Energy decay rates for solutions of Maxwell s system with a memory boundary condition


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1 Collect. Math. vv, n (yyyy, 6 c 7 Universitat de Barcelona Energy decay rates for solutions of Maxwell s system with a memory boundary condition Serge Nicaise Université de Valenciennes et du Hainaut Cambrésis LAMAV, Institut des Sciences et Techniques de Valenciennes 5933 Valenciennes Cedex 9 France Cristina Pignotti Dipartimento di Matematica Pura e Applicata, Università di L Aquila Via Vetoio, Loc. Coppito, 67 L Aquila, Italy Received November 6, 6. Revised April 3, 7 Abstract We consider the stabilization of Maxwell s equations with space variable coefficients in a bounded region with a smooth boundary, subject to dissipative boundary conditions of memory type on the boundary. Under suitable conditions on the domain and on the permeability and permittivity coefficients, we prove the exponential/polynomial decay of the energy. Our result is mainly based on the use of the multipliers method and the introduction of a suitable Lyapounov functional.. Introduction Let R 3 be an open bounded domain with a smooth boundary. In the domain, we consider the homogeneous Maxwell s system D curl (µb = in (, (. B curl (λd = in (, (. div D = div B = in (, (.3 D( = D and B( = B in (.4 λµd τ (t = k B(t ν t k(sb(t s νds on (,, (.5 Keywords: Maxwell s equations, variable coefficients, memory boundary conditions, stabilization. MSC: 93D5, 93D5, 35L.
2 Nicaise and Pignotti where D, B are threedimensional vectorvalued functions of t, x = (x, x, x 3 ; µ = µ(x and λ = λ(x are scalar functions in C ( bounded from below by a positive constant, i.e., λ(x λ >, µ(x µ >, x ; (.6 D, B are the initial data in a suitable space. Moreover, in the boundary condition (.5, ν denotes the outward unit normal vector to the boundary, k is a positive constant, k : [, R is a positive function of class C and D τ denotes the tangential component of the vector field D, that is D τ = ν (D ν. The integral boundary condition (.5 describes the memory effect, it means that the boundary is a medium with a high but finite electric conductivity [8, 3, ]. Frictional dissipative boundary condition (i.e. the case k = in (.5 was studied by many authors, see [7,, 6, 7,,, 4, 5]. On the contrary for boundary conditions with memory, only a few number of papers exists [8, 3, ]. In these papers, the authors consider Maxwell s equations with constant coefficients (or piecewise constant and prove the exponential decay of a suitable energy by combining the multipliers method with the use of Pazy s theorem. The main goal of the present work is to extend the previous results to the case of variable coefficients by using the multipliers method and by introducing a suitable Lyapounov functional. Note further that our choice of the energy allows to give a simple proof of the exponential decay. Under appropriate assumptions on k, we also prove the polynomial decay of the energy. Similar results already exist for the (linear or nonlinear wave equation with internal or boundary conditions with memory, let us quote [,, 5, 3, 4, 9, 4, 6, 7, 8]. These authors prove the exponential or polynomial decay of the energy by combining the multipliers method with the use of a suitable Lyapounov functional or integral inequalities. For semilinear problems blow up phenomena may occur, see for instance [5]. Note that the Lyapounov functional method was introduced by G. Chen [6] for the wave equation and by A. Haraux [] for semilinear hyperbolic problems with internal damping. It was further developed and adapted to boundary feedbacks by E. Zuazua and coauthors in [, 3, 8, 3]. The paper is organized as follows: Wellposedness of the problem is analysed in Section under appropriate conditions on k using semigroup theory. We further show that the same conditions on k guarantee the decay of the energy. Under some geometric assumptions on the domain and additional assumptions on k, we show in Section 3 the exponential stability of our system and in Section 4 the polynomial decay of the energy. Finally in Section 5 we end up with some examples, where the geometric assumptions are illustrated.. Wellposedness of the problem In this section, we consider the existence and regularity of solutions to our system (.(.5. We further show that our system is dissipative by proving that
