Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 2139792, 14 pages
https://doi.org/10.1155/2018/2139792

Research Article

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

Lianbing She,1,2 Yangrong Li,¹ and Renhai Wang¹

¹School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
²School of Mathematics and Information Engineering, Liupanshui Normal College, Liupanshui, Guizhou 553004, China

Correspondence should be addressed to Yangrong Li; liyr@swu.edu.cn

Received 27 September 2017; Accepted 19 December 2017; Published 17 January 2018

Academic Editor: Xiaohua Ding

Copyright © 2018 Lianbing She et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with pullback dynamics for the weakly damped Schrödinger equation with time-dependent forcing. An increasing, bounded, and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable. Also, we obtain the forward absorption, which is only used to deduce the backward compact-decay decomposition according to high and low frequencies. Based on a new existence theorem of a backward compact pullback attractor, we show that the nonautonomous Schrödinger equation has a pullback attractor which is compact in the past. The method of energy, high-low frequency decomposition, Sobolev embedding, and interpolation are quite involved in calculating a priori pullback or forward bound.

1. Introduction

This paper is concerned with the backward compact dynamics of space-periodic solutions for the nonautonomous complex-valued Schrödinger equation in :

The above equation for was introduced in [1] as a model for the propagation of solitons and laser beams. In such a case (without damping and forcing), it is easy to prove the energy conservation law; that is, (see, e.g., [2]). So no attractors exist. To obtain an attractor, we have to assume that the equation has a positive damping parameter .

The dynamical behavior of the damped Schrödinger equation was widely investigated by many physicists and mathematicians (see, e.g., [3–11]) but restricted in the autonomous case; that is, the force is time-independent (only space dependent).

This paper deals with dynamics for the nonautonomous Schrödinger equation; that is, the force is time-dependent. To the best of our knowledge, there is no literature treating nonautonomous dynamics (including random dynamics) for the Schrödinger equation, even in the simple case for the existence of a pullback attractor, although the theory and application of pullback attractors had been widely developed for many other PDEs (see [12–16]), and for pullback random attractors, see, for example, [17–20].

When one tries to look for a bounded pullback absorbing set for (1) (see Lemma 7), it seems to be assumed that is backward bounded in ; that is,

rather than the ordinary tempered integrable condition. Another difficulty is that we must treat the time-derivative , which is assumed to be backward tempered. All those assumptions are different from dealing with the pullback dynamics for other nonautonomous dissipative equations (see, e.g., [17, 21–28]).

On the other hand, the backward condition (2) permits us to consider further properties of the pullback attractor, for example, backward compactness, as considered in [29–31], where a pullback attractor is called backward compact if the union over the past is precompact.

So, in Section 2, we establish a new abstract theorem on a backward compact pullback attractor for a decomposable evolution process; that is, it has a backward compact-decay decomposition. For such a decomposable process, we show that the existence of a backward compact attractor is equivalent to the existence of an increasing, bounded, and pullback absorbing set (see Theorem 4).

We then apply the abstract result to the nonautonomous Schrödinger equation. In Section 3, the increasingly pullback absorption is verified if is assumed to be backward bounded and the time-derivative is backward tempered.

The difficulty arises from verifying the compact-decay decomposition according to the high and low frequency of the Fourier series. In this case, the pullback absorption may not be suitable for verifying such a decomposable property. So, in Section 4, we have to give an auxiliary result on the forward absorption. It may be possible to deduce a forward attractor (cf. [32, 33]), but we do not pursue this forward attractor in the present paper.

In Section 5, we present some techniques of splitting the solutions of (1) into high and low frequency parts and establish a new equation with initial value zero in the high-frequency part. Then the forward absorption obtained in Section 4 can be applied to prove that the new equation has a forward uniformly bounded solution in , which further proves that the component system is backward asymptotically compact in . Also, we prove that the difference of solutions from both equations in the high-frequency part is backward exponential decay and so obtain the compact-decay decomposition as required.

The final existence result of a backward compact attractor is given in Theorem 14. It is worth pointing out that the pullback-forward method (involving the high-low frequency decomposition) may be special for the nonautonomous Schrödinger equation, which is different from treating the pullback dynamics for other nonautonomous dissipative equations.

2. Backward Compact Dynamics for Decomposable Systems

Let be a Banach space equipped with the class of all bounded subset in . We consider a nonautonomous process on , which means is a continuous nonlinear mapping such that and for all with .

We assume that the process is decomposable in the following sense.

Definition 1. A nonautonomous process is said to have a backward compact-decay decomposition    if
(i) is backward asymptotically compact, that is, the sequence is precompact whenever with fixed , and is bounded in ;
(ii) is backward exponential decay, that is, for each and , there exist two positive constants and such that, for some ,

A nonautonomous set in means a set-valued mapping , which is said to have some topological properties (such as compactness, boundedness, and closedness) if the component set has the corresponding properties.

