We consider a coupled semilinear wave system posed in an inhomogeneous medium, with smooth boundary, subject to a nonlinear damping distributed around a neighborhood of the boundary according to the Geometric Control Condition. We show that the energy of the coupled system goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase-space. The approach involves refined techniques of microlocal analysis and follows ideas due to Burq and Gérard given in (Burq and Gérard (2001)).
This article addresses the uniform stability of a semilinear system of two coupled wave equations posed in an inhomogeneous medium and subject to a locally distributed nonlinear damping
where Ω is a bounded domain of (), with smooth boundary , and , are smooth functions and satisfy,
where , , α, β are positive constants and is a symmetric positive-definite matrix.
We shall assume the following conditions on f and h.
The nonlinear terms f and h are real-valued functions in which satisfy
which implies that, for a certain constant ,
Their primitives and verify
where , , and is the positive principal eigenvalue of the corresponding linear problem
In addition,
We shall assume the following conditions on g.
We consider a monotone increasing function such that
in which k and K are positive constants.
Inspired in [3 ,4] and [31], let ϕ be concave, strictly increasing function, with , such that
We could consider two different damping terms, and . However, to simplify the calculations, we decided to consider only one. When we have two different damping terms, the decay rate is given by the worst (slowest) scenario (see [3,4]).
We denote by ω a neighborhood of , which is the intersection of Ω with a neighborhood of in .
The nonnegative functions, responsible for the localized dissipative effect, satisfy the following conditions:
The function and satisfies
Let us assume that the following assumptions are also made:
ω geometrically controls Ω, i.e. there exists , such that every geodesic of the metric , where travelling with speed 1 and issued at , intersects ω in a time .
For every , the only solution v lying in the space , to system
where , is the trivial one .
Assumption 1.5 is the so called Geometric Control Condition (G.C.C.). It is well-known that it is necessary and sufficient for stabilization and control of the linear wave equation (see [9,13,16,17,22,40] and the references therein). In this direction, it is worth mentioning the work due to Betelú, Gulliver and Littman [10] who have discussed the question of closed geodesics in the interior of a region Ω. Such closed geodesics make control impossible since they are bicharacteristics that never reach the controlled boundary. In this paper, they show that, in the two-dimensional case, that the lack of closed geodesics in the interior is also sufficient for control. For this reason and since in the present article we do not have any control of the geodesics, because of the inhomogeneous medium, we consider ω a neighborhood of the whole boundary and to give conditions on the metric so that every geodesic of G enters the set ω, in a time . In a general setting, we must assume that and, also, we must assume the following conditions:
, ,
For all geodesic of the metric , with , there exists such that .
Notice that, if , then condition (i) above implies condition (ii), since the geodesics are straight lines. For a general metric, condition (i) can be verified without (ii) holds, if, for instance, G admits a trapped geodesic (see Fig. 1).
Function a is positive on the blue region around the boundary. Figure on the left side shows the geodesics (which are straight lines) when . Figure on the right side shows a trapped geodesic that does not meet the damped area in blue, which violates the (G.C.C.).
When we do not have any control on the geodesics of the metric , we have to assume damping everywhere on Ω, satisfying the assumptions (i) and (ii) above mentioned. The best way of doing so is to follow ideas firstly introduced in Cavalcanti et al. [16,17], namely, and , in a neighborhood, ω, of the boundary , while and in , where , for an arbitrary (see Fig. 2).
The demarcated region ω (in blue) and (in black) illustrates the damped region on the manifold , which can be considered with measure as small as desired, however totally distributed on Ω. The demarcated region (in white) illustrates the region without damping with measure arbitrarily large but also totaly distributed in Ω.
In the worst scenario aforementioned, it is possible to prove that every geodesic of the metric G meets the damped area (see Lemma 6.1 in Bortot et al. [12]). This ensures that every geodesic of the metric G enters in the effective dissipative region, or in other words there are no geodesic ‘trapped’ within the free sets of dissipative effects , for all .
It is important to observe that Assumption 1.5 is not obviously fulfilled for every matrix (For instance, the equator in the unit sphere is a periodic geodesic and we can consider that contains the equator as a domain in endowed with a Riemannian metric G). We shall give in the appendix of this manuscript examples where this situation occurs. An easy one happens when . In this case the geodesics are straight lines and necessarily they will meet the region ω.