3 Maxwell s system with a memory boundary condition 3 its energy decays. For that purpose, the function k( in the boundary condition (.5 needs to satisfy the following assumptions: k (t, k (t. (. Note that these assumptions are relatively standard, see [8, ]. With our assumptions (., problem (. (.5 can be formulated as a standard evolution equation u Au =, u( = u [] with an appropriate operator A, which generates a strongly continuous semigroup (proved using the same techniques as the ones from [, Theorem.]. This allows to obtain the following result: Theorem. Let the above assumptions on k be satisfied. Then, for all initial data D, B L ( 3 such that div D = div B = in, curl (µb, curl (λd L ( 3 and satisfying the compatibility condition λµd τ = k B ν on, there exists a unique solution (D, B of (. (.5 with the regularity D, B C (R ; L ( 3, curl (µb C(R ; L ( 3, curl (λd C(R ; L ( 3. We define the energy of our system (. (.5 by E(t := { λ D(t µ B(t } dx t k s (s t k(t B(t τ νdτ B(t τ νdτ d. dsd (. Note that our definition differs from the one in [3, ] and is inspired from the definition of the energy for the wave equation with memory boundary conditions [,, 5, 9, 7]. Let us further denote by E( the standard energy E(t := (λ D µ B dx. (.3 Proposition. For any regular solution (D, B to problem (. (.5, one has E (t = k B(t ν d k t (t B(t τ νdτ d t k s (s B(t τ νdτ dsd. (.4
4 4 Nicaise and Pignotti Proof. Differentiating (. we have E (t = {λdd µbb }dx t ( s ( s k (s B(t τ νdτ ( t ( t k(t B(t τ νdτ B (t τ νdτ k(t ( t B(t τ νdτ B( νd. B (t τ νdτ dsd d Then, using (., (. and integrating by parts, we get E (t = λµ(d ν Bd t ( s k (s B(t τ νdτ [B(t B(t s] νdsd ( t k(t B(t τ νdτ B(t νd. (.5 Identity (.5 can be rewritten as E (t = λµ(d ν Bd t ( s B(t ν k (s t s B(t τ νdτ dsd k (s d B(t τ νdτ dsd ds ( t k(tb(t ν B(t τ νdτ d, from which follows, integrating by parts, E (t = λµ(d ν Bd t ( s B(t ν k (s t s t ( t k (s k (t k(tb(t ν B(t τ νdτ dsd B(t τ νdτ dsd B(t τ νdτ d B(t τ νdτ d. Now, observe that the boundary condition (.5 can be rewritten as t λµd τ (t = k B(t ν k(tb(t τ νdτ t ( s k (s B(t τ νdτ ds. (.6 (.7
5 Maxwell s system with a memory boundary condition 5 Therefore, using (.7 and observing that (D ν B = D τ (B ν, we have ( t λµ(d ν Bd = k(tb(t ν B(t τ νdτ d t ( s B(t ν k (s B(t τ νdτ dsd k B ν d, that substituted in (.6 gives (.4. (.8 Our next purpose is to find sufficient conditions on k, on the coefficients λ, µ and on in order to guarantee the exponential/polynomial decay of the energy. 3. Stabilization result: exponential decay We assume that there exist a C vector field q and ρ > such that and λdiv q ξ λ µdiv q ξ µ 3 i,k= 3 i,k= q i x k ξ i ξ k q λ ξ ρλ ξ, x, ξ R 3, (3. q i x k ξ i ξ k q µ ξ ρµ ξ, x, ξ R 3. (3. Obviously, the vector field q and the parameter ρ may depend on the coefficients λ and µ and on the geometry of (see Section 5. We further assume that q ν > on. Therefore, for a suitable positive constant δ, one has q ν δ on. (3.3 Remark 3. Conditions (3. and (3. impose some restrictions on the coefficients λ and µ. They are more restrictive than the one taken in [3] using microlocal analysis and differential geometry arguments. Nevertheless in some particular cases, these conditions may become necessary according to differential geometry approach (see Example 3 below. We refer to the papers of Yao [9] and Macià and Zuazua [9] for the case of scalar wave equations with variable coefficients. The condition (3.3 has a different nature, it is a geometrical condition on. For the standard multiplier q(x = x x, it means that is strictly starshaped with respect to x. Moreover we make the following assumptions on the function k in the boundary condition (.5: for positive constants γ, γ. k(t, k (t γ k(t, k (t γ k (t, t >, (3.4
6 6 Nicaise and Pignotti Remark 3. Note that the assumptions (3.4 imply that the functions k and k are exponentially decaying to. These assumptions are quite reasonable, since in [3], the authors take k(t = exp ( σ(xt, for some σ(x >, while in [] the condition k H (R combined with their condition (. leads to (3.4. Now we recall a standard identity with multipliers: Proposition 3.3 Let (D, B be a regular solution of problem (. (.4 and let q be a C vector field. Then the following identity holds: (B D qdx = (div q(λ D µ B dx 3 q i (λd i D k µb i B k dx x k i,k= {(q λ D (q µ B }dx {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d. (3.5 Proof. See [5, Lemma.] that extends to variable coefficients λ, µ a previous identity proved by Komornik [6]. Remark 3.4 Note that identity (3.5 holds without assuming any particular boundary condition. We can estimate the boundary terms in (3.5 as follows. Lemma 3.5 Assume that (3.4 holds. Let (D, B be a regular solution of problem (. (.5 and let q be a C vector field verifying (3., (3., (3.3. Then, {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d C {k B(t ν t d k(t B(t τ νdτ d (3.6 t k s (s B(t τ νdτ } dsd, for a suitable positive constant C. Proof. Denoting by D ν and B ν the scalar normal components of D and B we have {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d = {(λ D µ B (3.7 (q ν µ(q BB ν λ(q DD ν }d.
