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The asymptotic property for nonlinear fourth-order Schrödinger equation with gain or loss

Boundary Value Problems20152015:177

https://doi.org/10.1186/s13661-015-0442-1

  • Received: 22 May 2015
  • Accepted: 18 September 2015
  • Published:

Abstract

We study the Cauchy problem of the nonlinear fourth-order Schrödinger equation with gain or loss: \(iu_{t}+\triangle^{2}u+\lambda|u|^{\alpha}u +i\varepsilon a(t)|u|^{\beta}u=0\), \(x\in R^{n}\), \(t\in R\), where \(2\leq\alpha\leq\frac{8}{n-4}\) and \(2\leq\beta\leq\frac{8}{n-4}\), ε is a real number, \(a(t)\) is a real function, and \(n>4\). We study the asymptotic properties of its local and global solutions as \(\varepsilon\rightarrow0\).

Keywords

  • nonlinear fourth-order Schrödinger equation with gain or loss
  • Fourier restriction norm method
  • Cauchy problem

1 Introduction

In this paper we study the following nonlinear fourth-order Schrödinger equation with gain or loss:
$$ \left \{ \textstyle\begin{array}{@{}l} iu_{t}+\triangle^{2}u+\lambda|u|^{\alpha}u +i\varepsilon a(t)|u|^{\beta}u=0,\quad x\in R^{n}, t\in R, \\ u(x,0)=u_{0}(x),\quad x\in R^{n}, \end{array}\displaystyle \right . $$
(1.1)
where \(u(x,t)\) are complex-valued function. We have \(2\leq\alpha\leq \frac{8}{n-4}\) and \(2\leq\beta\leq\frac{8}{n-4}\), ε is a real number, \(a(t)\) is a real function, and \(n>4\).
For the case \(\varepsilon=0\), the above equation is the nonlinear fourth-order Schrödinger equation,
$$ \left \{ \textstyle\begin{array}{@{}l} iu_{t}+\triangle^{2}u+\lambda|u|^{\alpha}u=0, \quad x\in R^{n}, t\in R, \\ u(x,0)=u_{0}(x),\quad x\in R^{n}. \end{array}\displaystyle \right . $$
(1.2)
For (1.2), in [1] we have obtained the local well-posedness result in the space \(C([-T,T], H^{2}(R^{n}))\) if \(n>4\) and \(2\leq\alpha\leq\frac{8}{n-4}\). We also get the global well-posedness result in the space \(C(R,H^{2}(R^{n}))\) if \(n>4\) and \(\lambda>0\), \(2\leq\alpha\leq\frac{8}{n-4}\) or \(\lambda<0\), \(2\leq\alpha\leq\frac{8}{n}\). For the energy-critical case, in [2] and [3], Pausader Benoit gives the global well-posedness and scattering for \(n\geq5\) and radial initial data. In [4], Miao et al. study the defocusing case and obtain the global existence for \(n\geq9\). In [5], Zhang and Zheng obtain the global solution and scattering for \(n=8\). Pausader Benoit also discusses the mass-critical case in [6].

For the case \(\varepsilon\neq0\), \(a(t)\) is the gain (loss) if \(a(t)<0\) (\(a(t)>0\)). In [7], the authors discuss the Schrödinger equation with gain. They have obtained the result: The value of \(a(t)\) will determine whether or not the solution will blow up. Feng et al. study the Schrödinger equation with gain/loss in [8] and [9]. They, respectively, give the limit behavior of solution as \(\varepsilon\rightarrow0\) and the global solution and blow-up result. As far as we know, there are fewer results about the fourth-order Schrödinger equation with gain. In this paper, we will discuss the local well-posedness and the global well-posedness of (1.1); especially, we will discuss the asymptotic behavior of the solution as \(\varepsilon\rightarrow0\).