For and considering , Assumption 1.6 is satisfied by the work of Ruiz [41]. According to Koch and Tataru [30] (see Theorem 8.15), in the more general case where and G may not be the identity, the unique continuation result follows locally. Hence, under the conditions specified in [30], Assumption 1.6 is fulfilled.
Although we do not consider in this article the case , it can be treated analogously, since all interpolation arguments used here fit well in this case and, moreover, if .
Main goal, methodology and previous results
The main objective of the present manuscript is to prove the existence and uniqueness for weak solutions to problem (1.1) and, in addition, that these solutions decay uniformly to zero, that is, setting
then for all mild solutions to problem (1.1), with the initial data taken in bounded sets of , one has
with , where the contraction semigroup is the solution of the differential equation
in which q is given in (4.4). This result is a local stabilization result. Indeed, the decay rate estimates given by (1.10) are uniform on every ball in with radius of the energy space but the result does not guarantee that the decay rate is global, i.e. whether (1.10) holds independently of the initial data.
Instead of studying the specific problem (1.1), we shall consider an auxiliary one, which we will describe in the sequel. For this purpose, let us induce in Ω a Riemannian metric g so that is a connected compact oriented n-dimensional Riemannian manifold with metric g of class and smooth boundary . Let be the Laplace–Beltrami operator in and its Riemannian connection. Let us consider the following coupled semilinear system subject to a damping:
The energy associated to problem (1.12) is given by
Inspired in Dehman, Gérard, Lebeau, [21] or Dehman, G. Lebeau and Zuazua [22], we give a direct proof of the inverse inequality to problem (1.12), namely, we prove that given there exists a positive constant C such that
provided the initial data are taken in bounded sets of and the nonlinearities f and g satisfy Assumption 1.1.
To prove (1.14) and, therefore, the stability result, we argue by contradiction and we find a sequence of weak solutions to problem (1.12) such that . In order to obtain a contradiction we shall prove that as .
From the contradiction argument, we will have that
and
when m goes to infinity.
We then use certain techniques of microlocal analysis to propagate the convergences (1.15) and (1.16) from to the whole set . To be more specific, we consider the microlocal defect measure μ, in short m.d.m., firstly introduced by Gérard [23], associated to the solution of the linear wave equation. By using properties associated to μ, we are able to prove that μ propagates along the bicharacteristic flow of the wave operator proving the desired convergence. The same argument can be repeated for . It is important to observe that, in the present article, in order to avoid certain technicalities that may arise when considering the propagation up to the boundary, we consider ω, the region where the damping is effective, as a neighborhood of the boundary and satisfying the (G.C.C.). Under these conditions, we can use the propagation results found in [14]. If we consider ω satisfying only the (G.C.C.), then we would need to consider the propagation up to the boundary, and, as a consequence, we would need to use the results firstly proved by Lebeau in [33] and Gérard–Leichtnam [25].
Inspired in the work [6], let us come back to our original problem (1.1) having in mind that and . Fix a coordinate system on , with Riemannian metric . Denote , let be the inverse matrix of and choose . The Laplace–Beltrami operator in this coordinate system is given by
where ∇ is the usual gradient correspondent to the Euclidean metric on the domain . Consequently,
and the analysis of (1.1) is a consequence of that one derived to problem (1.12).
There are a number of publications concerning the wave equation with frictional damping of the form or in the Euclidian setting or in the Riemannian one. In [35], dissipative systems with nonlinear localized damping are considered, weakening the usual geometrical condition on the localization of the damping and eliminating the usual assumption on the polynomial growth of the feedback at zero. In Cavalcanti et al. [15,16] and [17], the authors investigated the wave equation with localized nonlinear damping. In the first paper, the authors consider compact surfaces, while the other two articles deal with compact manifolds. Alabau-Boussouira in [4] proved sharp or quasi-optimal upper energy decay rates for a class of nonlinear feedbacks ranging from very weak non-linear dissipation to polynomial or polynomial-logarithmic decaying feedbacks at the origin. For the case of damped semilinear wave equations with a nonlinearity f, as considered in the present article, with , we refer to [20,26,36], and [42]. The case has been studied by Dehman, Lebeau and Zuazua in [22] and Joly and Laurent in [27]. The critical case has been investigated in [32].
