Electronic Journal of Qualitative Theory of Differential Equations

This paper deals with the stabilization of the linear biharmonic Schrödinger equation in an n-dimensional open bounded domain under Dirichlet–Neumann boundary conditions considering three infinite memory terms as damping mechanisms. We show that depending on the smoothness of initial data and the arbitrary growth at infinity of the kernel function, this class of solution goes to zero with a polynomial decay rate like t−n depending on assumptions about the kernel function associated with the infinite memory terms.


Problem setting
The fourth-order nonlinear Schrödinger equation (4NLS) or biharmonic cubic nonlinear Schrödinger equation i∂ t y + ∆y − ∆ 2 y = λ|y| 2 y, has been introduced by Karpman [12] and Karpman and Shagalov [13] to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity.Equation (1.1) arises in many scientific fields such as quantum mechanics, nonlinear optics, and plasma physics, and has been intensively studied with fruitful references (see [2,12,16] and references therein).
Over the past twenty years, equation (1.1) has been deeply studied from a different mathematical viewpoint, including linear settings which can be written generically as i∂ t y + α∆y − β∆ 2 y = f , ( with α, β ≥ 0 and different types of boundary conditions.For example, considering the problem (1.2) several authors treated this equation, see, for instance, [1,10,17,19,20,22] and the references therein.Inspired by these results for the linear problem associated with the 4NLS, a mathematical viewpoint problem is to study the well-posedness and stabilization for solutions of the system (1.2) in an appropriate framework.So, consider the equation (1.2) when α = β = 1 in a n-dimensional open bounded subset of R n .Our goal is to consider an initial boundary value problem (IBVP) associated with (1.2) when the source term f is viewed as an infinite memory term: Thus, the goal of this manuscript is to deal with the following system        i∂ t y(x, t) + ∆y(x, t) − ∆ 2 y(x, t)+(−1) j i ∞ 0 f (s)∆ j y(x, t − s)ds = 0, (x, t) ∈ Ω × R + , y(x, t) = ∇y(x, t) = 0, (x, t) ∈ Γ × R * + , y(x, −t) = y 0 (x, t), (x, t) ∈ Ω × R + , (1.3) where j ∈ {0, 1, 2}, Ω ⊂ R n is a n-dimensional open bounded domain with a smooth boundary Γ, and f : R + := [0, ∞) → R is the kernel (or relaxation) function.We point out that for each j the memory term present in (1.3) is modified.
In (1.3), the memory kernel f satisfies the following assumptions: For some positive constant c 0 , we have the following conditions f ′ < 0, 0 ≤ f ′′ ≤ −c 0 f ′ , f (0) > 0 and lim s→∞ f (s) = 0. (1.4) Under the Assumption 1, let us introduce the following energy functionals associated with the solutions of (1.3) with j ∈ {0, 1, 2} and g = − f ′ , so g ∈ C 1 (R + ) , g is non-negative and It is worth mentioning that the abuse of notation ∆ j 2 in (1.5) means the identity operator for j = 0, the ∇ operator for j = 1 and the Laplacian operator for j = 2.
Therefore, taking into account the action of the infinite memory term in (1.3), the following issue will be addressed in this article:

Historical background
Distributed systems with memory have a long history and have been first introduced in viscoelasticity by Maxwell, Boltzmann, and Volterra [3,4,15,18].In the context of heat processes with finite dimension speed, these systems have been introduced by Cattaneo [7] (a previous work of Maxwell had been forgotten).
In our context, to our knowledge, there is no result considering the system (1.3) in ndimensional case.However, considering the fourth-order Schrödinger system i∂ t u + ∆ 2 u = 0, (1.6) there are interesting results in the sense of control problems in a bounded domain of R or R n and, more recently, on a periodic domain T and manifolds, which we will summarize below.
The first result about the exact controllability of the linearized fourth order Schrödinger equation (1.6) on a bounded domain Ω of R n is due to Zheng and Zhou in [21].In this work, using an L 2 -Neumann boundary control, the authors proved that the solution is exactly controllable in H s (Ω), s = −2, for an arbitrarily small time.They used Hilbert Uniqueness Method (HUM) (see, for instance, [9,14]) combined with the multiplier techniques to get the main result of the article.More recently, in [22], Zheng proved a global Carleman estimate for the fourth-order Schrödinger equation posed on a finite domain.The Carleman estimate is used to prove the Lipschitz stability for an inverse problem associated with the fourth-order Schrödinger system.
Still, on control theory Wen et al. in two works [19,20], studied well-posedness and control problems related to the equation (1.6) on a bounded domain of R n , for n ≥ 2. In [19], they considered the Neumann boundary controllability with collocated observation.With this result in hand, the stabilization of the closed-loop system under proportional output feedback control holds.Recently, the same authors, in [20], gave positive answers when considering the equation with hinged boundary by either moment or Dirichlet boundary control and collocated observation, respectively.
To get a general outline of the control theory already done for the system (1.6), two interesting problems were studied recently by Aksas and Rebiai [1] and Gao [10]: Uniform stabilization and stochastic control problem, in a smooth bounded domain Ω of R n and on the interval I = (0, 1) of R, respectively.In the first work, by introducing suitable dissipative boundary conditions, the authors proved that the solution decays exponentially in L 2 (Ω) when the damping term is effective on a neighborhood of a part of the boundary.The results are established by using multiplier techniques and compactness/uniqueness arguments.Regarding the second work, the author showed Carleman estimates for forward and backward stochastic fourth order Schrödinger equations which provided the proof of the observability inequality, unique continuation property, and, consequently, the exact controllability for the forward and backward stochastic system associated with (1.6).
Recently, the first author [5] showed the global stabilization and exact controllability properties of the 4NLS on a periodic domain T with internal control supported on an arbitrary sub-domain of T.More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solution of the associated linear system, the authors proved that system (1.7) is globally exponentially stabilizable, considering f (x, t) = −ia 2 (x)u.This property together with the local exact controllability ensures that 4NLS is globally exactly controllable on T. Lastly, the first author showed in [6] the global controllability and stabilization properties for the fractional Schrödinger equation on d-dimensional compact Riemannian manifolds without boundary (M, g), Under the suitable assumption of the damping term a(x) they proved their result using microlocal analysis, being precise, they can prove propagation of regularity which together with the so-called Geometric Control Condition and Unique Continuation Property, shows the main results of the article.Is important to mention that when σ = 4 they have the equation (1.6).

Notations
Before presenting the main result let us give some notations and definitions.In what follows, the variables x, t, and s will be suppressed, except when there is ambiguity and, throughout this article, C will denote a constant that can be different from one step to the next in the proofs presented here.We will use the notations This approximation ensures that η t satisfies (1.9) To express the memory integral in (1.3) in terms of η t , we will denote g := − f ′ .Thus, according to (1.4), we have g ∈ C 1 (R + ) and Now on, rewrite (1.3) into i∂ t y(x, t) + ∆y(x, t) − ∆ 2 y(x, t) + i(−1) j ∞ 0 g(s)∆ j η t (x, s)ds = 0. (1.12) Define the following sets respectively 1 .Consider U = (y, η t ) T and U 0 (x, s) = (y 0 (x, 0), η 0 (x, s)) T where y ∈ L2 (Ω) and η t ∈ L j with Define the energy space as follows with inner product and norm respectively.Therefore, the systems (1.3) and (1.9) can be seen as the following initial value problem (IVP) Here, the operator A j is defined by Remark 1.3.Observe that for the fourth-order Schrödinger equation, the natural domain to be considered is H 2 0 (Ω) ∩ H 4 (Ω).However, since we are working with a more general operator, namely operator defined in (1.14) and (1.15), we need to impose A j (U) ∈ H j .However, note that the inclusion below is verified.So, the operator A j (U) is well-defined.