Definition 2. A nonautonomous set is called
(i) backward (resp., forward or globally) compact if and (resp., or ) are compact for each ;
(ii) increasing (resp., decreasing) if (resp., ) for all .

Definition 3. A backward compact set in is called a backward compact attractor for a nonautonomous process if is invariant, that is, for , and pullback attracting, that is, for any and ,

where the Hausdorff semi-distance .

A backward compact attractor must be unique and minimal, where the minimality means for any closed attracting set (see [30]).

For the purpose of applying to the Schrödinger equation, we need to establish a new existence theorem of a backward compact attractor for a backward compact-decay process, although other existence criteria were established in [29, 30].

Recall that a nonautonomous set is called pullback absorbing for if for each and there is a such that

Theorem 4. Suppose a nonautonomous process has a backward compact-decay decomposition for . Then the following statements are equivalent.
(i) There exists an increasing, bounded, and pullback absorbing set .
(ii) There is a backward compact attractor given by

Proof. The necessity is easily proved by setting , the -neighborhood of , for each . Since is obviously an increasing family in , it follows that is increasing. Since is precompact, it follows that is bounded (not necessarily precompact). Finally, it is easy to deduce the pullback absorption of from the pullback attraction of .
Conversely, suppose (i) is true; we show that is a backward compact attractor in three steps.
Step  1. We show the invariance. As usual (cf. [12]), it is easy to prove the positive invariance . On the other hand, let . We choose and such that

Let and . Then if is large enough. By the compact-decay decomposition, we know that is precompact and is exponential decay. Then, passing to subsequences, there is an such that

where for all . Hence, it further implies

Since is pullback absorbing and increasing, for each , we can choose a in such that is large enough; it follows that

By the process property, as ,

By (10)-(11), we have . By the continuity of , it follows from (7) and (9) that

which proves the negative invariance.
Step  2. We show the attraction of . If it is not true, then there are , , as and with such that

By using the compact-decay decomposition, we can take subsequences such that

where . Therefore,

Like we did in Step  1, we choose a subsequence of such that

and by (15),

The above two formulations imply , which contradicts with (13).
Step  3. It remains to show the precompact of with fixed . Let . Since is pullback absorbing and increasing, there is a such that

and so , which further implies

Hence is at least bounded. To prove the precompactness of , we take a sequence and then with . Let . Then invariance of implies

Since is proved to be bounded, there is a bounded sequence such that . By the backward compact-decay decomposition, we know is backward asymptotically compact on the bounded set . Then, passing to a subsequence, there is a such that

By using the decay property of , we know

The above limits imply

Hence, is precompact. In particular, is precompact. But is obviously closed and so it is compact. The proof is complete.

3. Pullback Absorption in Schrödinger Equations

We come back to the Schrödinger equation with time-dependent force as follows:

where , is an unknown complex-valued function.

3.1. Hypotheses and Existence of Solutions

Let be the space of complex-valued -functions whose norm is denoted by . Let be the space of all one-periodic -function with the norm . The norm in is denoted by . We then give some hypotheses on the time-space dependent force .

Assumption A1. is continuous and backward bounded:

Assumption A2. The time-derivative is backward tempered:

where as given in (24).

Assumption A1 implies and the finiteness of , where

The last inequality can be proved as follows.

Note that all functions () are finite, nonnegative, and increasing.

We will repeatedly use the following two energy inequalities.

Lemma 5. Let , , be the approximation solution of (24). Then

where

Proof. Equation (29) is easily obtained by multiplying (24) with (the complex conjugate of ) and taking the real part of the final equation.
To prove (30), we multiply (24) by and take the imaginary part; after some complex-valued calculations, we obtain

where is given by (31), and

By the Agmon inequality , we have . Hence,

We then rewrite the energy equation (32) as follows:

which is just the needed energy inequality.

Based on the above energy inequalities, one can obtain a priori estimate (for absorption, see the next subsection). Then it is similar as the autonomous case (see [5, 8]) to prove that (24) is well posed in . Namely, for each and , (24) has a unique solution and the solution is continuous. This well-posed property permits us to define an evolution process on by

3.2. Increasing, Bounded, and Pullback Absorbing Sets

In order to use the results of Theorem 4, we need to look for an increasing, bounded, and pullback absorbing set.

Lemma 6. Let A1 be satisfied. Suppose be a ball in ( is the radius) and , then there are such that, for all and ,

where are positive constants and are functions given in A1.

Proof. Let and . We apply the Gronwall inequality on (29) over the interval with and then find

Letting , we have

which yields (37) if we take . On the other hand, by (40) again,

where we use the fact that for . We then consider the sixth power to obtain

which yields

From this, it is easy to deduce (39) with and similarly obtain (38).