7 Maxwell s system with a memory boundary condition 7 We can rewrite (3.7 as {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d = {(λ D τ λdν µ B τ µbνq ν q[µ(b ν ν B τ B ν λ(d ν ν D τ D ν ]}d = (q ν(λdν µbνd {(λ D τ µ B τ (q ν µq B τ B ν λq D τ D ν }d, (3.8 where D τ and B τ are the tangential parts of D and B. From (3.8, applying Young s inequality and recalling assumption (3.3, we obtain {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d C { D τ B τ (3.9 }d, for a suitable constant C >. Then, using the boundary condition (.7 in (3.9, we have {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d { C B(t ν d k t (t B(t τ νdτ d t s k (s B(t τ νdτds } d. (3. Using CauchySchwarz s inequality and the assumptions on k, we can estimate { t k s } (s B(t τ νdτds t t [ k (s]ds [ k s (s] B(t τ νdτ ds (3. t C [ k s (s] B(t τ νdτ ds. Therefore, using (3. and (3.4, inequality (3. implies {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d C {k B(t ν t d k(t B(t τ νdτ d t k (s s B(t τ νdτ } dsd, for a suitable constant C >.
8 8 Nicaise and Pignotti By the estimate (3.6 and the assumptions (3., (3. on q, 3.5 gives (B D qdx ρ (λ D µ B dx C {k B(t ν t d k(t B(t τ νdτ d t k (s s B(t τ νdτ } dsd. (3. Now, we can give the exponential stability result (compare with [5, Lemma.5]. Theorem 3.6 Assume that (3., (3., (3.3 and (3.4 hold. Then, for any solution (D, B of problem (. (.5 (in the sense of Theorem., we have for suitable positive constants C, C. E(t C E(e C t, t >, (3.3 Proof. We first consider a regular solution (D, B of problem (. (.5. By assumption (3.4, (.4 implies E (t k γ B(t ν d γ t k(t B(t τ νdτ t k s (s B(t τ νdτ dsd. Define the Lyapounov functional Ẽ(t := E(t ˆγ d (3.4 (B D qdx, (3.5 where ˆγ is a positive constant chosen sufficiently small later on and q a C vector field verifying (3., (3. and (3.3. Differentiating (3.5 and using (3., we have d dtẽ(t ˆγρE(t k B(t ν d γ t k(t B(t τ νdτ d 4 γ t k s (s B(t τ νdτ dsd, 4 for ˆγ sufficiently small. By the definition of the energy E, we obtain for a positive constant c. d dtẽ(t c E(t, (3.6
9 Maxwell s system with a memory boundary condition 9 Now, to conclude it suffices to observe that for ˆγ sufficiently small, there exist positive constants c, c such that Hence by (3.7 and (3.6, we get c E(t Ẽ(t c E(t. (3.7 d CẼ(t, (3.8 dtẽ(t which means that Ẽ is exponentially decreasing. The claim follows by the first inequality in (3.7 for a regular solution. Since the estimate (3.3 is valid for finite energy solutions, by density it holds for any solution in the sense of Theorem.. Remark 3.7 Note that all the above results still hold if the function k depends also on the space variable, that is k := k(x, t, k : [, R, k C ( [,. The same remark applies to the following section. We have considered k := k(t only for the sake of simplicity. 4. Stabilization result: polynomial decay In this section we assume that the function k in the boundary condition (.5 verifies k(t, k (t γ [k(t] /p, k (t γ [ k (t] /(p, t >, (4. for positive constants γ, γ and for some real number p >. Note that the assumptions (4. imply that the functions k and k are polynomially decaying to as /( t p and /( t p respectively. Instead of Lemma 3.5 we need, in this case, the following result. Lemma 4. Assume that (4. holds. Let (D, B be a solution of problem (. (.5 and let q be a C vector field verifying (3., (3., (3.3. Then, there exist a positive constant C such that {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d C {k B(t ν d [k(t] /p t B(t τ νdτ d (4. t [ k (s] /(p s B(t τ νdτ } dsd.
