∞[.

Estimate (3.6) yields

(u_k)is bounded inL^∞_loc(0,∞;H₀¹(_Ω)), (3.7) (u⁰_k)is bounded inL^∞_loc(0,∞;L²(_Ω)), (3.8) µ₁(t)(u⁰²_k(t))is bounded inL¹(_Ω×(0,T)), (3.9) µ₁(t)(z²_k(x,ρ,t))is bounded inL_loc^∞ (0,∞;L¹(_Ω×(0, 1))), (3.10) µ₁(t)(z²_k(x, 1,t))is bounded inL¹(_Ω×(0,T)). (3.11) The second estimate.We first estimateu⁰⁰_k(₀). Replacingw_j byu⁰⁰_k(t)in (3.1) and takingt = _0, we obtain:

ku⁰⁰_k(₀)k₂≤ k_∆xu_0kk₂+µ₁(₀)ku_1kk₂+|µ₂(₀)|kz_0kk₂

≤ k_∆xu₀k₂+µ₁(0)ku₁k₂+|µ₂(0)|kz₀k₂

≤C.

Differentiating (3.1) with respect tot, we get

(u⁰⁰⁰_k (t) +_∆xu⁰_k(t) +µ₁(t)u⁰⁰_k(t) +µ₁⁰(t)u⁰_k(t) +µ₂(t)z⁰_k(1,t) +µ₂⁰(t)z_k(1,t),w_j) =0.

Multiplying byg⁰⁰_jk(t), summing overjfrom 1 tok, it follows that 1

2 d

dt ku⁰⁰_k(t)k²₂+k∇_xu⁰_k(t)k²₂+µ₁(t)

Z

Ωu⁰⁰²_k(t)dx+µ₁⁰(t)

Z

Ωu⁰⁰_k(t)u⁰_k(t)dx +µ2(t)

Z

Ωu⁰⁰_k(t)z⁰_k(x, 1,t)dx+µ⁰₂(t)

Z

Ωu⁰⁰_k(t)z_k(x, 1,t)dx=0.

(3.12)

Differentiating (3.4) with respect tot, we get

τz⁰⁰_k(t) + ^∂

∂ρz⁰_k,φ_j

=0.

Multiplying byh⁰_jk(t), summing overjfrom 1 tok, it follows that τ

2 d

dtkz⁰_k(t)k²₂+¹ 2

d

dρkz⁰_k(t)k²₂=0. (3.13) Taking the sum of (3.12) and (3.13), we obtain that

1 2

d dt

ku⁰⁰_k(t)k²₂+k∇_xu⁰_k(t)k²₂+τ Z ₁

0

kz⁰_k(x,ρ,t)k²_L2(_Ω)dρ

+µ₁(t)

Z

Ωu⁰⁰²_k(t)dx+ ¹ 2

Z

Ω|z⁰_k(x, 1,t)|²dx

=−µ₂(t)

Z

Ωu⁰⁰_k(t)z⁰_k(x, 1,t)dx−µ⁰₁(t)

Z

Ωu⁰⁰_k(t)u⁰_k(t)dx

−µ⁰₂(t)

Z

Ωu⁰⁰_k(t)z_k(x, 1,t)dx+¹

2ku⁰⁰_k(t)k²₂ Using(_H1),(_H2), Cauchy–Schwarz and Young’s inequalities, we obtain

1 2

d dt

ku⁰⁰_k(t)k²₂+k∇_xu⁰_k(t)k²₂+

Z ₁

0 τkz⁰_k(x,ρ,t)k²_L2(Ω)dρ

+µ₁(t)

Z

Ωu⁰⁰²_k(t)dx+ ¹ 2

Z

Ω|z⁰_k(x, 1,t)|²dx

≤ |µ₂(t)|ku⁰⁰_k(t)k₂kz⁰_k(x, 1,t)k₂+|µ⁰₁(t)|ku⁰⁰_k(t)k₂|ku⁰_k(t)k₂ +|µ⁰₂(t)|ku⁰⁰_k(t)k₂kz_k(x, 1,t)k₂+ ¹