We then consider the backward bound of solutions in as follows.

Lemma 7. Let A1, A2 be satisfied. Then for each ball in and , there exists , such that, for all and ,

where is a constant and thus is finite and increasing.

Proof. Let and . Applying the Gronwall inequality on (30) over , we get

By Lemma 6 and A2, there is a such that, for all and ,

On the other hand, by the definition of given in (31), we have

In particular, at the initial value, we have, if is large enough, then

Both (47) and (49) imply that, for all and (with a larger ),

By the Agmon inequality again, it follows from (31) that

which together with (50) and Lemma 6 imply that, for all and (with a larger ),

which shows the needed result.

Under the light of Lemma 7, we have the following increasing absorption.

Theorem 8. Let A1, A2 be true. Then the nonautonomous Schrödinger equation possesses an increasing, bounded, and pullback absorbing set in given by

Proof. By A1, A2, we can see that is an increasing and finite function with respect to ; this fact along with Lemma 7 shows that the nonautonomous set is increasing, bounded, and pullback absorbing in . In fact, by (45), the absorption is backward uniform.

Remark 9. It seems not to prove that the nonautonomous Schrödinger equation has a bounded absorbing set in if the assumption A1 is replaced by the weaker assumption that , although this weak assumption is enough for reaction-diffusion systems (see [29]), BBM equations (see [30]), and Navier-Stokes equations (see [31]).

4. Forward Absorption in Schrödinger Equations

This section establishes the forward absorption, which will be useful to deduce the compact-decay decomposition in the next section.

In this case, we need to strengthen Assumptions A1 and A2 as follows.

Assumption A1. is continuous such that .

Assumption A2. exists such that .

Assumption A3. exists such that .

Lemma 10. Let , be satisfied. Then for each ball in , there exists such that

where are constants depending on and .

Proof. Let and . We apply the Gronwall inequality on (29) over with and then find

Taking in (56), we have

Let . By , we know, for all and ,

We then consider the third power of (56) to obtain

which yields

Rewriting by , we deduce for all ,

and similarly we have

To prove (54), we apply the Gronwall inequality on (30) over to get

By (61)-(62) and Assumption A2, we easily deduce that, for ,

By the definition of given in (31), we have

In particular, if is large enough, then

By (61)-(62) and Assumption , for (with a larger ),

By the Agmon inequality again, it is similar as (51) to prove that

which together with (67) and (58) imply that, for all , , and (with a larger ),

which shows the needed result (54).
To prove (55), we multiply (24) by a test function with , then by the Agmon inequality,

Then (55) follows from (54).

5. Compact-Decay Decomposition

5.1. High-Low Frequency Decomposition

We expand (the solution of (24)) into its Fourier series

and split into low frequency part and high-frequency part with

Let be arbitrary but fixed and take the initial value in a ball of . Then we are concerned with two functions of :

By the forward absorption given in Lemma 10, we know there is a (depending on the radius of initial ball) such that

The high-frequency part satisfies the following equation.

Then we consider the following equation on with the initial value zero.

where (if exists) is actually a function w.r.t. , , , and . Sometimes, we write it as . We can decompose the evolution process by

for all , , and . In the sequel, the main task is to prove the asymptotic compactness of and the exponential decay of in for large .

5.2. The Uniformly Bounded Estimate for

To prove the existence of solutions for (76), we need to consider the approximation , which is the solution for the projection of (76) on the subspace

Then, by the standard Galerkin method and a priori estimate (see the next proposition), one can prove the existence of in if is large enough.

In the following proposition, we actually prove the result for . However, for the sake of simplicity, we omit the subscript and also omit the proof of convergence as .

Proposition 11. Suppose A, A are satisfied. Then there exists such that, for , (76) has a uniformly bounded solution in such that

where is a constant and is the forward absorbing entering time.

Proof. Multiplying (76) by , taking the imaginary part, after some computations, we obtain an energy equation:

with

We now consider the upper bound of ; by Assumption A, we have

By the classical interpolation and the Poincaré inequality on

the second term of can be bounded by

Note that forms a Banach multiplicative algebra in one-dimension; that is,

Then, by Lemma 10 and (74), the third term of can be bounded by

By (74) and the embedding , the fourth term is bounded by

By the Agmon inequality and (83), we have

Then the fifth term of is bounded by

Similarly, by (83) and the Hölder inequality, the sixth term of is bounded by

By (88), the rest terms can be bounded by

By the above estimates, we know that can be bounded by

Letting , where denotes the integer-valued function, we have, for ,

On the other hand, we similarly obtain a lower bound of .

where is uniformly bounded in view of (74). By (93)-(94), we have

Then it follows from (80) that

Since and , it follows from the Gronwall inequality on that

which along with (94) and Assumption A imply that, for all ,

Hence, satisfies the following twice inequality:

where , are positive constants. Let and . Then for all the discriminant of twice inequality is positive:

In this case, the equation has two positive roots.

