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

This paper deals with pullback dynamics for the weakly damped Schrödinger equationwith 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.


Introduction
This paper is concerned with the backward compact dynamics of space-periodic solutions for the nonautonomous complex-valued Schrödinger equation in R: The above equation for ,  = 0 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, ‖()‖ = ‖(0)‖ (see, e.g., [2]).So no attractors exist.To obtain an attractor, we have to assume that the equation has a positive damping parameter  > 0.
The dynamical behavior of the damped Schrödinger equation was widely investigated by many physicists and mathematicians (see, e.g., [3][4][5][6][7][8][9][10][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][13][14][15][16]), and for pullback random attractors, see, for example, [17][18][19][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  2 ; that is, sup rather than the ordinary tempered integrable condition.
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][30][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 highfrequency 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  2 , which further proves that the component system is backward asymptotically compact in  1 .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.

Backward Compact Dynamics for Decomposable Systems
Let (, ‖ ⋅ ‖  ) be a Banach space equipped with the class B of all bounded subset in .We consider a nonautonomous process  on , which means (, ) :  →  is a continuous nonlinear mapping such that (, ) = id  and (, ) = (, )(, ) for all  ⩾  ⩾  with  ∈ R.
We assume that the process is decomposable in the following sense.
A backward compact attractor must be unique and minimal, where the minimality means A ⊂ K for any closed attracting set K (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].
(ii) There is a backward compact attractor A = {A()} ∈R given by Proof.The necessity is easily proved by setting K() =  1 (∪ ⩽ A()), the 1-neighborhood of ∪ ⩽ A(), for each  ∈ R. Since ∪ ⩽ A() is obviously an increasing family in  ∈ R, it follows that K is increasing.Since ∪ ⩽ A() is precompact, it follows that K() is bounded (not necessarily precompact).Finally, it is easy to deduce the pullback absorption of K from the pullback attraction of A.
Conversely, suppose (i) is true; we show that A() fl (K(), ) is a backward compact attractor in three steps.
Step 2. We show the attraction of A.
Step 3. It remains to show the precompact of   fl ∪ ⩽ A() with fixed  ∈ R. Let  ⩽ .Since K is pullback absorbing and increasing, there is a  0 () such that and so (K(), ) ⊂ K(), which further implies Hence   is at least bounded.To prove the precompactness of   , we take a sequence   ∈   = ∪ ⩽ A() and then   ∈ A(  ) with   ⩽ .Let 0 ⩽   → ∞.Then invariance of A implies Since   is proved to be bounded, there is a bounded sequence {  } such that   = (  ,   −   )  .By the backward compact-decay decomposition, we know  1 is backward asymptotically compact on the bounded set   .Then, passing to a subsequence, there is a  ∈  such that By using the decay property of  2 , we know The above limits imply Hence,   is precompact.In particular, A() is precompact.But A() is obviously closed and so it is compact.The proof is complete.

Pullback Absorption in Schrödinger Equations
We come back to the Schrödinger equation with timedependent force as follows: where  > 0, (, ) is an unknown complex-valued function.

Hypotheses and Existence of Solutions.
Let  2 (Ω) be the space of complex-valued  2 -functions whose norm is denoted by ‖ ⋅ ‖.Let  1 be the space of all one-periodic  1function with the norm We then give some hypotheses on the time-space dependent force  = (, ).
Note that all functions   (⋅) ( = 1, 2, 3) are finite, nonnegative, and increasing.We will repeatedly use the following two energy inequalities.
To prove (30), we multiply (24) by −(  + ) and take the imaginary part; after some complex-valued calculations, we obtain where Φ 1 () is given by ( 31), and By the Agmon inequality 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  1 .Namely, for each  0 ∈  1 and  ∈ R, (24) has a unique solution (⋅, ,  0 ) ∈ ([, +∞),  1 ) and the solution (, , ⋅) :  1 →  1 is continuous.This well-posed property permits us to define an evolution process  on  1 by (36)

