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UDK 517.956.3

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ON A SYSTEM OF LINEAR THERMOELASTICITY WITH THE BESSEL OPERATOR

S. Mesloub and S. A. Messaoudi

Abstract. In this paper, we study an initial value problem for a one-dimensional system of thermoelasticity. Using an a priori bound and a density argument, we prove the existence and uniqueness of a generalized solution.

1. Introduction

In this paper we are concerned with a coupled system of thermoelasticity of the form

L_{1}u = u_{tt} − \frac{a}{r}(ru_{r})_{r} + brθ_{r} = f(r, t)  (1.1)
L_{2}θ = θ_{t} − \frac{κ}{r} (rθ_{r})_{r} + bru_{rt} = g(r, t)  (1.2)

in the bounded domain

Q = Ω × (0, T) = {(r, t) : 0 < r < 1, 0 < t < T } ,

where u is the displacement, θ is the difference absolute temperature, f is an external force, g is a heat supply, and a, b, κ are positive constants. This system can be regarded as a model for the radially symmetric deformation and temperature distribution in the unit disk.

For b = 0, system (1.1), (1.2) decouples and we obtain two independent equations; namely the hyperbolic wave equation and the parabolic heat equation “with the Bessel operator”. Both equations have been extensively investigated and several results concerning existence, uniqueness, and well-posedness have been established. For the parabolic case we mention here Cannon [3], Bouziani [2], Ionkin [5], Kamynin [6], P. Shi [15], Yurchuk [18], Mesloub [8], Mesloub and Bouziani [9], [10], [11], Kartynik [7]. For the hyperbolic case, we mention Mesloub and Bouziani [12], [13], Muravei and Philinovskii [14], Pulkina [16], [17] and Beilin [1].

AMS Subject Classification: 35K22, 58D25, 73B30

Keywords and phrases: Coupled system, thermoelasticiy, a priori bound.

This work has been funded and supported by the Research center project number Math/24-25/06 at King Saud University. The authors thank KSU for its support.

We supplement (1.1), (1.2) with the initial conditions

ℓ_{1}u = u(r, 0) = u_{0}(r),  0 < r < 1  (1.3)
ℓ_{2}u = u_{t}(r, 0) = u_{1}(r),  0 < r < 1  (1.4)
ℓ_{3}θ = θ(r, 0) = θ_{0}(r),  0 < r < 1  (1.5)

and the one-point boundary conditions

u(1, t) = 0,  0 < t < T  (1.6)
θ(1, t) = 0,  0 < t < T,  (1.7)

where the data functions satisfy, for compatibility,

u_{0}(1) = u_{1}(1) = θ_{0}(1) = 0.  (1.8)

Based on an a priori bound and on the density of the range of the operator generated by the problem in consideration, we prove the existence, uniqueness and the continuous dependence on the data of the strong solution of problem (1.1)– (1.7). We should note here that for the best of our knowledge, system (1.1), (1.2) has never been treated using this approach.

2. Functions spaces

Let L_{ρ}^{2}(Q) be the weighted L^{2}(Q) Hilbert space of square integrable functions on Q with scalar product

(u, θ)_{L_{ρ}^{2}(Q)} = ∫_{Q} ruθ dr dt,

and with the associated finite norm

∥u∥_{L_{ρ}^{2}(Q)}^{2} = ∫_{Q} ru^{2} dr dt,

and W_{2,ρ}^{1,1} be the weighted Hilbert space consisting of the elements u of L_{ρ}^{2}(Q) having first order generalized derivatives square summable on Q. W_{2,ρ}^{1,1} is equipped with the scalar product

(u, θ)_{W_{2,ρ}^{1,1}(Q)} = (u, θ)_{L_{ρ}^{2}(Q)} + (u_{r}, θ_{r})_{L_{ρ}^{2}(Q)} + (u_{t}, θ_{t})_{L_{ρ}^{2}(Q)},

and the associated norm is

∥u∥_{W_{2,ρ}^{1,1}(Q)}^{2} = ∥u∥_{L_{ρ}^{2}(Q)}^{2} + ∥u_{r}∥_{L_{ρ}^{2}(Q)}^{2} + ∥u_{t}∥_{L_{ρ}^{2}(Q)}^{2} .

We also use the weighted spaces on Ω, such as L_{ρ}^{2}(Ω) and W_{2,ρ}^{1}(Ω), whose definitions are analogous to the spaces on Q.