The problem of stabilization of coupled system has also been studied by several authors, as [1,2,5,18] and [29]. Alabau et al. [1] studied the indirect internal stabilization of weakly coupled system where the damping is effective in the whole domain. In [5] Alabau et al. consider the energy decay of damped hyperbolic system of wave-wave type which is coupled through the velocities; their interest is in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. Charles in [18] consider a coupled system of the wave in a one-dimensional bounded domain with nonlinear localized damping acting in their equations. Under certain conditions imposed on a subset where the damping term is efective, Kapitonov [29] proves uniform stabilization of the solutions of a pair of hyperbolic systems coupled in velocities.
The model proposed in this article is an extension for systems of the equation introduced by Cavalcanti et al. in [6], given by
in a connect compact oriented n-dimensional Riemannian manifold with smooth boundary. Inspired by works [9] and [39] the authors proved that the energy of the wave equation goes uniformly to zero for all initial data of finite energy phase-space.
Our purpose in this paper is to study the stabilization of system semilinear which is coupled through the velocities with a nonlinear localized damping acting in both equations. We shall to prove that the energy of the coupled system goes uniformly to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.
Our paper is organized as follows. In Section 2 we give some notation and we establish the well-posedness to problem (1.12). In the Section 3, Section 4 and in the appendix we give the proof of the stabilization which is our main result.
Well-posedness
We consider the Hilbert space
which is endowed with the inner product
where , , and ⊤ denotes the transpose.
If we denote , , and , then the initial boundary value problem (1.12) can be rewritten as the following first order problem:
in which and the operator is given by with component operators defined by
where , and the operator is given by
We observe that, in this case we have . In addition, the operator is given by
Now, we are ready to state the well-posedness result for problem (2.1), which ensures that problem (1.1) is globally well-posed. First we deal with the nonlinearities f, h in the conditions of Assumption 1.1, with the restriction if , where the operator defines a locally Lipschitz continuous operator. Then we consider, for the three dimensional case, the nonlinearities f, h as in Assumption 1.1 with and we observe that, the same arguments hold for the case , , the range in which Theorem 2.2 is valid, as proved in [11].
(Global Well-Posedness).
Assume that the hypotheses on the nonlinearities terms f, h specified in Assumption
1.1
are satisfied, with the restriction, if, and suppose that the initial data. Then problem (
2.1
) has a unique generalized solution. Moreover, if, then the solution is regular.
Let us show that A is maximal monotone and B is monotone, hemicontinuous and bounded, so from Corollary 1.1 in [7] we will have that is maximal monotone. In fact, since
for every , we have that A is monotone.
In order to prove the maximality of A, it is sufficient to prove that . Given , we have to show that there exists such that , that is,
and then
Defining
given by
it is easy to see that is bilinear, continuous and coercive. From the Lax–Milgram Theorem, we obtain the desired surjection. Therefore, the operator A is maximal monotone. Moreover, since and are real-valued nonnegative functions and by Assumption 1.2, we have that
Assumption 1.2 also implies that B is bounded and hemicontinuous. Then, from Corollary 1.1 in [7], it follows that is maximal monotone. Moreover, the nonlinear operator given in (2.4) is a locally Lipschitz continuous operator. Indeed, given a bounded set D in and , . Note that
Then, by Assumption 1.1, using Holder’s inequality and the fact is continuously embedded in for , we have that
and, analogously,
Hence, by Theorem 7.2 in [19], the Cauchy problem (2.1) has unique generalized solution on the interval . Furthermore, if , then the solution is regular. Let us see that . Indeed, given the energy functional defined in (1.13), it follows that
which shows that is non-increasing, with for all . On the other hand, from (1.5) we deduce
where . Thus,
Therefore the local solutions cannot blow-up in finite time and it follows that . □
In order to deal with the case , for three dimensional domains, we first recall some basic results.
(Strichartz estimates).
