Stabilization of a locally damped thermoelastic system

Stabilization of a locally damped thermoelastic system

Jáuber C. Oliveira^*; Ruy C. Charão

Departamento de Matemática, Universidade Federal de Santa Catarina 88040-900 Florianópolis, SC, Brasil E-mails: [e-mail], [e-mail]

ABSTRACT

We show that the solutions of a thermoelastic system with a localized nonlinear distributed damping decay locally with an algebraic rate to zero, that is, given an arbitrary R > 0, the total energy E(t) satisfies for t > 0: E(t) < C (1 + t)^–γ for regular initial data such that E(0) < R, where C and γ are positive constants. In the two-dimensional case, we obtain an exponential decay rate when the nonlinear dissipation behaves linearly close to the origin.

Mathematical subject classification: 35B40, 35L70.

Key words: thermoelastic system, nonlinear localized damping, algebraic decay rate, exponential decay rate.

1 Introduction

In this work we study decay properties of the solutions of the following initial boundary-value problem associated with the thermoelastic system:

where Ω is a bounded domain in

^N, N > 2. u(x, t) = (u¹ (x, t), ..., u^N (x, t)) is the vector displacement, Δ u = (Δ u¹ (x, t), ..., Δ u^N (x, t)) is the Laplacian operator of u, div u is the divergence of u and ∇ is the gradient operator. θ represents the temperature distribution. The vector function ρ is a dissipative term, localized in a neighborhood of part of the boundary of Ω. The coefficients a and b are related to the Lamé coefficients of Elasticity Theory and b² > a² > 0.

In this paper, we show the uniform stabilization of the total energy for the system (1.1)-(1.4) with algebraic rates, where the dissipative term ρ(x, u_t) is strongly nonlinear and effective only in a neighborhood of part of the boundary. When the nonlinear dissipative term ρ(x, s) behaves linearly for small s and the dimension is two, we obtain an exponential decay rate for the total energy of the system. If the dimension N > 3 and ρ(x, s) behaves linearly for all s, then the decay rate is also exponential. To prove these results we use ideas of [10], [8] and [13] to obtain some energy identities associated with localized multipliers in order to construct special difference inequalities for the associated energy. The main estimates in this work are obtained using Holmgren's Uniqueness Theorem and Nakao's Lemma.

Regarding works on the stabilization of thermoelastic systems, Dafermos [2] investigated the existence, uniqueness, regularity and stabilization (without rates) of the solution for the linear system in one dimension. Racke [17] considered a Cauchy problem for the three-dimensional nonlinear thermoelasticity equations and proved the global existence of smooth solutions for sufficiently small and smooth initial data. It was necessary to assume that certain nonlinear terms are quasilinear and have cubic nonlinearity.

Pereira-Perla Menzala [15] proved that the total energy of the linear thermoelastic system in an isotropic, non-homogeneous (bounded, n-dimensional) medium, with a linear dissipative term effective in the whole domain, decays to zero in an exponential rate. Rivera [23] later proved that the energy of the classical one-dimensional thermoelastic system decays to zero exponentially. Also, Henry-Lopes-Perisinotto [5] showed, using spectral analysis, that the three parts of the energy of that system decay exponentially to zero in the one-dimensional case, but such decay does not occur in higher dimensions.

Racke [18] used the energy method to prove the exponential decay to zero of displacement and temperature for the three-dimensional equations of linear thermoelasticity in bounded domains for inhomogeneous and anisotropic media assuming a linear damping force. Racke-Shibata-Zheng [20] considered the Dirichlet initial-boundary value problem in one-dimensional nonlinear thermoelasticity to prove that if the initial data are closed to the equilibrium then the problem admits a unique global smooth solution. They proved that as time tends to infinity, the solution is exponentially stable. They also used techniques based on the work of Rivera [23] to improve previous results of Racke-Shibata [19], which were based on spectral analysis to obtain decay rates for solutions of nonlinear thermoelasticity in one dimension. Rivera-Barreto [24] improved further the global in time unique existence result with exponential decay of the energy obtained by Racke-Shibata-Zheng [20] by assuming more general smoothness hypothesis on the initial data.