2ku⁰⁰_k(t)k²₂

≤ |µ₂(t)|²

2 ku⁰⁰_k(t)k²₂+¹

2kz⁰_k(x, 1,t)k²₂+ |µ₁⁰(t)|

4 ku⁰⁰_k(t)k²₂+|µ⁰₁(t)|ku⁰_k(t)k²₂ +|µ₂⁰(t)|

4 ku⁰⁰_k(t)k²₂+|µ⁰₂(t)|kz_k(x, 1,t)k²₂+¹

2ku⁰⁰_k(t)k²₂

≤c⁰ku⁰⁰_k(t)k²₂+|µ⁰₁(t)|ku⁰_kk²₂+|µ⁰₂(t)|kz_k(x, 1,t)k²₂+¹

2kz⁰_k(x, 1,t)k²₂.

≤c⁰ku⁰⁰_k(t)k²₂+Mµ₁(t)ku⁰_kk²₂+Mµ^˜ ₁(t)kz_k(x, 1,t)k²₂+¹

2kz⁰_k(x, 1,t)k²₂.

Integrating the last inequality over(0,t)and using (3.6), we get

ku⁰⁰_k(t)k²₂+k∇_xu⁰_k(t)k²₂+τ Z ₁

0

kz⁰_k(x,ρ,t)k²_L2(_Ω)dρ

≤

ku⁰⁰_k(0)k²₂+k∇_xu⁰_k(0)k²₂+τ Z ₁

0

kz⁰_k(x,ρ, 0)k²_L2(Ω)dρ

+2M Z _t

0 µ₁(s)ku⁰_k(s)k²₂ds +_{2 ˜}M

Z _t

0

µ₁(s)kz_k(x, 1,s)k²₂ds+2c⁰ Z _t

0

ku⁰⁰_k(s)k²₂ds

≤C+C⁰ Z _t

0

ku⁰⁰_k(s)k²₂+k∇_xu⁰_k(s)k²₂+τ Z ₁

0

kz⁰_k(x,ρ,s)k²_L2(_Ω)dρ

ds.

Using Gronwall’s lemma, we deduce that ku⁰⁰_k(t)k²₂+k∇_xu⁰_k(t)k²₂+τ

Z ₁

0

kz⁰_k(x,ρ,t)k²_L2(Ω)dρ≤Ce^C0T for allt∈IR⁺, therefore, we conclude that

(u⁰⁰_k)is bounded inL^∞_loc(0,∞;L²(_Ω)), (3.14) (u⁰_k)is bounded inL^∞_loc(0,∞;H₀¹(_Ω)), (3.15) (τz⁰_k)is bounded inL^∞_loc(0,∞;L²(_Ω×(0, 1))). (3.16) Applying Dunford–Pettis’ theorem, we deduce from (3.7), (3.8), (3.9), (3.10), (3.11), (3.14), (3.15) and (3.16), replacing the sequenceu_k with a subsequence if necessary, that

u_k →u weak-star in L_loc^∞ (0,∞;H²(_Ω)∩H₀¹(_Ω)), (3.17) u⁰_k →u⁰ weak-star in L^∞_loc(_0,∞;H¹₀(_Ω))_,

u⁰⁰_k →u⁰⁰ weak-star in L^∞_loc(0,∞;L²(_Ω)), (3.18) u⁰_k →χ weak in L²(_Ω×(0,T);µ₁(t)),

z_k →z weak-star in L^∞_loc(0,∞;H₀¹(_Ω;L²(0, 1)),

z⁰_k →z⁰ weak-star in L^∞_loc(0,∞;L²(_Ω×(0, 1))), (3.19) z_k(x, 1,t)→ψ weak in L²(_Ω×(0,T),µ₁(t))

for suitable functions

u∈ L^∞(0,T;H²(_Ω)∩H₀¹(_Ω)), z∈ L^∞(0,T;L²(_Ω×(0, 1))), χ∈ L²(_Ω×(0,T);µ₁(t)), ψ∈L²(_Ω×(0,T);µ₁(t)), for allT≥0. We have to show thatuis a solution of (P).