10 Nicaise and Pignotti Proof. Using CauchySchwarz s inequality we can estimate { t k s } (s B(t τ νdτds t t [ k (s] p/(p ds [ k (s] (p/(p s B(t τ νdτ ds. (4.3 By (4. the function k is decreasing as /(t p, then t [ k (s] p/(p ds, t >, is uniformly bounded (in t. So, (4.3 and the boundedness of k allow to obtain by (3. {(λ D µ B (q ν µ(q B(ν B λ(q D(ν D}d C {k B(t ν d [k(t] /p t B(t τ νdτ d t [ k (s] /(p s B(t τ νdτ } dsd, for a suitable constant C >. By the estimate (4. and the assumptions (3., (3. on q, (3.5 gives (B D qdx ρ (λ D µ B dx C {k B(t ν d [k(t] /p t t [ k (s] /(p s B(t τ νdτ d B(t τ νdτ } dsd. (4.4 We are ready to give the stabilization result. Theorem 4. Assume that (3., (3., (3.3 and (4. hold. Let (D, B be a regular solution of problem (. (.5 and let r be such that p < r < p p. Then, there exist a positive constant C (depending non linearly on D and B such that C E(t, t >. (4.5 ( t
11 Maxwell s system with a memory boundary condition Proof. By the assumption (4. and the identity (.4 we can obtain E (t k B(t ν d γ t [k(t] /p t s [ k (s] /(p γ Let Ẽ( be the functional defined by (3.5. By (4.6 and (4.4, for ˆγ sufficiently small, we have B(t τ νdτ B(t τ νdτ dsd. d dtẽ(t ˆγρE(t k B(t ν d γ [k(t] /p t B(t τ νdτ d 4 γ t [ k (s] /(p s B(t τ νdτ dsd. 4 d (4.6 (4.7 From Hölder s inequality (see [7, Lemma 4.], we may write t and then, { [ k (s] t s B(t τ νdτ dsd { t [ k (s] /(p s { t [ k (s] r s s B(t τ νdτ dsd B(t τ νdτ dsd } ( r(p [ k (s] B(t τ νdτ dsd t [ k (s] /(p s B(t τ νdτ dsd { We can estimate t [ k (s] r s { t [ k (s] r s { t [ k (s] r( t Moreover by (.4, we observe that B(t τ νdτ dsd B(t τ νdτ dsd } ( r(p } } },. (4.8 dsd } (4.9 ( r(p B(τ ν dτ. B(τ ν d k E (τ, τ >
12 Nicaise and Pignotti and then, since the energy E( is decreasing, t B(τ ν ddτ k E(, t >. (4. Therefore, from (4. and CauchySchwarz s inequality ( t d t B(τ ν dτ t B(τ ν dτd ct, (4. for a positive constant c. Moreover, by (4. recalling that r > p, it holds [ k (s] r ds <. (4. So, by (4.9, (4. and (4., we conclude that { t [ k (s] r s B(t τ νdτ dsd } ( r(p ct (4.3 Therefore, from (4.8 and (4.3, we have for all t >, { t [ k (s] ct s B(t τ νdτ dsd } t [ k (s] /(p s B(t τ νdτ dsd, (4.4 for a suitable positive constant c. By (4.4 and (4.7 it follows that, for all t >, Ẽ (t ct { E(t t ( t [ k (s] for a suitable constant c >. Now, observing that s [k(t] /p t }, B(t τ νdτ dsd ( r(p > p, B(t τ νdτ d (4.5
13 Maxwell s system with a memory boundary condition 3 using Hölder s inequality we can estimate { k(t t B(t τ νdτ d [k(t] { { t { [k(t] /p t { ct t t } t B(t τ νdτ d B(t τ νdτ } d } B(t τ νdτ } d B(t τ ν dτd [k(t] /p t } B(t τ νdτ d, where, in the last inequality, we have used (4.. Then, since E(t is bounded ( E(t E(t E(, by (4.6 we obtain E(t t { { c E(t t t [k(t] p t s t s [ k (s] k(t [ k (s] c[e(t], B(t τ νdτ d B(t τ νdτ dsd B(t τ νdτ d B(t τ νdτ dsd } } (4.6 (4.7 for some positive constant c. Therefore, by (4.5 and (4.7, Ẽ (t ct [E(t], for t >, which gives, for a suitable positive constant c, Ẽ (t ct [Ẽ(t], for t >. This implies that Ẽ is decaying to zero as /( t and so, (4.5 holds reminding that E(t cẽ(t.