Rivera [22] considered the linear, inhomogeneous thermoelasticity equations in one dimension for a bounded domain with several boundary conditions considered. It is proved that the solution (u, θ) has some partial derivatives decaying exponentially to zero in the L²-norm. Rivera [21] investigated the linear homogeneous and isotropic thermoelasticity equations with homogeneous Dirichlet boundary conditions in a general n-dimensional domain. The author shows that the curl-free part of the displacement and the thermal difference decay exponentially to zero as times goes to infinity. It is also proved that the divergence-free part of the displacement conserves its energy, which implies that if the divergence-free part of the initial data is not zero, then the total energy does not decay to zero uniformly.

Jiang, Rivera and Racke [7] proved the exponential decay for solutions of the linear isotropic system of thermoelasticity in bounded two- or three-dimensional domains with the hypothesis that the rotation of the displacement vanishes.

Lebeau-Zuazua [9] studied the linear system of thermoelasticity in two- and three-dimensional smooth bounded domain. They analyzed whether the energy of solutions decays exponentially to zero. They also prove that when the domain is convex, the decay rate is never uniform. Liu-Zuazua [12] established explicit formulas for the decay rate of the energy of a body in the framework of linear thermoelasticity when some part of the boundary of the body is clamped and on the rest there is some nonlinear velocity feedback. This result was obtained using the theory of semigroups, Lyapunov methods and multiplier techniques. Qin and Rivera ([16]) established the global existence, uniqueness and exponential stability of solutions to equations of one-dimensional nonlinear thermoelasticity with relaxation kernel and subject to Dirichlet boundary conditions for the displacement and to Neumann boundary conditions for the temperature difference.

Irmscher-Racke [6] obtained explicit sharp decay rates for solutions of the system of classical thermoelasticity in one dimension. They also considered the model of thermoelasticity with second sound and compared the results of both models with respect to the asymptotic behavior of solutions.

Our work generalizes, in the context of pure elasticity, the previous work of Bisognin-Bisognin-Charão ([1]), where it is proved a stabilization theorem with polynomial decay rate for the total energy of the system only in three-dimensions. We should mention that the work of Bisognin-Bisognin-Charão ([1]) generalized the work of Guesmia ([3]) that proved the stabilization of the total energy for pure system of elasticity with a nonlinear localized dissipation which does not couple the system of equations and behaves linearly far from the origin. Our work also generalizes the previously mentioned work of Pereira-Perla Menzala ([15]). Regarding the work of Jiang-Rivera-Racke ([7]), instead of assuming their condition that the rotation of the displacement vanishes and the dimension is two or three, we consider in the system a nonlinear localized weak dissipation in any dimension N > 2. Furthermore, according to the previous mentioned works, only part of the energy of the free thermoelastic system decays uniformly if N > 2. Thus, in this sense, our work also improves those results and other previous results for the thermoelastic system due to the fact that we have obtained exponential decay for the total energy when N = 2 and polynomial decay rate when N > 3 by including a weak localized nonlinear dissipative term ρ(x, s) in the system.

2 Hypotheses and Notation

Throughout this work the dot ( ∙ ) will represent the usual inner product betweentwo vectors in

^N. Let V₁ = (Ω)^N, V₂ = (Ω), U₁ = V₁∩ H² (Ω)^N, U₂ = V₂∩ H² (Ω). || denotes the norm of a vector in ^N, |∇u(x, t)|² = |∇uⁱ(x, t)|², ( , ) represents the inner product in L²(Ω)^N and || || denotes the corresponding norm. Also, ||∇u||² = ||∇uⁱ||² and = , where || ||₁ denotes the usual norm in H¹(Ω). We use |Ω| to represent the measure of Ω.

Now, we list the hypotheses which we use to establish existence and uniqueness of solutions.

(H0) Ω ⊂

^N is a bounded open set with smooth boundary (at least of class C²);

(H1) u_oЄ U₁, u₁Є V₁; θ_oЄ U₂

(H2) ρ: × ^N→ ^N is a function such that:

(a) ρ(x, s) ∙ s > 0, s Є ^N, x Є ;

(b) ρ and are continuous functions in × ^N;

(c) uⁱ

u^k > 0, ∀u Є ^N, ∀x Є , ∀s Є ^N, i.e., is positive semi-definite.