From (3.15) we have that (u⁰_k) is bounded in L^∞(0,T;H₀¹(_Ω)). Then (u⁰_k) is bounded in L²(0,T;H¹₀(_Ω)). Since(u⁰⁰_k)is bounded inL^∞(0,T;L²(_Ω)), then it is bounded inL²(0,T;L²(_Ω)), too. Consequently,(u⁰_k)is bounded inH¹(Q)_.

Since the embeddingH¹(Q),→ L²(Q)is compact, using the Aubin–Lions theorem [9], we can extract a subsequence(u_ς)of(u_k)such that

u⁰_ς→u⁰ strongly in L²(Q). (3.20) Therefore

u⁰_ς→u⁰strongly and a.e. inQ. (3.21)

Similarly we obtain

zς→zstrongly in L²(_Ω×(0, 1)×(0,T)) (3.22) and

zς→zstrongly and a.e. inΩ×(0, 1)×(0,T). (3.23) It follows at once from (3.17), (3.18), (3.19), (3.20) and (3.22) that for each fixedv∈L²(0,T;L²(_Ω)) andw∈ L²(0,T;L²(_Ω)×(0, 1))

Z _T

0

Z

Ω u⁰⁰_ς−_∆xu_ς+µ₁(t)u⁰_ς+µ₂(t)z_ς v dx dt

→

Z _T

0

Z

Ω u⁰⁰−_∆xu+µ₁(t)u⁰+µ₂(t)z

v dx dt,

(3.24)

Z _T

0

Z ₁

0

Z

Ω

τz⁰_ς+ _∂ρ^∂zς

w dx dρdt→

Z _T

0

Z ₁

0

Z

Ω

τz⁰+ _∂ρ^∂z

w dx dρdt (3.25) asς→+_∞. Thus the problem (P) admits a global weak solutionu.

Uniqueness. Let (u₁,z₁) and (u2,z2) be two solutions of problem (2.10). Then (w, ˜w) = (u₁,z₁)−(u₂,z₂)satisfies























w⁰⁰(x,t)−_∆xw(x,t) +µ₁(t)w⁰(x,t) +µ₂(t)w˜(x, 1,t) =0, inΩ×]0,+_∞[, τw˜⁰(x,ρ,t) +w˜ρ(x,ρ,t) =0, inΩ×]0, 1[×]0,+_∞[

w(x,t) =0, on∂Ω×]0,+_∞[

˜

w(x, 0,t) =w⁰(x,t), onΩ×]0,+_∞[

w(x, 0) =0, w⁰(x, 0) =0, inΩ

˜

w(x,ρ, 0) =_{0, inΩ}×]_{0, 1}[

(3.26)

Multiplying the first equation in (3.26) byw⁰, integrating overΩand using integration by parts, we get

1 2

d

dt(kw⁰k²₂+k∇_xwk²₂) +µ₁(_t)kw⁰k²₂+µ₂(_t)(w˜(_{x, 1,t})_,w⁰) =_{0. (3.27)} Multiplying the second equation in (3.26) by ˜w, integrating overΩ×(_{0, 1})and using integra- tion by parts, we get

τ 2

d

dtkw˜k²₂+¹ 2

d

dρkw˜k²₂=0. (3.28)

Then

τ 2

d dt

Z ₁

0

kw˜k²₂dρ+ ¹

2(kw˜(x, 1,t)k²₂− kw⁰k²₂) =_{0. (3.29)} From (3.27), (3.29), using the Cauchy–Schwarz inequality we get