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  1 ( is the radius) and  ∈ R, then there are  0 =  0 () such that, for all  0 ∈   and  ⩾  0 , Letting  = , we have which yields (37) if we take  0 = 2 ln( + 1)/.On the other hand, by (40) again, where we use the fact that ‖(r, ⋅)‖ 2 ⩽  1 () for r ⩽  ⩽ .We then consider the sixth power to obtain which yields From this, it is easy to deduce (39) with  0 () = (2 ln 2 + 6 ln  − ln )/ and similarly obtain (38).
We then consider the backward bound of solutions in  1 as follows.
Proof.Let  ⩽  and  ⩾ 0. Applying the Gronwall inequality on (30) over [ − , ], we get By Lemma 6 and A2, there is a On the other hand, by the definition of Φ 1 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  0 ∈   and  ⩾  0 (with a larger  0 ), By the Agmon inequality again, it follows from (31) that 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 K = {K()} ∈R in  1 given by Proof.By A1, A2, we can see that  1 () is an increasing and finite function with respect to ; this fact along with Lemma 7 shows that the nonautonomous set K is increasing, bounded, and pullback absorbing in  1 .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  1 if the assumption A1 is replaced by the weaker assumption that  3 () < ∞, although this weak assumption is enough for reaction-diffusion systems (see [29]), BBM equations (see [30]), and Navier-Stokes equations (see [31]).

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.

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  ⩽ 0 be arbitrary but fixed and take the initial value  0 in a ball   of  1 .Then we are concerned with two functions of  ⩾ 0:  =  () =  (; ) =    ( + , ,  0 ) ,  =  () =  (; ) =    ( + , ,  0 ) . (73) By the forward absorption given in Lemma 10, we know there is a  1 > 0 (depending on the radius of initial ball) such that sup The high-frequency part () satisfies the following equation.for all  ⩾  1 ,  ⩽ 0, and  0 ∈  1 .In the sequel, the main task is to prove the asymptotic compactness of V and the exponential decay of  in    1 for large .

The Uniformly Bounded Estimate for 𝑍.
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    1 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 A1  , A2  are satisfied.Then there exists  0 =  0 (, ) such that, for  ⩾  0 , (76) has a uniformly bounded solution  in    1 such that where  is a constant and  1 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 Ψ 2 (); by Assumption A1  , we have By the classical interpolation and the Poincaré inequality on the second term of Ψ 2 () can be bounded by Note that  1 forms a Banach multiplicative algebra in onedimension; that is, Then, by Lemma 10 and (74), the third term of Ψ 2 () can be bounded by (87) By the Agmon inequality and (83), we have Then the fifth term of Ψ 2 () is bounded by Similarly, by ( 83) and the Hölder inequality, the sixth term of Ψ 2 () is bounded by By (88), the rest terms can be bounded by By the above estimates, we know that Ψ 2 () can be bounded by (92) , where [⋅] denotes the integervalued function, we have, for  ⩾   0 , On the other hand, we similarly obtain a lower bound of Ψ 1 ().
We show that  1 ⩽ 2 2 .Indeed, where the last inequality is obviously true.Now the twice inequality (99) has the solution Since ( 1 ) = 0 and the mapping  → () is continuous on which proves the required bound.

Further Regularity Estimates.
We further show a regularity result for  in  2 .
Proposition 12. Suppose A1  , A2  , and A3 are satisfied.Then there exists  1 =  1 (, ) ⩾  0 such that, for  ⩾  1 , the solution  of ( 76) is uniformly bounded in    2 : where  is a constant and  1 is the forward absorbing entering time.
Proof.Just like we did in the above proposition, we need to firstly show the bound of    in  1 and then let  → +∞ to obtain the bound of   in  1 .Thus, for the sake of convenience, we omit the detail of letting  → +∞ and also drop the superscript  of   and write  =   , V = V  =  +   ,   =     in this proof.
The first thing we shall do is to obtain the following equation by differentiating (76): with initial value   ( 1 ) = 0, where ,   , and  are given by In order to show the needed result (105), we divide it into three steps.

Backward Compact Attractors
Finally, we state and prove the main result.We need to spilt the evolution process .Let  ⩾  2 be fixed, where  2 is the level given in Proposition 13.Then for all  ⩾ 0,  ∈ R, and  0 ∈  1 , we write  =  1 +  2 with where  1 =  1 (‖ 0 ‖  1 ) is the forward entering time.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  1 is backward asymptotically compact.Indeed, let  ∈ R and take some sequences   ⩽ ,   → +∞, and  0, ∈   ⊂  1 .Without loss of generality, we assume   ⩾ max{ 1 , }, where  1 is the forward entering time.Then by (133) we have which implies  2 is backward decay.
Remark 15. (I) In fact, under Assumptions A1  -A2  , 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 ,   ∈ (R,  2 ) and they are time-periodic, then  satisfies both conditions A1  and A2  .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.