(d) There exist positive constants K₀, K₁, K₂ and K₃ and numbers p, r, -1 < r < +∞, -1 < p < if N > 3 and -1 < p < +∞ if N = 1 or 2, such that: K₂a(x) |s|^r+2< ρ(x, s) ∙ s and |ρ(x, s)| < K₀a(x) (|s|^r+1 + |s|), for |s| < 1, K₃a(x) |s|^p+2< ρ(x, s) ∙ s and |ρ(x, s)| < K₁a(x)(|s|^p+1 + |s|), for |s| > 1, where the function a = a(x) is such that a: → ⁺ belongs to L^∞ (Ω).

Remark 1. If a(x) is a continuous function on , then ρ(x, s) = a(x)|s|^ps, s Є ^N, is an example of a function which satisfies (a)-(d) with r = p.

In order to study the stabilization of the total energy for this system, we specify where the damping is effective in the domain, that is, where the dissipative term is localized. We choose x_o in ^N and we define

Γ (x_o) = {x Є ∂ Ω: (x - x_o) ∙ η(x) > 0},

where η(x) denotes the outward unit normal vector at x Є ∂Ω.

Now, let ω ⊂ be a neighborhood of Γ (x_o). Then, in addition to the hypothesis (H2)(d) that the function a = a(x) is nonnegative, we also assume that

a(x) >a_o > 0 , in ω.

The energy E(t) of the system (1.1) is defined by

where {u = u(x, t), θ = θ(x, t)} is the solution of (1.1)-(1.4). We denote the energy difference E(t) - E(t + T) by ΔE. Also, let E₁ denote the part of E without the term 1/2 ∫_Ωθ², and E₂ = E - E₁.

In order to reduce the size of the formulas, we use the following notation.

Remark 2. The multiplication of (1.1) by u_t, (1.2) by θ followed by adding the resulting equations and integrating over Q (t) produces:

The hypothesis H2(a) implies that the energy is a nonincreasing functionof t.

3 Results

Regarding the existence and uniqueness of solution for the problem (1.1)-(1.4), the following result holds.

Theorem 3.1 (Existence and Uniqueness). Under the hypotheses of the previous section, the initial-boundary value problem (1.1)-(1.4) has a unique solution u = u(x, t), θ = θ(x, t) such that for each T > 0, u Є C([0, T], U₁), u_tЄ C([0, T], V₁), u_ttЄ C([0, T], L²(W)^N), Є θ C([0,T], U₂) and θ' Є C([0,T], V₂).

Proof. We sketch the proof based on Semigroups for the nonlinear case. We consider the operator A defined by

where A₁u = -a²Δ u - (b² - a²) ∇ div u, A₂θ = -Δθ, B θ = -∇θ, and C v = -div v. The domain of A : D(A) ⊂ H → H is given by

U₁ × V₁ × U₂,

where

H = V₁ × L²(Ω)^N × L²(Ω)

is a Hilbert space with the inner product

〈(z₁, z₂, z₃),(w₁, w₂, w₃)〉 = ∫_Ω{z₂w₂ + a²

+ (b² - a²) div (w₁) div (z₁) + z₃w₃} dx.

Then, the original problem is equivalent to

where

F (w₁, w₂, w₃) = (0, ρ(∙, w₂), 0)

We use the theorem of Hille-Yosida to prove that the operator A (densely defined) generates a Cº-semigroup of contractions. The hypotheses of the Hille-Yosida are verified after proving that for each positive λ, there exists (λI - A)^-1Є (H) and ||(λI - A)^-1 || < .

Next, using hypothesis H2(b), it is easy to prove that F : H → H is Lipschitz continuous on bounded sets. As a consequence, one obtains a local (generalized) solution on a interval [0, T_max]. Then, by proving that ||U||_X< C E(0), where is used the fact that the total energy of the system is decreasing due to hypothesis (H2)(a), it is possible to show that the solution can be extended to the interval [0, ∞). Finally, by using hypothesis (H2)(b), one proves that F is continuously differentiable. It follows that the generalized solution of (3.2)-(3.3) is a classical solution on [0, ∞). The uniqueness follows from the hypothesis (H2)(c) on the function ρ.