1 2

d dt

kw⁰k²₂+k∇_xwk²₂+τ Z ₁

0

kw˜k²₂dρ

+µ₁(t)kw⁰k²₂+¹

2kw˜(x, 1,t)k²₂

=−µ₂(t)(w˜(x, 1,t),w⁰) + ¹ 2kw⁰k²₂

≤ |µ₂(t)|kw˜(x, 1,t)k₂kw⁰k₂+ ¹ 2kw⁰k²₂. Using Young’s inequality, we obtain

1 2

d dt

kw⁰k²₂+k∇_xwk²₂+τ Z ₁

0

kw˜k²₂dρ

≤ckw⁰k²₂,

wherecis a positive constant. Then integrating over(0,t), using Gronwall’s lemma, we con- clude that

kw⁰k²₂+k∇_xwk²₂+τ Z ₁

0

kw˜k²₂dρ=0.

Hence, uniqueness follows.

4 Asymptotic behavior

From now on, we denote byc various positive constants which may be different at different occurrences. We multiply the first equation of (2.10) byφ⁰E^qu, whereφis a bounded function satisfying all the hypotheses of Lemma2.1. We obtain

0=

Z _T

S E^qφ⁰ Z

Ωu

u⁰⁰−_∆u+µ₁(t)u⁰+µ₂(t)z(x, 1,t)

dx dt

=

E^qφ⁰ Z

Ωuu⁰dx T

S

−

Z _T

S

(qE⁰E^q−1φ⁰+E^qφ⁰⁰)

Z

Ωuu⁰dx dt

−2 Z _T

S E^qφ⁰ Z

Ωu⁰²dx dt+

Z _T

S E^qφ⁰ Z

Ω u⁰²+|∇u|² dxdt +

Z _T

S

E^qφ⁰µ₁(t)

Z

Ωuu⁰dx dt+

Z _T

S

E^qφ⁰µ₂(t)

Z

Ωuz(x, 1,t)dx dt.

Similarly, we multiply the second equation of (2.10) byE^qφ⁰ξ(t)e^−2τρz(x,ρ,t)_{and get} 0=

Z _T

S E^qφ⁰ Z

Ω

Z ₁

0 e^−2τρξ(t)z

τz_t+z_ρ

dx dρdt

= 1

2E^qφ⁰ξ(_t)τ Z

Ω

Z ₁

0 e^−2τρz²dx dρ T

S

−¹ 2

Z _T

S

Z

Ω

Z ₁

0 E^qφ⁰ξ(_t)_τe^−2τρ0_z²_{dx dρdt} +

Z _T

S E^qφ⁰ Z

Ω

Z ₁

0

ξ(t) 1

2

∂

∂ρ e^−2τρz²

+τe^−2τρz²

dx dρdt

= 1

2E^qφ⁰ξ(t)τ Z

Ω

Z ₁

0 e^−2τ(t)ρz²dx dρ T

S

− ^τ 2

Z _T

S E^qφ⁰ξ(t)⁰

Z

Ω

Z ₁

0 e^−2τρz²dx dρdt +¹

2 Z _T

S E^qφ⁰ξ(t)

Z

Ω

e^−2τz²(x, 1,t)−z²(x, 0,t)dx dt+

Z _T

S E^qφ⁰ξ(t)τ Z ₁

0

Z

Ωe^−2τρz²dx dρdt.

Taking their sum, we obtain A

Z _T

S E^q+1φ⁰dt≤ −

E^qφ⁰ Z

Ωuu⁰dx T

S

+

Z _T

S

(_qE⁰_E^q−1φ⁰+_E^qφ⁰⁰)

Z

Ωuu⁰dx dt +2

Z _T

S E^qφ⁰ Z

Ωu⁰²dx dt−

Z _T

S µ₁(t)E^qφ⁰ Z

Ωuu⁰dx dt

−

Z _T

S µ₂(t)E^qφ⁰ Z

Ωuz(x, 1,t)dx dt− 1

2E^qφ⁰ξ(t)τ Z

Ω

Z ₁

0 e^−2τρz²dx dρ T

S

+ ¹ 2

Z _T

S E^qφ⁰ξ(t)⁰τ Z

Ω

Z ₁

0

e^−2τρz²dx dρdt

− ¹ 2

Z _T

S E^qφ⁰ξ(t)

