Indirect linear locally distributed damping of coupled systems

abstract: The aim of this paper is to prove indirect internal stabilization results for different coupled systems with linear locally distributed damping (coupled wave equations, wave equations with different speeds of propagation). In our case, a linear local damping term appears only in the first equation whereas no damping term is applied to the second one (this is indirect stabilization, see [11]). Using the piecewise multiplier method we prove that the full system is stabilized and that the total energy of the solution of this system decays polynomially.


Introduction and main result
Let Ω be a non-empty bounded open set in IR N of class C 2 and Γ = ∂Ω its boundary and let a ∈ C 0 (Ω) be a positive function in Ω.
We consider the following system of two coupled wave equations with homogeneous Dirichlet conditions on the boundary : In this paper, we only consider a feedback which depends on the velocity in a linear way, i.e. a(x)u 1 .One can remark that this local damping term appears only A. BEYRATH in the first equation.Our purpose is to prove that the full system is polynomially stable, i.e. that the energy of the solution decays polynomially for sufficiently smooth initial data.We refer to [3] for the proof that this system is not exponentially stable.
The problem of stabilization of the wave equation in a bounded domain using a locally distributed damping has been studied by several authors.When the feedback depends on the velocity in a linear way, Zuazua [14] proved thanks to the multiplier method that the energy decays exponentially when the damping region contains a neighborhood of ∂Ω or a neighborhood of Γ(x 0 ) := {x ∈ ∂Ω, (x−x 0 ).ν ≥ 0}, where ν is the outward unit normal to Ω and x 0 ∈ IR N .Martinez [12], using the piecewise multiplier method introduced by Liu [11], weakened the usual geometrical conditions on the localization of the damping.
The problem of stabilization of weakly coupled systems has also been studied by several authors.Under certain conditions imposed on the subset where the damping term is effective, Kapitonov [5] proves uniform stabilization of the solutions of a pair of hyperbolic systems coupled in velocities.Alabau and al. [3] studied the indirect internal stabilization of weakly coupled systems where the damping is effective in the whole domain.They prove that the behavior of the first equation is sufficient to stabilize the total system and to have polynomial decay for sufficiently smooth solutions.Alabau [1] proves indirect boundary stabilization (polynomial decay) of weakly coupled equations.She establishes a polynomial decay lemma for non-increasing and nonnegative function which satisfies an integral inequality.
Our purpose in this paper is to study the indirect internal stabilization of coupled systems with a local damping term applied only to the first equation and to prove that the full system is polynomially stabilized.We therefore generalize the result of [3] to the case of a non-coercive feedback operator in the case of wave equations.
We denote by A the unbounded operator in the energy space : The problem (1.1) can then be reformulated under the abstract form The existence and the regularity of the solution of (1.1) is given by the following theorem.
Theorem 1 For all Proof : To establish the well-posedness of our problem, we prove that A is a maximal monotone operator in the energy space i.e.A is monotone and The proof is standard and thus is left to the reader.
We define the partial energies : and the full energy : Let us denote by C generic positive constants which do not depend on the initial data.α is supposed to be here a sufficiently small positive number.However the results are valid for sufficiently small negative α as well.
We make the following geometric assumptions.These assumptions have been introduced in [11] for the piecewise multiplier method (see also [12]).
• Let ω be an open subset of Ω.We denote by ν the outward unit normal vector to its boundary.If Ω j ⊂ IR N is a lipschitz domain, then we denote by ν j the outward unit normal vector to its boundary ∂Ω j .
• Assumptions on the local damping : a ∈ C 0 (Ω) and there exists a constant γ which satisfies : Geometric assumptions for Ω and ω : We assume that there exist > 0, domains Ω j ⊂ Ω, 1 ≤ j ≤ J with a lipschitz boundary ∂Ω j and points x j ∈ IR N so that : We obtain then our main result : , the full energy of the solution U of system (1.1) decays polynomially, i.e.
Moreover if the initial data are more regular, i.e.U 0 = (u 0 1 , u 0 2 , u 1 1 , u 1 2 ) ∈ D(A n ) for a certain positive integer n, then the following inequality holds :