□

3.1 Boundedness of the Laplacian L ² -norm

In the previous section, we proved the existence of a unique solution (u, θ) for the thermoelastic system in the class

u Є C ([0, ∞); H²(Ω)^N∩ (Ω)^N) ∩ C¹ ([0, ∞); (Ω)^N) ∩ C² ([0, ∞); L² (Ω)^N)

and

θ Є C ([0, ˞); H²(Ω) ∩ (Ω)) ∩ C¹ ([0 ∞); (Ω))

In order to prove that

we write

where Lu is the differential operator defined by

Lu = -a²Δu - (b² - a²) ∇ div u.

To show (3.4), we only need to show that each term in the right-hand side of the equation (3.5) is in L^∞ (0, ∞; L²(Ω)^N). This is proved in the next Lemmas. Then, the uniform boundedness of L²-norm of the Laplacian of u follows from the elliptic regularity of the operator L.

Lemma 3.2. The L²-norm of ρ(∙, u₁) is bounded by a constant that depends only on the initial data u₀, u₁, θ₀.

Proof. We have ||ρ (x, u₁)||²< |ρ(x, u₁)|²dx + |ρ(x, u₁)|²dx, where

Ω₁: = {x Є Ω : |u₁| < 1}

and

Ω₂ = Ω - Ω₁.

If r > 0, then, due to our hypothesis on the function ρ,

which is bounded (u₁Є L²(Ω)^N).

Let N > 3 and 0 < p < , then due to the hypotheses on the function ρ

The last estimate is due to facts that 2 < 2 p + 2 < and (Ω) is continuously embedded in L^q (Ω) for q Є [2, ]. Thus, the lemma holds for the case 1: N > 3, r > 0 and 0 < p < .

The proof of the other cases is similar.

□

Now, we obtain an estimate for the L²-norm of ||u_tt(0)||.

Lemma 3.3. There is a positive constant C = C(u₀, u₁, θ₀) such that ||u_tt(0)||< C and ||θ_t(0)|| < C.

Proof. We take L² inner-product between the first equation in the thermoelastic system with u_tt(t) and evaluate at t = 0 to obtain

Since u_tt(0) Є L²(Ω)^N, u₀Є H²(Ω)^N and θ₀ Î H²(Ω), we have the following estimate

The result follows now from the previous lemma and our hypothesis on the initial data. The proof of the estimate for θ is similar.

□

Lemma 3.4.

(i) u_tt belongs to L^∞(0, ∞; L²(Ω)^N) and for each t the L²-norm of u_tt(t) is bounded by a constant that depends only on the initial data u₀, u₁, θ₀.

(ii) For all t Є (0, ∞),

Proof. We differentiate once the first equation of the thermoelastic system with respect to t, multiply each member of the resulting equation by u_tt and integrate over Ω. Therefore, we have

By the hypothesis on the initial data (u₁Є (Ω)^N), it follows that a² + (b² - a²) div u₁ belongs to L²(Ω)^N. So, after integrating this equation over [0, t], we obtain

Now, differentiating the second equation of the thermoelastic system withrespect to t, multiplying by θ_t and integrating over Ω, we obtain

||θ_t||² + 2 ||∇θ_t||² - 2 (∇θ_t, u_tt) = 0.

Integrating over [0, t], we obtain

Adding equations (3.10) and (3.11) and using (H2)(c), we obtain

The result follows from the Lemma 3.3.

□

Lemma 3.5. The L²-norm of ρ(., u(t)) is bounded by a constant that depends only on the initial data u₀, u₁, θ₀.

Proof. This proof is similar to the proof of Lemma 3.2, therefore we present the proof for the three-dimensional case only.

||ρ(x, u)||²< |ρ(x, u)|²dx + |ρ(x, u)|²dx, where for each t,

Ω₁(t) := {x Є Ω: |u_t(t)| < 1}

and

Ω₂(t) = Ω - Ω₁(t).

Let

and

In order to prove the result, it is sufficient to estimate I₁(t) considering the cases r > 0 and -1 < r < 0, and estimate I₂(t) considering the cases 0 < p < 2 and -1 < p < 0. If r > 0, then, due to H2(d),

The last inequalities are due to the fact that ||u_t||² is part of the energy and the fact that the energy is decreasing.

If 0 < p < 2, then due to H2(d),

The last term in this estimate is bounded for all t due to the previous lemma.

where the last inequality follows from Hölder's inequality. If -1 < p < 0, then

Lemma 3.6. ∇ θ Є L^∞ (0, ∞; L²(Ω )^N).