3. Reformulation of the problem

We reformulate problem (1.1)–(1.7) as the problem of solving the operator equation

AU = H,  (3.1)

where U, AU and H are respectively the pairs:

U = (u, θ),  (3.2)
AU = (L_{1}u, L_{2}θ),  (3.3)

and

H = (H_{1}, H_{2}).  (3.4)

The right-hand sides of (3.3) and (3.4) are respectively defined by

L_{1}u = {L_{1}u, ℓ_{1}u, ℓ_{2}u} , L_{2}θ = {L_{2}θ, ℓ_{3}θ} ,  (3.5)

and

H_{1} = {f, u_{0}, u_{1}} , H_{2} = {g, θ_{0}} .  (3.6)

The operator A is considered from a space B = B_{1}×B_{2} into the space H = H_{1}×H_{2}, where B is a Banach space consisting of all functions (u, θ) ∈ (L_{ρ}^{2}(Q))^{2} satisfying conditions (1.6)–(1.7) and having the finite norm

∥U∥_{B}^{2} = sup_{0≤τ≤T} (∥u(., τ)∥_{W_{2,ρ}^{1,1}(Ω)}^{2} + ∥θ(., τ)∥_{L_{ρ}^{2}(Ω)}^{2}) + ∥θ_{r}∥_{L_{ρ}^{2}(Q)}^{2}  (3.7)

and H = H_{1} × H_{2} is the completion of the Hilbert space {L_{ρ}^{2}(Q) × W_{2,ρ}^{1}(Ω) × L_{ρ}^{2}(Ω)} × {L_{ρ}^{2}(Q) × L_{ρ}^{2}(Ω)} with respect to the norm

∥H∥_{H}^{2} = ∥f∥_{L_{ρ}^{2}(Q)}^{2} + ∥u_{0}∥_{W_{2,ρ}^{1}(Ω)}^{2} + ∥u_{1}∥_{L_{ρ}^{2}(Ω)}^{2} + ∥g∥_{L_{ρ}^{2}(Q)}^{2} + ∥θ_{0}∥_{L_{ρ}^{2}(Ω)}^{2} . (3.8)

Let D(L), be the domain of definition of the operator A, defined by:

D(A) = {(u, θ) ∈ (L_{ρ}^{2}(Q))^{2}/ u_{t}, θ_{t}, u_{tt}, u_{r}, θ_{r}, u_{rr}, θ_{rr}, u_{tr}, θ_{tr} ∈ L_{ρ}^{2}(Q)} ,

satisfying conditions (1.6)–(1.7).

4. Uniqueness of the solution

Theorem 4.1. For any function U = (u, θ) ∈ D(A), there exists a positive constant C independent of U, such that

∥U∥_{B} ≤ C ∥AU∥_{H} .  (4.1)

Proof. Consider the scalar products:

(u_{t}, L_{1}u)_{L_{ρ}^{2}(Q^{τ})} and (θ, L_{2}θ)_{L_{ρ}^{2}(Q^{τ})},  (4.2)

where Q^{τ} = (0, τ) × Ω. We have from (4.2)

(u_{t}, u_{tt})_{L_{ρ}^{2}(Q^{τ})} − a((ru_{r})_{r}, u_{t}))_{L^{2}(Q^{τ})} + (θ, θ_{t})_{L_{ρ}^{2}(Q^{τ})}

− κ((rθ_{r})_{r}, θ))_{L^{2}(Q^{τ})} + b(ru_{rt}, θ)_{L_{ρ}^{2}(Q^{τ})} + b(ru_{t}, θ_{r})_{L_{ρ}^{2}(Q^{τ})}

= (u_{t}, L_{1}u))_{L_{ρ}^{2}(Q^{τ})} + (θ, L_{2}θ)_{L_{ρ}^{2}(Q^{τ})}. (4.3)

Using conditions (1.3)0-(1.7), we can evaluate the first five terms on the left-hand side of (4.3) as follows

(u_{t}, u_{tt})_{L_{ρ}^{2}(Q^{τ})} = \frac{1}{2} ∥u_{t}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} − \frac{1}{2} ∥u_{1}∥_{L_{ρ}^{2}(Ω)}^{2} ,  (4.4)