Proof of the main result
We first verify that (1.1) is a dissipative problem : Lemma 2.1 For every initial data in H, the energy of the corresponding solution center Proof : This is a well-known result.Let U 0 ∈ D(A) and denote by U the solution of (1.1).Differentiating E 1 (u 1 (t)) and E 2 (u 2 (t)) and using the first and second equations of (1.1), we have : which leads to : Then integrating on [S; T ], we obtain : By density of D(A) in H, this result holds for every U 0 ∈ H.One can prove the following elementary inequality between E 1 (u 1 , t), E 2 (u 2 , t) and E(U, t) for α sufficiently small.Its proof is left to the reader : Lemma 2.2 For every initial data in H and for every U solution of the system (1.1), we have the following estimates, provided that α is sufficiently small To prove Theorem 2, we first estimate ) dt using piecewise multiplier techniques to obtain Proposition 1. Then we estimate easily Finally, summing these two estimations, we use the result of Alabau [1] to conclude.
Proposition 2.1 Assume the hypothesis H 1 .Then there exists a constant C > 0, such that for all We need several intermediate steps to prove this estimate.The first proposition is based on the use of the piecewise multiplier method for a single wave locally damped equation.
The multiplier method was introduced by K. Liu [11] for a single wave damped equation when the damping term is locally distributed.P. Martinez [12] weakens the usual geometrical conditions on the localization of the damping.

Step 1. We first prove an intermediate estimation concerning the partial energy
Then the following inequality holds Proof : Let Θ be an open subset of Ω and h a vector field of class C 1 from Θ to IR N .We set M (u 1 ) = 2h.∇u 1 + (N − 1)u 1 .As usual, we use the multiplier M (u 1 ) in the first equation (1.1) and integrate on [S; T ] × Θ.
In the different following estimates, we omit to write the differential elements to simplify the expressions.
= 0, we integrate by parts to obtain : Hence after integrating by parts and Considering all the boundary integrals on the left-hand side, we deduce the following equality : The main problem is to estimate the boundary terms of previous equality.Usually, we choose Θ = Ω and an adequate vector field h so that the estimate of the boundary integral is possible using the boundary condition : it is easy on the part of the boundary where {u = 0} ∩ {m.ν ≤ 0} and on the other part, we choose h = 0.In our case, we estimate the boundary integral on each subset Ω j .We use the vector field h j as introduced in Martinez in [12].
We choose Θ = Ω j and h First we show that the boundary integral in the above expression is negative.We remark that by construction: We deduce that the boundary integral term in (2.4) is equal to: By definition of Γ j (x j ), we deduce that T S ((∂Ωj \Γj (xj ))∩∂Ω) Ψ j (m j .νj )(∂ν j u 1 ) 2 ≤ 0.
Hence, since Ψ j = 0 on Q 0 , using the previous inequality in (4) and summing the resulting inequalities on j, we obtain : Using now the definition of h, Computing (2.3) with Θ = Ω and (2.5), we have : Using then the definition of partial energy E 1 (u 1 ), we obtain the following inequality : We now estimate the right-hand side terms as follows : It is easy to verify that ||M (u and thus we have : where C is a positive constant.We can then replace the second term of inequality We estimate now the third term of (2.6) as follows : where C is a positive constant and α is supposed a sufficiently small positive number.
Using the same argument, we estimate the last term of (2.6) by Choosing now α and δ small enough, we conclude the proof of Proposition 2.

2.2.
Step 2. In this section, we want to get rid of |∇u 1 | 2 in the estimate of Proposition 2.
Lemma 2.3 There exists a positive constant C such that, for all δ > 0 : Multiplying the first equation of the initial system by ξu 1 and applying the Green's formula, we get : 1 ∆ξ dx dt and so we have Finally, using the definition of the function ξ, we obtain the following inequality : We need to estimate every term of the right-hand side of this inequality to prove Lemma 3.
The first term of (2.7) can be easily estimated as follows : For the second term of (2.7), using the definition of ξ, we obtain : For the last term : Hence, using these estimates in (2.7), we get the inequality announced in Lemma 3.

