Regularity properties of the cubic biharmonic Schrödinger equation on the half line

Abstract

In this paper we study the regularity properties of the cubic biharmonic Schrödinger equation posed on the right half line. We prove local well-posedness and obtain a smoothing result in the low-regularity spaces on the half line. In particular we prove that the nonlinear part of the solution on the half line is smoother than the initial data obtaining a full derivative gain in certain cases. Moreover, in the defocusing case, we establish global well-posedness and global smoothing in the higher order regularity spaces by making use of the global-wellposedness result of Özsarı and Yolcu (Commun Pure Appl Phys 18(6):3285–3316, 2019) in the energy space. Also this paper improves the well-posedness result of Özsarı and Yolcu (Commun Pure Appl Phys 18(6):3285–3316, 2019) in the case of cubic nonlinearity.

Appendix

In this section, we reserve some useful inequalities to be used in the text when necessary. Firstly we start with the lemma which is a consequence of the proof of Theorem 1.3 in [34].

Lemma 7.1

when $\mu =1$ $(\text {defocusing nonlinearity})$, the solutions of the Eq. (1) satisfy the following a priori estimate

$$\begin{aligned} \left\Vert u\right\Vert _{H^2({\mathbb {R}}^+)}\le C(\left\Vert g\right\Vert _{H^2},\left\Vert h_1\right\Vert _{H^1},\left\Vert h_2\right\Vert _{H^1}). \end{aligned}$$

Next Lemma is useful in the proofs of Proposition 4.6 and Proposition 4.7.

Lemma 7.2

For $m,n,k\in {\mathbb {R}}$ we have

$$\begin{aligned} |m^4-n^4+k^4-(m-n+k)^4|\gtrsim |m-n||n-k|(m^2+n^2+k^2). \end{aligned}$$

Proof

Let $g(m,n,k):=m^4-n^4+k^4-(m-n+k)^4$. Then

$$\begin{aligned} g(m,n,k)&=(m-n)\big [(m^2+n^2)(m+n)-(m-n)^3-4(m-n)^2k-6(m-n)k^2-4k^3\big ]\\&=(m-n)(n-k)\big [4m^2+2n^2+4k^2-2mn-2nk\big ]\\&=(m-n)(n-k)\Big [\frac{5}{2}(m+n)^2+m^2+k^2+2(n-\frac{1}{2}m-\frac{1}{2}k)^2\Big ] \end{aligned}$$

which gives the desired estimate. $\square$

Lemma 7.3

(See [6]) For $-\frac{1}{2}\le s \le \frac{1}{2}$, we have

$$\begin{aligned} \left\Vert fg\right\Vert _{H^s}\lesssim \left\Vert f\right\Vert _{H^{\frac{1}{2}+}}\left\Vert g\right\Vert _{H^s} \end{aligned}$$

Finally we have the following lemmas we use throughout the text. For proofs of the first and the second of these, see [13] and [16] respectively.

Lemma 7.4

If $\beta \ge \gamma \ge 0$ and $\beta +\gamma >1$ then

$$\begin{aligned} \int _{{\mathbb {R}}}\frac{dx}{\langle x-a_1\rangle ^{\beta } \langle x-a_2\rangle ^{\gamma }}\lesssim \langle a_1-a_2\rangle ^{-\gamma }\varphi _{\beta }(a_1-a_2) \end{aligned}$$

where

$$\begin{aligned} \varphi _{\beta }(a)=\sum _{|n|\le |a|}\frac{1}{\langle n \rangle ^{\beta }} \sim \left\{ \begin{array}{ll} 1 &{}\quad \beta >1 \\ \log (1+\langle a\rangle ) &{} \quad \beta =1 \\ \langle a \rangle ^{1-\beta } &{}\quad \beta <1. \end{array}\right. \end{aligned}$$

Lemma 7.5

For fixed $\rho \in (\frac{1}{2},1)$, we have

$$\begin{aligned} \int \frac{1}{\langle x\rangle ^{\rho }\sqrt{|x-a|}}dx\lesssim \frac{1}{\langle a\rangle ^{\rho -\frac{1}{2}}}. \end{aligned}$$