22 S. Mesloub, S. A. Messaoudi

−a((ru_{r})_{r}, u_{t}))_{L^{2}(Q^{τ})} = \frac{a}{2} ∥u_{r}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} − \frac{a}{2} ∥\frac{∂u_{0}}{∂r} ∥_{L_{ρ}^{2}(Ω)}^{2} ,  (4.5)
(θ, θ_{t})_{L_{ρ}^{2}(Q^{τ})} = \frac{1}{2} ∥θ(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} − \frac{1}{2} ∥θ_{0}∥_{L_{ρ}^{2}(Ω)}^{2} ,  (4.6)
−κ((rθ_{r})_{r}, θ))_{L^{2}(Q^{τ})} = κ ∥θ_{r}∥_{L_{ρ}^{2}(Q^{τ})}^{2} ,  (4.7)
b(ru_{rt}, θ)_{L_{ρ}^{2}(Q^{τ})} = −b(ru_{t}, θ_{r})_{L_{ρ}^{2}(Q^{τ})} − 2b(u_{t}, θ)_{L_{ρ}^{2}(Q^{τ})}.  (4.8)

Substitution of equalities (4.4)–(4.8) into (4.3), yields

\frac{a}{2} ∥u_{r}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥u_{t}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥θ(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + κ ∥θ_{r}∥_{L_{ρ}^{2}(Q^{τ})}^{2}

= \frac{1}{2} ∥u_{1}∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{a}{2} ∥\frac{∂u_{0}}{∂r} ∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥θ_{0}∥_{L_{ρ}^{2}(Ω)}^{2}

+ 2b(u_{t}, θ)_{L_{ρ}^{2}(Q^{τ})} + (u_{t}, L_{1}u))_{L_{ρ}^{2}(Q^{τ})} + (θ, L_{2}θ)_{L_{ρ}^{2}(Q^{τ})}. (4.9)

Using Cauchy-Schwarz inequality, we estimate the last three terms in the right-hand side of (4.9) as follows

2b(u_{t}, θ)_{L_{ρ}^{2}(Q^{τ})} ≤ b ∥θ∥_{L_{ρ}^{2}(Q^{τ})}^{2} + b ∥u_{t}∥_{L_{ρ}^{2}(Q^{τ})}^{2} ,  (4.10)
(u_{t}, L_{1}u))_{L_{ρ}^{2}(Q^{τ})} ≤ \frac{1}{2} ∥u_{t}∥_{L_{ρ}^{2}(Q^{τ})}^{2} + \frac{1}{2} ∥L_{1}u∥_{L_{ρ}^{2}(Q^{τ})}^{2} ,  (4.11)
(θ, L_{2}θ)_{L_{ρ}^{2}(Q^{τ})} ≤ \frac{1}{2} ∥θ∥_{L_{ρ}^{2}(Q^{τ})}^{2} + \frac{1}{2} ∥L_{2}θ∥_{L_{ρ}^{2}(Q^{τ})}^{2} .  (4.12)

Combining (in)equalities (4.9)–(4.12), we obtain

\frac{a}{2} ∥u_{r}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥u_{t}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥θ(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + κ ∥θ_{r}∥_{L_{ρ}^{2}(Q^{τ})}^{2}

≤ \frac{1}{2} ∥u_{1}∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{a}{2} ∥\frac{∂u_{0}}{∂r} ∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥θ_{0}∥_{L_{ρ}^{2}(Ω)}^{2} (b + \frac{1}{2}) ∥u_{t}∥_{L_{ρ}^{2}(Q^{τ})}^{2}

+ (b + \frac{1}{2}) ∥θ∥_{L_{ρ}^{2}(Q^{τ})}^{2} + \frac{1}{2} ∥L_{2}θ∥_{L_{ρ}^{2}(Q^{τ})}^{2} + \frac{1}{2} ∥L_{1}u∥_{L_{ρ}^{2}(Q^{τ})}^{2} . (4.13)

It is easy to check that

\frac{1}{2} ∥u(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} ≤ \frac{1}{2} ∥u_{0}∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥u∥_{L_{ρ}^{2}(Q^{τ})}^{2} + \frac{1}{2} ∥u_{t}∥_{L_{ρ}^{2}(Q^{τ})}^{2} ,  (4.14)