Step 3. Now we want to get rid of the new term
Lemma 2.4 There exists a positive constant C such that, for all η > 0 : Multiplying the first equation of initial system (1.1) by z, we have : Then, using the boundary conditions and the system verified by z, we have : We first give some well-known results which will be used to estimate the different right-hand side terms of inequality (2.8).

A. BEYRATH
For each t, we consider the solution z of the following elliptic problem : in Ω z = 0 on ∂Ω Hence using the Green's formula, we have : Then using Poincaré's inequality, we have : .
In a similar way with the derived system, we obtain : 1 dx.We are now ready to estimate the different right-hand side terms of inequality (2.8) as follows.
Thanks to , we get : 2  1 dx and Young's inequality, we have for every η > 0 : For the third term of (2.8), we have : At last using Poincaré's inequality, we have : Choosing α small enough and using these estimates in (2.8), we obtain : u 2 1 dx dt, we obtain the announced inequality of Lemma 4.

2.4.
Step 4. We can now conclude the proof of Proposition 1.
u 2 1 dx dt.Using both lemmas 3 and 4 in Proposition 2, we easily obtain : Finally, for cη < 1, we have proved the claim announced in Proposition 1, that is : Step 5. Let us now estimate the three integral terms on the right hand side of Proposition 1 to obtain a first inequality between the partial energy E 1 (u 1 (t)) and the full energy E(U (t)).

A. BEYRATH
The estimation of C T S ω u 2 1 dx dt is easy because we are on the subset where the damping is effective : on ω, a(x) ≥ γ > 0. Then : Integrating on [S; T ], we obtain : Thus we have : We estimate the second term as follows : (2.9) To estimate the term C T S Ω u 2 2 dx dt, we multiply the first equation of the initial system (1.1) by u 2 and the second equation by u 1 .We compute then and integrate on Ω×[S; T ] and thanks to the initial data we obtain the following equality : We estimate as usual the right hand side terms : Let us send now the term with u 2 to the left hand side to obtain : Hence using (2.9) we obtain : Computing (2.10) in Proposition 1, we get that : Hence using Lemma 2 and for α small enough we obtain the announced result : 2.6.Step 6.Let us now give an estimation of the second partial energy It is easy to obtain this estimation because the damping term doesn't appear on the second equation of the initial system.Multiplying the second equation of system (1.1) by u 2 , integrating on Ω×[S; T ] and using Green's formula, we have : Using then the definition of E 2 (u 2 (t)) and for α small enough, we get that : Let us estimate the right-hand side terms : Using the fact that the full energy is non-increasing, we obtain easily that : For the second and third terms, we have : The only term whose estimation is not easy is the last term T S Ω u 2 2 dx dt : Using inequality (2.10) with the derivatives, we obtain : The second and third terms of the right-hand side are easily estimated by C α 2 E(U (S)).
For the first term, we use (2.12) to obtain : C Using this last estimate in (2.13), we obtain for sufficiently small α : The constant of the last term does not depend on α, so we need to use (2.12) to estimate this term and finally we obtain : We can now complete the proof of Theorem 2. We add both estimates of the partial energies E 1 (u 1 (t)) and E 2 (u 2 (t)) obtained in steps 5 and 6, and so we have : Using then the following result of Alabau [1], we prove the polynomial energy decay of the solution of the system (1.1) : If E is a non-increasing function which verifies (13) for all U ∈ D(A), then the full energy of the solution U of system (1.1) decays polynomially, i.e.
∀ t > 0, E(U (t)) ≤ C t (E(U (0)) + E(U (0))) Moreover if the initial data are in D(A n ) for a certain positive integer n, then the following inequality holds: This completes the proof of Theorem 2. Remark : We obtain the same result for the system of two coupled wave equations with different speeds of propagation with locally distributed damping.
Let Ω be a non-empty bounded set in IR N of class C 2 and Γ = ∂Ω its boundary.
where a ∈ C 0 (Ω) is a positive function in Ω, c 1 and c 2 are two different constants in IR * .