Letbe a compact Riemannian manifold with boundary,andsatisfyingThere existssuch that for everyand every, the solution u ofsatisfies the estimate
It follows from Theorem 2.2 and a fixed point argument that, given , fixed, and , the problem
has a unique solution in , with depending on the initial data. Moreover, the associated energy to the problem (2.10) is defined as
with and, by Gronwall’s inequality, that is, the solution cannot blow-up in finite time, which allows us to extend the solution to and, in particular, the solution is global if . Considering fixed, then for any initial data such that and we also have and the following Strichartz type estimates
for such that and and the constant depending on and Ω. (See [6]).
We shall also employ the following truncations of the source terms (first used in [37]),
where is a smooth cut-off function such that , if , if and . Then, we have the following result proved in [38].
For each, the functionsand, given in (
2.13
), define a globally Lipschitz continuous operatorwith Lipschitz constant depending on k.
Letandbe a three dimensional Riemannian manifold. Then, consideringand, there exists a unique solutionfor the problem
Let us consider the truncated functions , defined in (2.13) and take a sequence of regular initial data such that
By Theorem 7.2 in [19], for each there exists solution for the problem
Moreover, defining
where and , we have
Since controls the norm , it follows that we can extend the solution globally in time, that is . Moreover, (2.18) implies that, for every fixed, and are bounded sequences in and , respectively. It follows that there exist subsequences, still denoted by , , and a function such that
From Aubin’s lemma, we obtain
By (1.2) and (2.21), we have
Then, there exists such that
Since , it follows that the operator given by is linear and bounded. Then
At this point it is important to observe that, for every and fixed, and it can be estimated by a constant which does not depend on k; in addition, and , as .
Indeed, firstly note that, since , we can make use of Strichartz, as in Remark 5, for the first equation of problem (2.16), to obtain the estimate
Since , then and . Therefore, the space is reflexive and separable. Moreover, where . Thus, taking into account (1.4), we deduce that
Observe that, is uniformly bounded in the reflexive space , there exists a subsequence, still denoted with the same index, and a function , such that
Since , it follows that .
In the same way, since we can make use of Strichartz, as in Remark 5, for the second equation of problem (2.16), to obtain the estimate
As before, is uniformly bounded in . Thus, there exists a subsequence, still denoted with the same index, and a function such that
Then, passing the limit in (2.16) we get
with , . Applying the Strichartz estimates, separately for each equation of the problem (2.28), we have that, and
and
Since , from (2.24) we obtain that is bounded in . Taking into account the convergence (2.21), we deduce that a.e. in as . Therefore, there exists a positive contant such that for all . Hence,
That is,
This allows us to conclude that .
Now we observe that,
We first show that
Indeed, note that,
where
with .
Therefore,
Noting that , and as , from Lebesgue’s Dominated Convergence Theorem we deduce that
An analogous procedure shows that
Define for . Observe that . Applying the Lebesgue’s Dominated Convergence Theorem to the sequence we deduce that
and
Combining (2.37) and (2.38) the convergence (2.32) follows.
To show that
we first note that, since for every , we have
Using Hölder and interpolation inequalities, we obtain
with . Hence, using Hölder again with , and , we have
Indeed, from (2.23) and (2.29) we obtain , for every fixed , and by (2.21) we have .
By using this procedure to treat the other terms in (2.39), we may conclude that
The same argument shows that
Therefore,
which implies that is a Cauchy sequence in . It follows that
Analogously, we have that
and then is a Cauchy sequence in . Hence
Since a and b are non-negatives and by using (1.2), we conclude that
Defining
we have
and then, by Gronwall’s inequality,
The convergences (2.15), (2.40) and (2.42) imply
This last limit allows us to conclude that and . Indeed, first observe
Since g is a monotone increasing function, it easy to see that is a monotone and hemicontinuous operator. Therefore, is maximal monotone. Thus, with this in hand and observing that, by (2.44) we have, for every fixed
and
Taking into account the Lemma 2.3 of [8], we can conclude that and , in view of the fact that the same properties are valid for .
Observe that, for every and then the uniform convergence in (2.44) implies that and passing to the limit in (2.16) we get
Moreover, considering the energy
associated to the problem (2.48), we have and then , for every ; hence, the solution cannot blow-up in finite time, which implies that the solution can be globally extended in time.