Z

Ω

e^−2τ(t)z²(x, 1,t)−z²(x, 0,t)dx dt,

(4.1)

where A = 2 min{1,e^−2τ1}. Using the Cauchy–Schwarz and Poincaré’s inequalities and the definition ofEand assuming thatφ⁰ is a bounded non-negative function on IR⁺, we get

E^q(t)φ⁰ Z

Ωuu⁰dx

≤cE(t)^q+1. By recalling (2.14), we have

Z _T

S

qE⁰E^q−1φ⁰ Z

Ωuu⁰dx

dt≤c Z _T

S E^q(t)|E⁰(t)|dt≤c Z _T

S E^q(t)(−E⁰(t))dt

≤cE^q+1(S),

Z _T

S E^qφ⁰⁰ Z

Ωuu⁰dx dt ≤c Z _T

S E^q+1(t)(−φ⁰⁰)≤cE^q+1(S)

Z _T

S

(−φ⁰⁰)dt≤cE^q+1(S),

and

Z _T

S

E^qφ⁰ Z

Ωu⁰²dxdt≤c Z _T

S

E^qφ⁰ 1 µ₁(t)

Z

Ωµ₁(t)u⁰²dxdt

≤

Z _T

S E^q φ⁰

µ₁(t) −E⁰

dt. (4.2)

Define

φ(t) =

Z _t

0 µ₁(s)ds. (4.3)

It is clear thatφis a non-decreasing function of classC¹on IR⁺,φis bounded and

φ(t)→+_∞ast →+_∞. (4.4)

So, we deduce, from (4.2), that Z _T

S E^qφ⁰ Z

Ωu⁰²dxdt≤c Z _T

S E^q(−E⁰)dt≤cE^q+1(S), (4.5) By the hypothesis(H1), Young’s and Poincaré’s inequality and (2.14), we have

Z _T

S E^qφ⁰ Z

Ωuu⁰dx dt

≤c Z _T

S E^qφ⁰kuk₂ku⁰k₂dt

≤cε⁰ Z _T

S E^qφ⁰kuk²₂dt+c(ε⁰)

Z _T

S E^qφ⁰ku⁰k²₂dt

≤ε⁰c∗ Z _T

S E^qφ⁰k∇_xuk²₂dt+c(ε⁰)

Z _T

S E^qφ⁰ku⁰k²₂dt

≤ε⁰c∗ Z _T

S E^q+1φ⁰dt+cE^q+1(S). Recalling thatξ⁰ ≤0 and the definition ofE, we have

Z _T

S

(E^qξ(_t))⁰τ Z

Ω

Z ₁

0 e^−2τρz²dx dρdt≤

Z _T

S

(E^q)⁰ξ(_t)τ Z

Ω

Z ₁

0 e^−2τρz²dx dρdt

≤c Z _T

S E^q|E⁰|dt

≤c Z _T

S E^q(−E⁰(t))dt

≤cE^q+1(S),

Z _T

S E^qξ(t)

Z

Ωe^−2τz²(x, 1,t)dx dt≤ c Z _T

S E^qξ(t)

Z

Ωz²(x, 1,t)dx dt

≤ c Z _T

S E^q −E⁰ dt

≤ cE^q+1(S),

Z _T

S E^qξ(t)

Z

Ωz²(x, 0,t)dx dt=

Z _T

S E^qξ(t)

Z

Ωu⁰²(x,t)dx dt

≤cE^q+1(S),

where we have also used the Cauchy–Schwarz inequality. Combining these estimates and choosingε⁰ sufficiently small, we conclude from (4.1) that

Z _T

S E^q+1φ⁰dt≤CE^q+1(S). This ends the proof of Theorem2.2.

Acknowledgements

This work is partially supported by KFUPM under Grant #I N121029.

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