14 4 Nicaise and Pignotti 5. Examples Let us finish our paper with some examples that illustrate the assumptions (3., (3. and (3.3. Example. If we take λ and µ verifying for any x, λ(x λ(x (x x cλ(x, (5.8 µ(x µ(x (x x cµ(x, (5.9 for a given point x and a positive constant c, then (3. and (3. hold for the standard multiplier q(x := x x. In particular (5.8 and (5.9 hold if λ(x (x x, µ(x (x x, x. If the domain is strictly starshaped with respect to x, then (3.3 also holds. Example. If λ µ and q := λ (x(x x, then (3. and (3. become λ (x x ξ λ ξ ( λ ξ((x x ξ ρ ξ, x, ξ R 3. In particular, this is verified if x x λ < 5 (inf λ. As λ is always supposed to be positive in, the condition (3.3 holds if the domain is strictly starshaped with respect to x. Example 3. equivalent to that is If λ µ x and if we take q := x, then (3. and (3. are ( x ξ x ( x ξ ρ( x ξ, ( x ξ ρ( x ξ. Consequently if { x R 3 : x < }, then there exists ρ > such that this condition is satisfied. Moreover the condition (3.3 will hold if the domain is strictly starshaped with respect to. For this example note that if contains the unit sphere, i.e., { x R 3 : x }, then (3. and (3. reveal to be necessary because the wave equation u tt div (λµ u is not exactly controllable from the boundary since the unit sphere is a close geodesic (see [9]. From these examples we see that our results are limited by the conditions (3., (3. and (3.3. To extend such results to a larger class of coefficients λ and µ and allow less geometrical constraints on the domain, a combination of microlocal analysis and differential geometry should be used. But, in authors opinion, this combination does not work for boundary conditions with memory.
15 Maxwell s system with a memory boundary condition 5 References. M. Aassila, M.M. Cavalcanti, and J.A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a starshaped domain, SIAM J. Control Optim. 38 (, D. Andrade and J.E. Muñoz Rivera, Exponential decay of nonlinear wave equation with a viscoelastic boundary condition, Math. Methods Appl. Sci. 3 (, M.M. Cavalcanti, V.N. Domingos Cavalcanti, and M.L. Santos, Uniform decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, in System modeling and optimization, IFIP Int. Fed. Inf. Process. 66, 39 55, Kluwer Acad. Publ., Boston, MA, M.M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential Integral Equations 8 (5, M.M. Cavalcanti and H.P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 4 (3, 3 34 (electronic. 6. G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. (9 58 (979, M. Eller, J.E. Lagnese, and S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math. (, M. Fabrizio and A. Morro. A boundary condition with memory in electromagnetism, Arch. Rational Mech. Anal. 36 (996, A. Guesmia, Stabilisation de l équation des ondes avec conditions aux limites de type mémoire, Afrika Math. (999, A. Haraux, Semilinear hyperbolic problems in bounded domains, Math. Rep. 3 (987, i xxiv and 8.. A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal. (988, B.V. Kapitonov, Stabilization and exact boundary controllability for Maxwell s equations, SIAM J. Control Optim. 3 (994, B.V. Kapitonov and G.P. Menzala, Uniform stabilization for Maxwell s equations with boundary conditions with memory, Asymptot. Anal. 6 (, M. Kirane and N.E. Tatar, A memory type boundary stabilization of a mildly damped wave equation, E. J. Qual. Theory Differ. Equ., 6 (999, M. Kirane and N.E. Tatar, Nonexistence results for a semilinear hyperbolic problem with boundary condition of memory type, Z. Anal. Anwendungen 9 (, V. Komornik, Boundary stabilization, observation and control of Maxwell s equations, Panamer. Math. J. 4 (994, V. Komornik, Exact controllability and stabilization. The multiplier method, Masson, Paris; John Wiley and Sons, Ltd., Chichester, V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69 (99, F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal. 3 (, 6.. R. Nibbi and S. Polidoro, Exponential decay for Maxwell s equations with a boundary memory condition, J. Math. Anal. Appl. 3 (5, S. Nicaise, M. Eller, and J.E. Lagnese, Stabilization of heterogeneous Maxwell s equations by linear or nonlinear boundary feedback, Electron. J. Differential Equations (, 6 (electronic.. S. Nicaise and C. Pignotti, Boundary stabilization of Maxwell s equations with spacetime variable coefficients, ESAIM Control Optim. Calc. Var. 9 (3, (electronic. 3. S. Nicaise and C. Pignotti, Internal and boundary observability estimates for heterogeneous Maxwell s system, Appl. Math. Optim. 54 (6, K.D. Phung, Contrôle et stabilisation d ondes électromagnétiques, ESAIM Control Optim. Calc. Var. 5 (, (electronic.
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