Proof. We use the fundamental identity:

Then, due to the hypothesis H2(a), we have

However,

Therefore, using Cauchy-Schwartz's inequality, we obtain

The terms ||u_t||², ||∇u||², ||div u||², ||θ||² are all bounded by a constant (independent of t) times the initial energy, since they are part of the energy and the energy is bounded by the initial energy. The remaining terms are boundeddue to Lemma 3.4. We conclude that ||∇ θ|| is bounded for all t, which concludes the proof.

□

In summary:

< C with C a positive constant which does not depend on t and depends only on the initial data, ||a(.)||_∞ and |Ω|.

3.2 Theorem on stabilization

From now on, we will study the asymptotic behavior of the total energy of the system. Under certain hypothesis we will obtain exponential decay of the energy, which means that there exist positive constants M₀ and k₀ such that E(t) < M₀ exp(-k₀t) if t > 0.

The main result of this paper is the following theorem of stabilization for the total energy of the system.

Theorem 3.7 (Stabilization). Let R > 0 and initial data in D(A) satisfying E(0) < R. Then, under the previous hypotheses, the total energy for the solution u = u(x, t), θ = θ (x, t) of the problem (1.1)-(1.4) has the following asymptotic behavior in time

where C = C(R, r, p) is a positive constant (which depends on the initial data, ||a||_∞ and |Ω|) and the decay rate γ is given according to the following cases:

Case 1. If N = 2 and

(a) -1 < r < 0 and -1 < p < +∞ then γ = ;

(b) r > 0 and -1 < p < +∞, then γ = .

Case 2. If N > 3 and

(a) r > 0 and 0 < p < , then γ = min;

If r = 0 and 0 < p < , then γ = ;

If r > 0 and p = 0, then γ = ;

(b) r > 0 and -1 < p < 0, then γ = min ;

If r = 0 and -1 < p < 0, then γ = ;

(c) -1 < r < 0 and 0 < p < , then

If -1 < r < 0 and p = 0, then γ = ;

(d) -1 < r < 0 and -1 < p < 0, then

The decay rate is exponential in the following cases:

(a) N = 2 , r = 0 and -1 < p < +¥,

(b) N > 3 and r = p = 0.

4 Energy identities

In the next lemma, we obtain energy identities that will be used in the following sections to obtain estimates in terms of energy differences.

Next, we use the notation

where h is a vector field in ^N.

Lemma 4.1 (Energy identities). Let u(x, t), θ(x, t) be the solution of (1.1)-(1.4), m Є W^1,∞ (Ω) and h: → ^N a vector field of class C¹. Then, the following identities hold.

where η = η(x) is the outward unit normal at x Є ∂ Ω.

Proof. Equations (4.4), (4.1) and (4.2) are proved by multiplying (1.1), respectively, by the following multipliers M(u) = u, M(u) = m(x)u and M(u) = h : ∇u, followed by integration over Q(t). Equation (4.3) is a special case of equation (4.2) for h = x - x_o. We used the fact that u = 0 on ∂Ω × [0, ∞) and, as a consequence, the following vector identity:

□

on ∂Ω × [0, ∞).

5 Energy estimates

In this and the next sections, the symbol C may denote different positive constants. These constants depend on, at most, |Ω|,|a|_∞ and the initial data.

The idea to obtain the stabilization of the energy (to zero) is to show an estimate for the energy, as follows:

E(s)^1+δ < C (E(t) - E(t + T)) for all t > 0,

for some positive δ and a fixed T > 0, which may be large.

Then the asymptotic behavior is obtained using the following Lemma:

Lemma 5.1 (Nakao [13]). Let Φ(t) be a nonnegative function on ⁺ satisfying

Φ(s)^1+δ < C₁ {Φ(t) - Φ(t + T)}

with T > 0, δ > 0 and C₁ a positive constant. Then Φ(t) has the decay property

Φ(t)< C₁Φ(0) (1+t)^-1/δ, t > T

where C ₁ is a positive constant.

If δ = 0, then Φ(t) has the decay property (exponential decay)

Φ(t) < C₁Φ(0) exp^{-κ t},

for positive constants κ and C ₁ .

We also include the following lemma, which will be used to estimate an integral involving the dissipative term ρ.