Main result
As mentioned, some valuable efforts in the last years focus on the well-posedness and stabilization problem for the fourth-order Schrödinger system.So, in this article, we present a new way to ensure that, in some sense, the Problems 1.1 and 1.2 can be solved for the system (1.3) in n-dimensional case.To do that, we use the ideas contained in [11], so additionally to the Assumption 1 we have also assumed the memory kernel satisfying the following: Assume there is a positive constant α 0 and a strictly convex increasing function Additionally, when (1.17) is not verified, we will assume that y 0 satisfies, The next theorem is the main result of the article.
Remark 1.5.Let us give some remarks about the Assumption 2.
i. Thanks to the relation (1.18), we have that (1.19) is valid, for example, if is bounded with respect to s.
Remark 1.6.Now, we will present the following remarks related to the main result of the article.
i.When (1.17) is verified, note that G n (0) = 0, so (1.20) implies Since we have that D(A 2 j ) is dense in H j , when j = 0, and D(A 4 j ) is dense in H j when j = 1, 2 (see Lemma A.1 in A), we have that (1.24) is valid for any U 0 ∈ H j .Therefore, in this case, (1.21) gives G n (s) = s n and from (1.20) we get showing that the energy (1.5) associated with the solutions of the system (1.13) have a polynomial decay rate.
ii.Given (1.18) verified, the relation of (1.20) is weaker than the previous case.For example, when g = g 2 defined by (1.23), we see that G(s) = s p with p > q 2 +1 q 2 −3 satisfies the Assumption 2.Moreover, and so, Therefore, the energy (1.5) associated with the solutions of the system (1.13) satisfies showing that the decay rate of (1.20) is arbitrarily near of

Novelty and structure of the work
Among the main novelties introduced in this article, we give an affirmative answer to the Problems 1.1 and 1.2, providing a further step toward a better understanding of the stabilization problem for the linear system associated with (1.1) in the n-dimensional case.Here, we have used the multipliers method and some arguments devised in [11].
Since we are working with a mixed dispersion we can consider three different memory kernels acting as damping control to stabilize equation (1.3) in contrast to [5], for example, where interior damping is required and no memory is taken into consideration, in a onedimensional case.Moreover, if we also compare with the linear Schrödinger equation (see e.g.[8]) we have more kernels acting to decay the solution of the equation (1.3) since we have more regularity with the mixed dispersion, which is a gain due the bi-Laplacian operator.
In addition to this, recently, using another approach, the authors in [6] showed that the system (1.8) is stable, however considering a damping mechanism and some important assumptions such as the Geometric Control Condition (GCC) and Unique Continuation Property (UCP).Here, we are not able to prove that the solutions decay exponentially, however, with the approach of this article, the (GCC) and (UCP) are not required.The drawback is that we only provide that the energy of the system (1.3), with memory terms, decays in some sense as explained in the Remark 1.6.
A natural issue is how to deal with the 4NLS system given in (1.1).The main point is that we are not able to use Strichartz estimates or Bourgain spaces to obtain more regularity for the solution of the problem with memory terms, therefore, Theorem 1.4 for the system (1.1) with memory terms remains open.Now, let us present the outline of our paper.In Section 2 we prove a series of lemmas that are paramount to prove the main result of the article.With the previous section in hand, Theorem 1.4 is shown in Section 3. Finally, for the sake of completeness, in Appendix A, we present the existence of a solution for the system (1.13) in the energy space H j .

Auxiliary results
In this section, we will give some auxiliary lemmas that help us to prove the main result of the article.In this way, the first result shows identities for the derivatives of E j given by (1.5).Lemma 2.1.Suppose the Assumption 1.Then, the energy functional satisfies Proof.Observe that (2.1) is a direct consequence of (A.3), and the result follows.
Next, we will give a H 1 -estimate for the solution of (1.12).