We show that . Indeed,

where the last inequality is obviously true. Now the twice inequality (99) has the solution

Since and the mapping is continuous on , it follows that

which proves the required bound.

5.3. Further Regularity Estimates

We further show a regularity result for in .

Proposition 12. Suppose A, A, and A3 are satisfied. Then there exists such that, for , the solution of (76) is uniformly bounded in :

where is a constant and is the forward absorbing entering time.

Proof. Just like we did in the above proposition, we need to firstly show the bound of in and then let to obtain the bound of in . Thus, for the sake of convenience, we omit the detail of letting and also drop the superscript of and write , , in this proof.
The first thing we shall do is to obtain the following equation by differentiating (76):

with initial value , where , , and are given by

In order to show the needed result (105), we divide it into three steps.
Step  1. We show . Let be a test function taken from such that . Then by Assumption A,

which implies that . Since is a bounded operator from to , it follows that for all and .
The rest term of can be rewritten as

Note that we have the inverse inequality on . By Lemma 10 and Proposition 11, we know , where . Then, by the multiplicative algebra (see (85)), we have, for all ,

Therefore, and so with .
Step  2. We give the estimates of , where it is easy to obtain

By Assumption A3, for the test function , we have

Hence, for all and .
To estimate the rest terms, we note that , which can be obtained from (76) and the uniform bound of . By Lemma 10, and so . For the second term of , by the multiplicative algebra (85) again, we have

where . Hence and so in view of the boundedness of the operator from to . It is similar to obtain the estimates for the rest terms in and so

Step  3. Finally, we bound . Multiplying (106) by , taking the imaginary part of the resulting equation and then integrating over , we obtain an energy equation:

where

By the bound of in and in , and by the inequality on the space (see (88)), we have, for ,

Letting , then we have

On the other hand, we have the lower bound of :

Letting , we see, that for ,

Therefore, we substitute (119)-(121) into (115) and use (114) to find

Applying the Gronwall inequality on and observing , we get

which along with A and (121) imply the needed result (105).

5.4. Exponential Decay for Large Times

Let , which is a solution of (75). Let be the corresponding solution of (76). We need to prove the exponential decay of .

Proposition 13. Suppose A, A are satisfied. Then there exists such that, for ,

Proof. Recall that and . Then it follows from (75) and (76) that is the solution of

Multiplying (125) by , taking the imaginary part of the resulting equation and then integrating over , we obtain an energy equation:

with

By the boundedness of , in and in , we have the upper bound of :

and also the lower bound of :

if is large enough. By (126)–(129), we have

if and . By the Gronwall inequality on , we obtain

By Lemma 10, . Note that and so , which easily implies that is bounded. Therefore, by (129), we have

which completes the proof.

6. Backward Compact Attractors

Finally, we state and prove the main result. We need to spilt the evolution process . Let be fixed, where is the level given in Proposition 13. Then for all , , and , we write with

where is the forward entering time.

Theorem 14. (a) Suppose A1, A2. Then the nonautonomous Schrödinger equation has an increasing, bounded, and pullback absorbing set .
(b) Suppose A, A, A3. Then the nonautonomous Schrödinger equation has a backward compact pullback attractor given by

Proof. The assertion (a) follows from Theorem 8. To prove (b), by the abstract result (i.e., Theorem 4), it suffices to prove that the evolution process has a backward compact-decay decomposition.
We need to prove that is backward asymptotically compact. Indeed, let and take some sequences , , and . Without loss of generality, we assume , where is the forward entering time. Then by (133) we have

By Lemma 10, we know that

which means that is a bounded sequence in the finite-dimensional space and so, passing to a subsequence, it is convergent. By Proposition 12, we know

which means that is bounded in . By the Sobolev compact embedding, it is precompact in and so is .
On the other hand, by Proposition 13, we know, for , and ,

which implies is backward decay.

Remark 15. (I) In fact, under Assumptions A-A, it is possible to prove the existence of a forward attractor (see [32, 33]), which may be different from the pullback attractor. In the present paper, we only impose the forward absorption to deduce a backward compact-decay composition, which seems not to be deduced from the pullback absorption for the nonautonomous Schrödinger equation.
(II) Suppose and they are time-periodic, then satisfies both conditions A and A. In this case, by using the theoretical result as given in [17], one can obtain a periodic pullback attractor and maybe a periodic forward attractor.

Conflicts of Interest

There are no conflicts of interest to declare.

Acknowledgments

The authors were supported by National Natural Science Foundation of China Grant 11571283 and L. She was supported by Natural Science Foundation of Education of Guizhou Province: KY103.

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