Lemma 5.2 (Gagliardo-Nirenberg). Let 1 < r < p < ∞, 1 < q < p and 0 < m. Then,

for v Î W^m,p (Ω) ∩ L^r(Ω), Ω ⊂ ^N, where C₂ is a positive constant and

provided that 0 < q < 1.

Now, we begin the estimates for the energy of (1.1).

Lemma 5.3. Let β > 0 such that -1 > 0 and γ = 2 min{γ₁, γ₂}, where γ₁ = ( - 1) and γ₂ = (1 + β (1 - )). Then,

for all t> 0, T > 0, where M_o = |x - x_o|.

The proof is obtained multiplying the identity (4.3) by β and adding with the identity (4.4). The details are similar to Oliveira-Charão ([14]) for an incompressible vector wave equation.

It is necessary to estimate the boundary integral which appear in the above lemma.

Lemma 5.4. There exists a constant C > 0 such that

Proof. Let h : ^N→ ^Nbe a vector field of class C¹over which satisfies:

where η = η(x) is the outward unit normal vector at x Є ∂ Ω, ⊂ ^N is an open set such that Γ(x_o) ⊂ ∩ ⊂ w ⊂ .

Let m Є W^1,∞ (Ω) be a function such that

For the existence of such functions, see [4, 10].

Using identity (4.2), the properties of h, m and the summation convention, we have:

Therefore,

Since, by Poincare's inequality,

we estimate ∫_Q(t) m(x) [a² |∇u|² + (b² - a²) (div u)²] dxds using identity (4.1) and the properties of m:

Now, using Young's inequality in the last integral, the fact that isbounded and absorbing the term with the divergence in the left-hand side, we obtain

For the last estimate we have also used the inequality:

Combining the previous estimates, the proof of Lemma 5.4 follows.

□

Lemma 5.5. Let u(x, t) be the solution of (1.1)-(1.4). There exists a constant T > 0, which depends on E(0), such that

for all t > 0.

Proof. It follows from the identity in Remark 2 that

This fact allows us to get an estimate for the total energy E(t).

Because the energy decreases, we have: T E(t + T) < E(s) ds.

These facts and the last two lemmas imply that:

Thus, if we choose a fixed T such that T > 2 C₁ + 1, the lemma is proved.

Lemma 5.6. Let (u, θ) be the solution of (1.1)-(1.4). Then, for T given by the previous lemma, we have:

If r > 0, 0 < p < and N > 3, then

If N = 2, the estimate (5.9) holds for r > 0, p > 0.

If r > 0, -1 < p < 0 and N > 3, then

If r > 0, -1 < p < 0 and N = 2, then

If -1 < r < 0, 0 < p < and N > 3, then

If N = 2, the estimate (5.12) holds for -1 < r < 0, p > 0.

If -1 < r < 0, -1 < p < 0 and N > 3, then

If -1 < r < 0, -1 < p < 0 and N = 2, then

The numbers r and p appear in hypothesis (H2)(d) on the growth of the functionρ.

Proof. By hypotheses on the growth of ρ, we have

for t > 0, where

Ω₁ = Ω₁(t) = {x Є Ω, |u_t(x, t)| < 1}, Ω₂ = Ω\Ω₁.

We will use the following estimate as a consequence of Gagliardo-Nirenberg's Lemma, Poincare's inequality and the boundedness of the L²-norm of Δu,

with θ = .

Case 1: Estimating I₁ for r > 0, N > 2:

Using Poincare's inequality we obtain

because

where C depends on |Ω|, and the fixed number T. We have used the fact that E(t) is a non increasing function of t.

Using the hypotheses H2(a), H2(d) and identity (2.2) in Remark 2, it follows that

Case 2: Estimating I₁ for -1 < r < 0, N > 2:

Using Hölder's inequality, Poincaré's inequality in (Ω)^N and hypotheses H2(a), H2(d), we obtain

Case 3: Estimating I₂ for 0 < p < , N > 3 ( or p > 0 if N = 2):

where we have used Poincaré's inequality in (Ω)^N, Hölder's inequality and the hypothesis in (H2)(d) on the boundedness of a(x).

Now, the hypothesis H2(d), identity (2.2) in Remark 2 and estimate (5.15) imply

Case 4a: Estimating I₂ for -1 < p < 0, N > 3:

Thus,

where l' is the conjugate exponent of l.