Uniqueness. Assume that there exists another solution for the problem (2.48). Setting and , we have
It follows that
On the other hand,
Analogously, we have
Therefore, we have that
Defining
we obtain
By Gronwall’s inequality, we can conclude that
Therefore, , , which concludes the proof. □
The observability inequality
Consider the following coupled semilinear problem subject to a nonlinear damping:
The energy associated to problem (3.1) is given by
Using integration by parts, we deduce the energy identity
Before we enunciate our main result, we will prove, in the next two subsections, the following estimate:
Forand, there exists a constantsuch that inequalityholds for every weak solutionof the damped problem (
3.1
) if the initial data satisfies.
The subcritical case
By standard density arguments it is enough to work with regular solutions. Let us assume that (3.4) does not hold. Then, let be a sequence of initial data and let be the corresponding sequence of solutions, with uniformly bounded in m, satisfying
For some eventual subsequence, we have that . If and since
passing to the limit in (3.22) we infer
and for , , by (3.33) we deduce that
which implies by Assumption 1.6 that and, consequently, .
Now, let us consider , that is, . Note that we can write
Thus
Note that
since
Therefore
Analogously, we have that
So, passing to the limit in (3.22), we obtain
and for , , we infer
which implies, by Assumption 1.6, that and, consequently, . As a consequence in all the convergences in (3.25)–(3.27).
Remember that our objective is to prove that converges to zero, where
is the energy associated to system (3.1). For this purpose, consider
First, we will prove that
From de above convergences we know that
so let us consider be the microlocal defect measure (m.d.m.) associated to (which is assured by Theorem A.1 in the Appendix). Then, by (3.28), (3.32) and (3.36) and, having in mind that , we have that
From this, we derive
Taking into account that ω geometrically controls Ω, we deduce two facts:
The is contained in the characteristic set of the wave equation .
propagates along the bicharacteristic flow of this operator, which means, particularly, that if some point does not belong to the the whole bicharacteristic issued from is out of .
Indeed, from (3.43) and Theorem A.2 we deduce item (i).
Furthermore, from Proposition A.1 and Theorem A.4 found in the Appendix, we deduce that in is a union of curves like
where is a geodesic associated to the metric.
Since by (3.29) we have strongly in then, from Remark 7 we have that in and, consequently, .
On the other hand, let and let x be a geodesic of G defined near . Once the geodesics inside , enter necessarily in the region ω, then, for any geodesic of the metric G, with , there exists such that does not belong to the , so that does not belong as well and item (ii) follows. Once the time and the geodesic x were taken arbitrary, we conclude that is empty. Therefore, by Remark 7 we deduce
Similarly, since
then we consider the microlocal defect measure (m.d.m.) associated to (which is assured by Theorem A.1). Then by (3.28), (3.32) and (3.37), and having in mind that , we conclude that
From this, we obtain
Analogously, we obtain
Let and . To conclude the proof, let us see that multiplying the first equation of (3.22) by and the second equation of (3.22) by we have, respectively,
and
Integration by parts in (3.48) implies
By (3.27), (3.28), (3.32), (3.36) and (3.45), and having in mind that , from (3.50) we deduce that
Furthermore, since
then from (3.52) and (1.5) we conclude that
Analogously, by (3.27), (3.28), (3.32), (3.37), (3.47) and (3.49), we conclude that
which implies, in addition to all the convergences above, that
Then, by the decrease of the energy, we obtain
This implies, together with the energy identity
and (3.28), that
Thus, by energy identity again, by (3.28) and (3.53), we have that
Therefore, , as desired to prove. □
The critical and supercritical cases
In this subsection we shall prove the observability inequality which reads to problem (3.1) when for . For sake of simplicity, as usual in the literature, in the forthcoming computations we assume . For , the result can be proved analogously. We will prove
where and C are positive constants and provided that . To prove it we shall combine Strichartz estimates for the wave equation with the following unique continuation property, proved in [27].
Letsatisfying Assumption
1.1
and let. Assume that f is analytic inand that ω is an open subset of Ω satisfying Assumption
1.5
. Then there existssuch that the only solution w ofwithis.
Arguing as in [27], we can extend Proposition 3.1 for f belonging to a generic set of .