Lemma 2.2.
There exist positive constants c k,j , j ∈ {0, 1, 2} and k ∈ {1, 2} such that the following inequality Proof.We use the multipliers method to prove (2.2).First, multiplying the equation (1.12) by y, integrating over Ω and taking the real part we get taking into account the boundary conditions in (1.3) and (1.9), for y(t, •) ∈ H 2 0 (Ω), for all t ∈ R + .
Note that the last term of the left-hand side of (2.3) can be bounded using the generalized Young's inequality giving for any δ > 0. In addition to that, the first term of the left-hand side of (2. (2.6) We now split the remainder of the proof into three cases.and taking δ = 1 2 > 0, the inequality (2.2) holds with c 1,1 = 2C(δ) and c 2,1 = 2. Case 3. j = 2 Finally, just take any δ > 0 such that δ < 1.Therefore, using (2.6) we get (2.2) for c 1,2 = C(δ) and c 2,2 = 1, achieving the result.
We need now define the following higher-order energy functionals for ) in the case when j = 1, 2, and )) for k ∈ {1, 2} when j = 0.In addition to that, the linearity of the operator A j together with (2.1) gives With this in hand, let us control the last term of the right-hand side of (2.2) in terms of the E ′ j,1 and the L j -norms of the ∆ j 2 η t tt .
Proof.Differentiating (1.9) with respect to t, multiplying the result by g(s), and integrating on [0, ∞) we have taking into account the third relation in (1.10).So, we get (2.11) Now, let us bound the right-hand side of (2.11).To do that, reorganize the terms of the (RHS) and note that (2.12) The generalized Young inequality gives for any δ > 0 that Substituting both inequalities into (2.12)yields Now replacing (2.13) into (2.11)we have thanks to Poincaré inequality.Here, Now, just in the case j = 2, we need an estimate H 2 -for the solution of (1.12) similar to the estimate (2.2).This estimate is reported in the following lemma.Lemma 2.4.When j = 2, there exist positive constants c k,2 , k ∈ {1, 2}, such that the following inequality (2.17)

Taking into account that
Im and, thanks to the generalized Young inequality, we have that ( As a consequence of (2.10), the last term of the right-hand side of (2.16) can be bounded as follows.

.21)
Proof.Using the Poincaré inequality in the first term of the right-hand side of (2.10), and taking ϵ = c * ϵ, where c * is the Poincaré constant, the result follows.
The next lemma combines the previous one to get an estimate in H j for solutions of (1.12).
Before presenting the main result of this section, the next result ensures that the following norms ∥∆ j 2 η t ∥, ∥η t ∥, and ∥η t tt ∥ can be controlled by the generalized energies E k,j (0) and the initial condition y 0 , for t ≥ s ≥ 0. The result is the following one.Lemma 2.7.Considering the hypothesis of the Lemma 2.6, the following inequality holds where M j,0 (t, s) := (2.25) Additionally, for j = 0, we have 0) , when s > t ≥ 0. Consequently, (2.24) is verified.Now, for j = 0, since ∥y∥ 2 is part of E 0 (see (1.5)), and the energy E 0 is non-increasing, we observe, using Hölder inequality, that for s > t ≥ 0. Thus, (2.26) follows.
Therefore, inequality (2.27) follows using the previous inequality with j = 0, and thanks to the relation (2.26), the result is proved.
The next result is the key lemma to establish the stabilization result for the biharmonic Schrödinger system (1.3).