Now, choosing

we have l' = and

since u_t Є L^∞ ([0, ∞); (Ω)^N) L^∞ ([0, ∞); (Ω)^N) due to Lemma 3.4 and Sobolev's inequality. The positive constant C depends also on the initial data.

Case 4b: Estimating I₂ for -1 < p < 0, N = 2:

Then, using Poincaré's inequality for u Є (Ω) and the hypotheses on the function ρ(x, s), it follows that

Now, using the inequality of Gagliardo-Nirenberg (Lemma 5.2) with θ = it follows that

Then, by Remark 2 and the boundedness of Δu (proved in a previous section), we obtain

6 Main estimates for stabilization

Using Young's inequality and Lemmas 5.5, 5.6, we obtain the next result.

Proposition 6.1. The energy for the solution of problem (1.1)-(1.4) satisfies

i = 1,2,3,4, where

Now, using Poincare's and Young's inequalities, we obtain (with ∈₀ = )

In the last inequality we have used the identity mentioned in Remark 2.

Using the above estimate in (6.1), we obtain

Thus, the natural dissipation of the system reduced the proof of the theorem of stabilization to the following estimate.

Proposition 6.2. Let R > 0 fixed and {u, θ} be the solution of the problem (1.1)-(1.4). Let u_o, u₁ and θ_o be such that E(0) < R. Then, there exists a constant C > 0 such that

where T > 0 is given by Lemma 5.5, the constant C depends on R. Here i = 1, 2, 3, 4 according to the cases previously described.

Proof. We present the proof by contradiction for N > 3 (the proof for N = 2 is easier). Suppose there exists a sequence of solutions with the following corresponding initial data and a sequence Ω ⁺ such that

Let

and

Then, from (6.4), we obtain

Let vⁿ (x, t) = and ηⁿ(x, t) = , 0 < t < T. Furthermore, from (6.5), we have that

for all n Є .

The estimate (6.1) together com (6.7) and (6.8) imply that

But, I_n(t_n) is bounded due to (6.7). Therefore, we obtain that

for all 0 < t < T and for all n Є , where C > 0 does not depend on t and n. Therefore,

for all 0 < t < T and for all n Є . However, from Poincaré's inequality and from the estimate (6.9), it follows that

for all 0 < t < T and for all n Є .

Thus, there exists a constant C > 0 such that

for all 0 < t < T and for all n Є .

We conclude that

for all n Є .

Now, we claim that

First we prove this claim for the case where r > 0 and 0 < p < .

For the case r > 0 and 0 < p < , proceeding as we did previously when we defined regions Ω₁(t), Ω₂(t) to estimate I₁ and I₂, we obtain

where C depends on T, |Ω| and ||a||_∞. Now, using the definition of D₁(t)

By (6.6),

where

However, {λ_n}_{n> 1} is a bounded sequence:

Hence, (6.7) and the definition of I_n imply that

as m goes to ∞.

The remaining cases are treated similarly. We have proved that

Thefore, (6.12) is proved.

At this point, we shall prove that

To prove (6.16), we use Cauchy-Schwarz's inequality to obtain

Then,

when n → ∞.

Then, (6.16) holds.

Also, we need to show that

From (6.7) we have that

then

since vⁿ → v weak- in L²(0, T; [L²(Ω)]^N).

So, (6.19) is proved.

Finally, we prove that

From Cauchy-Schwarz's inequality, we have that

Therefore, (6.20) holds.

Now, we pass to the limit of . From (6.11) it follows that there are function v(t), η(t) and subsequences of the sequences vⁿ and ηⁿ, which we continue to represent by vⁿ and ηⁿ such that

vⁿ (t) v(t) weak in W^{1, ∞} (0, T; [L² (Ω)]^N) ∩ L^∞ (0, T; (Ω)]^N)

ηⁿ (t) η(t) weak in L^∞(0, T; L² (Ω)).

Therefore, the functions v(t) and η(t) satisfy:

i) v Є W^{1, ∞} (0, T; [L² (Ω)]^N) ∩ L^∞ (0, T; (Ω)]^N) and η Є L^∞(0, T; L² (Ω));

ii) v_tt - a²Δv - (b² - a²) ∇(div v) = 0,

iii) η_t - Δη + div v_t = 0,

iv) |v_t|²dxds = 0,

v) |v|²dxds = 1.