If we sum side to side (4.13) and (4.14), we get

∥u_{r}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + ∥u_{t}(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + ∥u(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + ∥θ(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + ∥θ_{r}∥_{L_{ρ}^{2}(Q^{τ})}^{2}

≤ c (∥u_{1}∥_{L_{ρ}^{2}(Ω)}^{2} + ∥θ_{0}∥_{L_{ρ}^{2}(Ω)}^{2} + ∥u_{0}∥_{W_{2,ρ}^{1}(Ω)}^{2} + ∥L_{2}θ∥_{L_{ρ}^{2}(Q^{τ})}^{2}

+ ∥L_{1}u∥_{L_{ρ}^{2}(Q^{τ})}^{2} + ∥u∥_{L_{ρ}^{2}(Q^{τ})}^{2} + ∥u_{t}∥_{L_{ρ}^{2}(Q^{τ})}^{2} + ∥θ∥_{L_{ρ}^{2}(Q^{τ})}^{2}) , (4.15)

where

c = \frac{max(a, 2b + 2)}{min(a, 1, 2κ)}.

On a system of linear thermoelasticity 23

Applying Gronwall’s inequality [4] to (4.15), we obtain ∥u(r, τ)∥_{W_{2,ρ}^{1,1}(Ω)}^{2} + ∥θ(r, τ)∥_{L_{ρ}^{2}(Ω)}^{2} + ∥θ_{r}∥_{L_{ρ}^{2}(Q^{τ})}^{2} ≤ ce^{cT} (∥u_{1}∥_{L_{ρ}^{2}(Ω)}^{2} + ∥θ_{0}∥_{L_{ρ}^{2}(Ω)}^{2} + ∥u_{0}∥_{W_{2,ρ}^{1}(Ω)}^{2} + ∥f∥_{L_{ρ}^{2}(Q^{τ})}^{2} + ∥g∥_{L_{ρ}^{2}(Q^{τ})}^{2}) . (4.16) Replacing the left-hand side of (4.16) by its upper bound with respect to τ over (0, T), gives the desired estimate (4.1), with C = \sqrt{c} exp(\frac{cT}{2} ).

It can be proved in a standard way that the operator A: B = B_{1} × B_{2} → H = H_{1} × H_{2} is closable. Let \overline{A} be its closure.

Proposition 4.2. The operator A: B → H has a closure. Proof. The proof can be established in a similar way as in [12]. These are some consequences of Theorem 4.1. Corollary 4.3. There exists a positive constant C such that

∥U∥_{B} ≤ C ∥\overline{A}U∥_{H} ,  ∀U ∈ D(\overline{A}).  (4.17)

Inequality (4.17) leads to the following results:

Corollary 4.4. A strong solution of (1.1)–(1.7) is unique and depends continuously on H = (H_{1}, H_{2}) ∈ H, where H_{1} = {f, u_{0}, u_{1}} and H_{2} = {g, θ_{0}} .

Corollary 4.5. The range R(\overline{A}) of \overline{A} is closed in H and R(\overline{A}) = \overline{R(A)}. This last corollary shows that in order to prove that problem (1.1)–(1.7) has a

strong solution for arbitrary (H_{1}, H_{2}) ∈ H, it is sufficient to prove that the range of A is dense in H; that is \overline{R(A)} = H.

5. Solvability of the problem Proposition 5.1. If, for some function W = (w_{1}, w_{2}) ∈ (L_{ρ}^{2}(Q))^{2} and for all

elements U ∈ D_{0}(A) = {U/U ∈ D(A) : ℓ_{1}u = ℓ_{2}u = ℓ_{3}θ = 0}, we have

(L_{1}u, w_{1})_{L_{ρ}^{2}(Q)} + (L_{2}θ, w_{2})_{L_{ρ}^{2}(Q)} = 0,  (5.1)

then W vanishes almost everywhere in Q.