Assume that (3.55) does not hold, then, there exists a sequence of weak solutions to problem (3.1), such that and
From (3.57) we obtain
Let us take the following sequence of problems into account
Now, we define:
It is not difficult to check that there exists such that , for all , and that the sequence is bounded. Hence, in order to achieve a contradiction, it is enough to prove that converges to zero.
Indeed, first, taking (3.58) and (3.60) into consideration we infer
Thus, by (3.61) and since and we have, respectively, that
Since is non-increasing and remains bounded, we obtain a subsequence of such that
Let us consider the following subsequence of normalized problems
For some eventual subsequence, we have that . We shall divide our proof into two cases: or .
Case (i): . Passing to the limit in (3.66) we arrive at
and then, by Proposition 3.1 we have that .
Case (ii): . Now, let us consider , that is, when . We can write
Thus,
We claim that
Since we can make use of Strichartz, as in the Remark 5, for the first equation of problem (3.59), to obtain the estimate
for every . Since , we conclude that
Observing that the Strichartz estimates applied to the first equation of the problem (3.66) ensure that
where the constant does not depend on k.
Hence, from the above calculation, as can be controlled by the norm we can conclude that is a bounded sequence in .
Therefore, taking into account (3.68), the above limitations and the fact that the affirmation (3.69) follows.
Thus,
Analogously, we have
So, passing to the limit in (3.66) we arrive at
and for , from (3.73) we infer
which implies that and, consequently, . Thus, in both the cases we have . Remember that our main objective is to prove that converges to zero. For this purpose let
First, we will prove that
From the above convergences we know that
So, let us consider be the microlocal defect measure associated with in , i.e., is the m.d.m. associated with and in .
Observe that is linearizable in the sense defined by P. Gerard in [24]. Indeed, remember that is a sequence of solutions to problem (3.66) and consider the following problem of linear wave with initial data
Thus, for we have that
Due to the convergences result in (3.61), (3.62) and the cases (i) and (ii) above we have that
On the other hand, by Gronwall’s inequality we obtain that
for all . Then
Therefore, we can conclude that is linearizable in the sense defined by P. Gerard in [24], thus propagates as the -m.d.m. associated to the linear wave equation with the same initial data. Then, by the theorem of microlocal elliptic regularity from the m.d.m. due to Gérard [23] we have
and that propagates along the bicharacteristic flow of the wave operator.
On the other hand
From the above we deduce that
which implies that is zero in . Then, since the geodesics inside , in view of main assumption, enters necessarily in the region ω, for all geodesic of the metric , we have by propagation that . As a consequence,
which means that strongly in and observing (3.76) we get
Analogously, since
we can consider be the microlocal defect measure associated with in . Furthermore, due to the convergences result in (3.61), (3.62) and the cases (i) and (ii) above, is linearizable in the sense defined by P. Gerard in [24]. Then similarly we obtain that
Then using an equipartition of the energy we can conclude that as we desired to prove. □
Uniform decay rate estimates
Before proving the main result of this section, we will need to define some functions. For this purpose, we will follow the ideas first introduced in the literature by Lasiecka and Tataru [31]. We will repeat them briefly. Let ϕ be concave, strictly increasing function such that and
Note that such function ϕ can be straightforwardly constructed, given the hypotheses on function g given in (1.2). Next we define
Since r is monotone increasing, it follows that is invertible for all . For where , we set,
where c is a positive constant that will be determined later.
The function z is easily seen to be positive, continuous and strictly increasing with . Finally, let
Assume that ω geometrically controls Ω, that Assumptions
1.1
–
1.5
hold and, in addition, that the Lemma
3.1
are in place. Letbe the mild solution of problem (
3.1
). Given, there exists asuch thatwith, where the contraction semigroupis the solution of the differential equationwhere q is as given in (
4.4
) provided that. Here the constant c (from definition (
4.3
) is.
Let
and
Using the Assumptions 1.2, 1.3, by Jensen’s inequality we obtain
where . Defining
we obtain that
where the function z is defined in (4.3). Proceeding verbatim as considered in [31] we obtain the decay rate estimate given in (4.5) as desired to prove. □
Footnotes
Acknowledgements
The second author is partially supported by the CNPq grant 300631/2003-0. The third, the fourth and the fifth authors are supported by CAPES.
Appendix
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