We observe that {η, v_t} is also solution of equations ii), iii). Since v_t = 0 a.e. in ω × (0, T) and (v_t)" -a²Δ(v_t) - (b² - a²) ∇(div v_t) = 0 with homogeneous Dirichlet boundary conditions, it follows from a consequence of Holmgren's Uniqueness Theorem ([11], p. 88) that v_t = 0 in Ω × (0, T), therefore v(x, t) = F(x) a.e. in Ω × (0, T). Thus, v satisfies -a²Δv -(b² - a²) ∇¸ v = 0 with homogeneous Dirichlet boundary conditions. From the uniqueness of this Dirichlet problem, it results that v = 0 a.e. in (0, T) × Ω. This contradicts the item (v).

□

The following estimate is a consequence of the previous proposition.

Proposition 6.3.

where D_i(t) (i = 1,2,3,4) are given in Proposition 6.1.

7 Proof of the theorem of stabilization

Now, it remains to estimate the integral ∫_Qω(t)|u_t|²dx ds in terms of ΔE (energy difference).

Proof. Case a:r> 0 and 0 < p < if N > 3 (r > 0, p > 0 if N = 2):

Using the hypothesis on ω in (H2)(d), Hölder's inequality and the definitions of Ω₁, Ω₂, we obtain

where the last C depends on |Ω|, T and ||a||_∞.

Then, due to Remark 2, we have

From estimates (6.21) and (7.1), and the expression for D₁(t) we obtain

Then, using the fact that E(t) is nonincreasing function of t, we obtain that

where

is such that 0 < K₁< 1 and C is a positive constant which depends on the initial data.

We have obtained the following inequality

If we set 1 + γ = , then γ = and applying Nakao's Lemma to (7.3) we obtain for N > 3 that

with We see from (7.2) that if r = p = 0, then according to Nakao's Lemma, the decay rate is exponential. If r = 0, p > 0, then the decay rate depends only on p : γ₁ = ; and if r > 0, p = 0, then the decay rate depends only on r : γ₁ = .

If N = 2, r > 0 and p > 0 then γ₁ = . If N = 2, r = 0 and p > 0 then the decay rate is exponential.

Case b: r> 0 and -1 < p < 0 and N > 2:

In this case we have

Then, in the same way as in case (a), we get

Now, assuming that N > 3 and considering (6.21), (7.5) and the definition of D₂(t) we have

If we define

then we have 0 < K₂ < 1 and

.

Similarly to the case (a), using Nakao's Lemma, we obtain

with

From (7.6), it follows that if r = 0 then γ₂ = .

When N = 2, (6.21), (7.5) and the definition of D₂(t) imply that

If r = 0, then E(t) < C ΔE. Thus, by Nakao's Lemma, we obtain exponential decay of the energy.

If r > 0, we obtain

with

Applying Nakao's Lemma, we obtain

with γ₂ = .

Case c: -1 < r < 0 and 0 < p < (-1 < r < 0, p > 0 if N = 2):

We have

and it follows that

Thus, from (6.21), (7.8) and the definition of D₃(t), we get

Therefore

with

is such that 0 < K₃ < 1.

We conclude by Nakao's Lemma,

with

From (7.10) it follows that γ₃ = if p = 0 and N > 3 or N = 2 and p > 0.

Case d: -1 < r < 0, -1 < p < 0 and N > 2:

We have

Assuming that N > 3, it follows from (6.21), the definition of D₄(t) and the previous inequality that

Thus, we get

with

Then, using Nakao's Lemma, we obtain

with

Now we consider N = 2. It follows from (6.21), the definition of D₄(t) and the inequality at the beginning of this case that

Thus,

with K₄ = .

Then, using Nakao's Lemma, we obtain

with γ₄ = .

Now, the proof of the Theorem 3.7 is complete.

□

Acknowledgments. The authors would like to thank the referees for thevaluable comments and important suggestions which have substantially improved this paper.

Received: 21/IX/07. Accepted: 22VII/08.

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Corresponding author.

Publication Dates

• Publication in this collection
  29 Oct 2008
• Date of issue
  2008

History

• Received
  21 Sept 2007
• Accepted
  22 July 2008