Proof. Since relation (5.1) holds for any element of D_{0}(A), we then take an element U = (u, θ) with a special form given by

U = \{ \begin{matrix}(0, 0), & 0 ≤ t ≤ s \\ (∫_{s}^{t}(τ − t)u_{ττ} dτ, ∫_{s}^{t} θ_{τ} dτ), & s ≤ t ≤ T \end{matrix}  (5.2)

such that (u_{tt}, θ_{t}) is a solution of the system

\{ \begin{matrix}ru_{tt} & = E_{1}(r, t), \\ rθ_{t} & = E_{2}(r, t), \end{matrix}  (5.3)

where E_{1}(r, t) = ∫_{t}^{T} w_{1}(r, τ) dτ, and E_{2}(r, t) = ∫_{t}^{T} w_{2}(r, τ) dτ. It is clear that

\{ \begin{matrix}w_{1} & = −ru_{ttt} \\ w_{2} & = −rθ_{tt}. \end{matrix}  (5.4)

By virtue of relations (5.2) and (5.3), the function U = (u, θ) ∈ (L_{ρ}^{2}(Q))^{2}. In fact U possesses a higher order of smoothness.

Using lemma 3.2 of [11], the function W = (w_{1}, w_{2}) represented by (5.4) is in (L_{ρ}^{2}(Q))^{2}.

Replacing the functions w_{1} and w_{2} given by (5.4) in the relation (5.1), we obtain

− (u_{tt}, u_{ttt})_{L_{ρ}^{2}(Q)} + a(u_{ttt}, (ru_{r})_{r})_{L^{2}(Q)} − b(rθ_{r}, u_{ttt})_{L_{ρ}^{2}(Q)}

− (θ_{t}, θ_{tt})_{L_{ρ}^{2}(Q)} + κ(θ_{tt}, (rθ_{r})_{r})_{L^{2}(Q)} − b(rθ_{tt}, u_{tr})_{L_{ρ}^{2}(Q)} = 0. (5.5)

Taking into account the special form of U given by (5.2) and (5.3), using conditions (1.6)–(1.7), and integrating by parts each term of (5.5), gives

−(u_{tt}, u_{ttt})_{L_{ρ}^{2}(Q)} = \frac{1}{2} ∥u_{tt}(r, s)∥_{L_{ρ}^{2}(Ω)}^{2} ,  (5.6)
a(u_{ttt}, (ru_{r})_{r})_{L^{2}(Q)} = \frac{a}{2} ∥u_{rt}(r, T)∥_{L_{ρ}^{2}(Ω)}^{2} ,  (5.7)
−b(rθ_{r}, u_{ttt})_{L_{ρ}^{2}(Q)} = b(rθ_{rt}, u_{tt})_{L_{ρ}^{2}(Q)},  (5.8)
−(θ_{t}, θ_{tt})_{L_{ρ}^{2}(Q)} = \frac{1}{2} ∥θ_{t}(r, s)∥_{L_{ρ}^{2}(Ω)}^{2} ,  (5.9)
κ(θ_{tt}, (rθ_{r})_{r})_{L^{2}(Q)} = κ ∥θ_{rt}∥_{L_{ρ}^{2}(Q_{s})}^{2} ,  (5.10)
−b(rθ_{tt}, u_{tr})_{L_{ρ}^{2}(Q)} = −b(rθ_{rt}, u_{tt})_{L_{ρ}^{2}(Q_{s})} − 2b(θ_{t}, u_{tt})_{L_{ρ}^{2}(Q_{s})}.  (5.11)

Combining equalities (5.5)–(5.11), we get

\frac{1}{2} ∥θ_{t}(r, s)∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{1}{2} ∥u_{tt}(r, s)∥_{L_{ρ}^{2}(Ω)}^{2} + \frac{a}{2} ∥u_{rt}(r, T)∥_{L_{ρ}^{2}(Ω)}^{2} + κ ∥θ_{rt}∥_{L_{ρ}^{2}(Q_{s})}^{2}

= 2b(θ_{t}, u_{tt})_{L_{ρ}^{2}(Q_{s})}, (5.12)

where Q_{s} = Ω × [s, T]. By using Cauchy-Schwarz inequality and discarding the two last terms of the left-hand side of (5.12), we obtain

∥θ_{t}(r, s)∥_{L_{ρ}^{2}(Ω)}^{2} + ∥u_{tt}(r, s)∥_{L_{ρ}^{2}(Ω)}^{2}

≤ 2b ∫_{s}^{T} (∫_{0}^{1} rθ_{t}^{2}(r, t) dr + ∫_{0}^{1} ru_{tt}^{2}(r, t) dr) dt. (5.13)

From (5.13), it follows that

−\frac{d}{ds}\{e^{2bs} ∫_{s}^{T} (∫_{0}^{1} rθ_{t}^{2}(r, t) dr + ∫_{0}^{1} ru_{tt}^{2}(r, t) dr) dt\} ≤ 0.  (5.14)

