The Property of the Solution about Cauchy Problem for Fourth-Order Schrödinger Equation with Critical Time-Oscillating Nonlinearity
We study the property of the solution in Sobolev spaces for the Cauchy problem of the following fourth-order Schrödinger equation with critical time-oscillating nonlinearity , where , , and is a periodic function. We obtain the asymptotic property of the solution for the above equation as under some conditions.
In this paper we study the Cauchy problem of the following fourth-order Schrödinger equation with a time oscillating critical nonlinearity where is a -periodic function and is a real constant. We define the solution of (1) to be .
In this paper we plan to study the behavior of as . We define to be the average value of on . Naturally, we think of the following equation:
We define the solution of (2) to be . We will study the relation of and as .
Definition 1. For two integers and , we say that is an admissible pair if the following condition is satisfied: To begin with, we give the following local well-posed results by similar techniques in .
Proposition 2. Let . Then there exists a unique -solution on a maximal time interval for (1). Moreover, for any admissible pair and , . If (or), then or.
Energy-critical fourth-order Schrödinger equations are very important equations which arise in many physical application fields [2, 3]. They have been discussed not only by physical researchers but also from mathematical viewpoint by many authors [4–8]. In particular, if the coefficient of nonlinear term is a periodic function in time, condensation and blowup phenomenon may appear from physical experiments. Naturally, from mathematical point, we will analyze why does it happen? What is the condition when condensation or blowup phenomenon appears? What is the property if condensation appears? For the Schrödinger equation, in  Fang and Han studied the Schrödinger equation with time-oscillating critical nonlinearities. They gave the asymptotic property of local solution and global solution for large frequencies (as ). As far as we know, there are few results about the asymptotic property of the solutions for the energy-critical fourth-order Schrödinger equations with periodic coefficient in time. So we will utilize the ideals and techniques of  to study the asymptotic behavior of the solution for (1) as . The difficulty of our work is how to seek power index. The index is dependent to space dimension. And the order of equation is four. This will make it more difficult to look for proper index for us. The aim of this paper is to study the property of the solution for the Cauchy problem of fourth-order Schrödinger equation with time-oscillating critical nonlinearities.
The main results of this work are the following theorems.
Theorem 3. Suppose that . For arbitrary initial data , we define and , respectively, as the maximal solution of (1) and (2) on maximal time . Then we have where and is arbitrary admissible pair.
Theorem 4. Suppose that . Let ,. Assume that is the global solution of (2), and . Then the solution of (1) is global if is sufficiently large. Moreover, we have for any admissible pair .
The succeeding section is devoted to establishing the dispersive estimates for the linear equation related to (1) and (2), and we will present the nonlinear estimates of nonlinearity. In Section 3 we present the proofs of Theorems 3 and 4.
2. Notations and Preliminaries
Given and a function space on , we denote by and , respectively, the following norm and the corresponding function space on .
For , and for , Later we will particularly take . For simplicity of the notations, we, respectively, abbreviate and as, respectively, and . In particular, for the case , we abbreviate as . We also abbreviate . In the following, we will introduce our four working spaces.
For any time interval , we denote
The fundamental solution of the linear equation related to (1) and (2) is given by the following oscillatory integral:
We denote by the fundamental solution operator So (1) and (2) have the following integral forms, respectively:
Lemma 5 ((Strichartz estimates) (see )). Assume , , and is a solution on of the following initial value problem: then for all admissible pairs and , we have
Lemma 6. Let be a compact time interval containing . Then we have
Proof. By the definition of , we obtain
Using Hardy-Littlewood-Sobolev inequality, we have which completes the proof.
Lemma 7. For any compact time interval , we have
Proof. Using Sobolev embedding (see ) , we have
Similarly, using Sobolev embedding and noting that is an admissible pair, we have
which completes the proof of the first inequality.
Next we prove the second inequality.
Using interpolation inequality (see ), we obtain Using Sobolev embedding, we obtain So we have which completes the proof of the second inequality.
Assume that the nonlinear function . Then we have the following two lemmas.
Lemma 8. Let be a compact time interval. Then, we have
Proof. Using Lemma of  and Lemma 7, we have For the second inequality, we have For , using Lemma of  and Lemma 7, we obtain For , using Lemma of  (, , ) and Sobolev inequality, we have From (28), using interpolation inequality and Lemma 7, we have From (26), (27), and (29), we have Similarly, we can prove which completes the proof.
Using Lemma 8, we immediately obtain the following lemma.
Lemma 9. Let . Then we have
Using the proof techniques in , similarly we can obtain the following lemma.
Lemma 10. For any admissible pair , , , we have
Lemma 11. Let ,. For any initial value , , assume that then we have where is admissible pair.
Proof. First we prove that .
By (34), there must be constants and such that From (11) and (12), we have where and .
By Lemma 10, we have Using Hölder inequality and Sobolev embedding inequality, we obtain where .
Using (38) and (39), we have
In the following we will prove that .
Since , we can divide the time interval into subintervals ,, where ,, such that in each part .
On , since , we have
For the case , we have
For the case , we have
On , we have
Similarly for the case , we have
For the case , we have
By induction, we have for .
So we have
Using the above estimate and (40), we have
In the following we discuss the estimate .
By (11) and (12), we have where and .
Using (11)-(12) and Lemma 5, we obtain
Noting that so .
Using Lemma 5 and Lemma 10, we have
Using Hölder inequality, Sobolev embedding, and Lemma 7, we obtain