Lemma 2.8.
There exist positive constants d j,k , for each j ∈ {0, 1, 2} and each k ∈ {0, 2} such that the following inequality holds G 0 (ϵ 0 E j (t)) for any ϵ 0 > 0. Here, E j,0 = E j , E ′ j,0 = E ′ j (0) and G 0 defined as in Theorem 1.4.Proof.Suppose, first, that the relation (1.17) is satisfied.So, thanks to the relation (2.9), we have On the other hand, suppose now that (1.18) and (1.11) are verified.Let us assume, without loss of generality, that E j (t) > 0 and g ′ < 0 in R + .Let τ j,k (t, s), θ j (t, s), j ∈ {0, 1, 2}, k ∈ {0, 2} and ϵ 0 be a positive real number which will be fixed later on, and Additionally, thanks to the continuity of K we have K(0) = 0. We claim that the function K is non-decreasing.Indeed, since G is convex we have that G −1 is concave and G −1 (0) = 0, implying that for 0 ≤ s 1 < s 2 , proving the claim.Now, note that thanks to the fact that K is non-decreasing and by (2.24), (2.26), (2.28), and (2.27), we get (2.31) Denote the dual function of G by G * (s) = sup τ∈R + {sτ − G(τ)}, for s ∈ R + .From the Assumption 2 we have Observe also that in particular and Therefore, we obtain, by using the previous equality in (2.31), that Thanks to the fact that (G ′ ) −1 is non-decreasing we get, where m 0 = sup s∈R + g(s) G −1 (−g ′ (s)) .Note that (1.18) and (1.19), yields that Thus, using that τ j,k (t, s) = 1 m 0 M j,k (t,s) and relation (2.1), we have that Finally, multiplying the previous inequality by gives for c * * defined by (2.15).From the definition of E j given by (1.5) we found that 2 Thanks to the inequality (2.23), we have 2 G 0 (ϵ 0 E j (t)) G 0 (ϵ 0 E j (t)) Observe that H 0 (s) = G 0 (s) s is non-decreasing and E j is non-increasing for each j, thus is non-increasing for each j, and therefore by (3.2) we get G 0 (ϵ 0 E j (0)) For ϵ 0 > 0 small enough we have Thus, dividing (3.3) by c 1 > 0 yields that where G 0 (ϵ 0 E j (0)) Now, integrating (3.4) on [0, t], t ∈ R * + , and observing that G 0 (ϵ 0 E j (t)) is non-increasing gives Because G 0 is invertible and non-decreasing, we deduce that Suppose, for induction hypothesis, that for some n ∈ N * , we have that (1.20) is verified when U 0 ∈ D(A 2n+2 j ) for j ∈ {1, 2} and U 0 ∈ D(A 2n j ) for j = 0.For j ∈ {1, 2}, let us take ) and for j = 0, take U 0 ∈ D(A
Now, for j = 0, we found So, it follows from the induction hypothesis that: there exists α j,n such that Now, since U t and U tt are solution of (1.13) with initial conditions U t (0) ∈ D(A 2n+2 j ) and U tt (0) ∈ D(A 2n+2 j ), respectively, the induction hypothesis guarantees the existence of β n,t > 0 and γ n,t > 0, such that respectively.Thus, as G ′ n s are non-decreasing for dj,n = max{3α j,n , 3β j,n , 3γ j,n }, we get Finally, how t ∈ [T, 2T], we have and from (3.4) we found the following , where α j,n+1 := max 1 ϵ 0 , 2d j,n .In other words, there is α j,n+1 > 0 such that (1.20) holds for n + 1.By the principle of induction we have that (1.20) is verified for all n ∈ N * , showing Theorem 1.4.

A Well-posedness via semigroup theory
This section is devoted to proving that the system (1.13) is well-posed in the energy space H j .To do that, first, let us present some properties of A j , defined by (1.14)-(1.15)and its adjoin A * j defined by for j ∈ {0, since (1.10) is verified.So, A j is dissipative.Similarly, A * j defined by (A.1) is dissipative.Now, let us prove that D(A j ) is dense on H j .Since we showed that A j is dissipative, we need to prove that the image of I − A j is H j , since H j is reflexive.To do that, pick ( f 1 , f 2 ) ∈ H j = L 2 (Ω) × L 2 g (R + ; H j 0 (Ω)), we claim that there exists (y, η t ) ∈ D(A j ) such that (y, η t ) − (i∆y − i∆ 2 y + (−1) j+1 ∞ 0 g(s)∆ j η t (•, s)ds, y − η t s ) = ( f 1 , f 2 ).
Or equivalently, we claim that there exits (y, η t ) ∈ D(A j ) satisfying  which is a direct consequence of the Lax-Milgram theorem.Therefore, (y, η t ) ∈ D(A j ) is a strong solution of (I − A j )(y, η t ) = ( f 1 , f 2 ) and I − A j is surjective, showing the result.Similarly, it is shown that D(A * j ) defined by (A.2) is dense in H j .
The main result of this section is a consequence of the Lemma A.1 and can be read as follows.
Theorem A.2. Suppose that Assumption 1 and (1.9) are verified.Thus, for each j ∈ {0, 1, 2}, the linear operator A j defined by (1.14) is the infinitesimal generator of a semigroup of class C 0 and, for each n ∈ N and U 0 ∈ D(A n j ), the system (1.13) has unique solution in the class U ∈ )).