Integrating (5.14) over [s, T] and using the fact that

∫_{s}^{T} (∫_{0}^{1} rθ_{t}^{2}(r, t) dr + ∫_{0}^{1} ru_{tt}^{2}(r, t) dr) dt |_{s=T} = 0,

it follows that

e^{2bs}(∫_{s}^{T} ∫_{0}^{1} rθ_{t}^{2}(r, t) dr dt + ∫_{s}^{T} ∫_{0}^{1} ru_{tt}^{2}(r, t) dr dt) ≤ 0.  (5.15)

Hence, we deduce from (5.15) that W = (w_{1}, w_{2}) = (0, 0) almost everywhere in Q_{s}. Proceeding in this way step by step, we prove that W = 0 almost everywhere in Q.

We now prove the following theorem which gives the existence of a strong solution of problem (1.1)–(1.7).

Theorem 5.2. For any (f, g) ∈ (L_{ρ}^{2}(Q))^{2} and any u_{0} ∈ W_{2,ρ}^{1}(Ω), u_{1} ∈ L_{ρ}^{2}(Q),

θ_{0} ∈ L_{ρ}^{2}(Q), there exists a unique strong solution U = \overline{A}^{−1}H = \overline{A^{−1}}H of the problem (1.1)–(1.7), where H= (H_{1}, H_{2}) ∈ H, H_{1} = {f, u_{0}, u_{1}}, H_{2} = {g, θ_{0}}, U = (u, θ) and

∥U∥_{B} ≤ C ∥AU∥_{H} ,

for a positive constant C, independent of U.

Proof. It is sufficient to prove that the range R(A) of A is dense in H = H_{1}×H_{2}. Suppose that for some Ψ = (G_{1}, G_{2}) = ({w_{1}, w_{3}, w_{4}} , {w_{2}, w_{5}}) ∈ H, the

orthogonal of R(A), so that

(AU, Ψ)_{H} = ({L_{1}u, L_{2}θ} , {G_{1}, G_{2}})_{H}
= ({(L_{1}u, ℓ_{1}u, ℓ_{2}u), (L_{2}θ, ℓ_{3}θ), ({w_{1}, w_{3}, w_{4}} , {w_{2}, w_{5}})})_{H}
= (L_{1}u, w_{1})_{L_{ρ}^{2}(Q)} + (ℓ_{1}u, w_{3})_{W_{ρ}^{1}(Ω)} + (ℓ_{2}u, w_{4})_{L_{ρ}^{2}(Ω)}
+ (L_{2}θ, w_{2})_{L_{ρ}^{2}(Q)} + (ℓ_{3}θ, w_{5})_{L_{ρ}^{2}(Ω)} = 0.  (5.16)

We must show that Ψ = 0. Putting U ∈ D_{0}(A) in (5.16), we get

(L_{1}u, w_{1})_{L_{ρ}^{2}(Q)} + (L_{2}θ, w_{2})_{L_{ρ}^{2}(Q)} = 0,  ∀U ∈ D(A).  (5.17)

Hence proposition 5.1 implies that w_{1} = w_{2} = 0.

The relation (5.17), implies that

(ℓ_{1}u, w_{3})_{W_{ρ}^{1}(Ω)} + (ℓ_{2}u, w_{4})_{L_{ρ}^{2}(Ω)} + (ℓ_{3}θ, w_{5})_{L_{ρ}^{2}(Ω)} = 0,  (5.18)

for all U ∈ D(A).

Since the three quantities in (5.18) vanish independently and since the ranges of the trace operators ℓ_{1}, ℓ_{2}, and ℓ_{3} are respectively everywhere dense in the spaces W_{2,ρ}^{1}(Ω), L_{ρ}^{2}(Q), and L_{ρ}^{2}(Q), therefore it follows, from (5.18), that w_{3} = w_{4} = w_{5} = 0. Hence \overline{R(A)} = H.

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(received 13.04.2004)

S. Mesloub, King Saud University, College of Sciences, Department of Mathematics, P.O.Box 2455, Riyadh 11451, Saudi Arabia

E-mail: mesloubs@yahoo.com

S. A. Messaoudi, Department of Mathematics, KFUPM, Dhahran 31261, Saudi Arabia

E-mail: messaoud@kfupm.edu.sa