2 The preliminary estimates

First, we denote by \(U(t)\) (\(t\in R\)) the fundamental solution operator of the fourth-order Schrödinger equation [10], i.e.,
$$U(t)\varphi(x)=F^{-1} \bigl(e^{-it\xi^{4}}\hat{\varphi}(\xi) \bigr) \quad \mbox{for } \varphi\in S^{\prime}(R), $$
where φ̂ denotes the Fourier transformation of φ, and \(F^{-1}\) represents the inverse Fourier transformation.
Thus the equivalent integral equations [11] of (1.1) and (1.2) are, respectively,
$$ u_{\varepsilon}(t)=U(t)u_{0}+i\lambda\int_{0}^{t}U(t-s) \bigl(|u_{\varepsilon}|^{\alpha}u_{\varepsilon}\bigr) (s)\,ds - \varepsilon\int_{0}^{t}U(t-s)a(s) \bigl(|u_{\varepsilon}|^{\beta}u_{\varepsilon}\bigr) (s)\,ds $$
(2.1)
and
$$ u(t)=U(t)u_{0}+i\lambda\int_{0}^{t}U(t-s) \bigl(|u|^{\alpha}u \bigr) (s)\,ds. $$
(2.2)
Second, we introduce the following notations. For any given \(T>0\), we define the space \(L^{q}(0,T;W^{2,r}(R^{n}))\) with the norm
$$\|u\|_{L^{q}(0,T;W^{2,r})}:= \biggl(\int_{0}^{T}\bigl\| u( \cdot,t)\bigr\| ^{q}_{W^{2,r}(R^{n})}\,dt \biggr)^{\frac{1}{q}}. $$
For two integers \(8\leq q\leq\infty\) and \(2\leq r<\infty\), we say that \((q,r)\) is an admissible pair if the following condition is satisfied:
$$\frac{2}{q}=\frac{n}{4} \biggl(1-\frac{2}{r} \biggr). $$

For simplicity, in this paper, we will use C to denote various constants which may be different from line to line.

We have the following Strichartz estimate (see [1]): For any admissible pair \((q,r)\)
$$ \bigl\| U(t)\varphi(x)\bigr\| _{L^{q}(0,l;L^{r})}\leq C\|\varphi\|_{L^{2}} $$
(2.3)
and
$$ \biggl\| \int_{0}^{t}U(t-s)f(x,t)\,ds\biggr\| _{L^{q}(0,l;L^{r})} \leq C\|f\|_{L^{\gamma ^{\prime}}(0,l;L^{\rho^{\prime}})}, $$
(2.4)
where \((\gamma,\rho)\) is an arbitrary admissible pair, and ′ represents the conjugate number.

From Theorem 4.5 of [1], we have the following results.

Proposition 2.1

(subcritical case)

Assume that \(n>4\), \(a\in L^{\infty}(0,\infty)\), \(2\leq\alpha<\frac{8}{n-4}\), and \(2\leq\beta<\frac{8}{n-4}\), \((\gamma_{1},\rho_{1})=(\alpha+2,\frac{2n(\alpha+2)}{n(\alpha+2)-8})\), \((\gamma_{2},\rho_{2})=(\frac{8(\beta+2)}{n\beta},\beta+2)\). For any \(u_{0}\in H^{2}(R^{n})\), there exists δ such that the Cauchy problem (1.1) has a unique solution \(u_{\varepsilon}\) in the space \(L^{\infty}(0,\delta;H^{2}(R^{n}))\cap L^{\gamma_{1}}(0,\delta;W^{2,\rho_{1}}(R^{n})) \cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}}(R^{n}))\). Moreover,
$$\|u_{\varepsilon}\|_{L^{\infty}(0,\delta;H^{2}) \cap L^{\gamma_{1}}(0,\delta;W^{2,\rho_{1}}) \cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}})}\leq2\|u_{0}\|_{H^{2}}. $$

Proposition 2.2

(critical case)

Assume that \(n>4\), \(a\in L^{\infty}(0,\infty)\), \(\alpha=\frac{8}{n-4}\), \(2\leq\beta<\frac{8}{n-4}\), \((\gamma^{*},\rho^{*})=(\frac{2n}{n-4},\frac{2n^{2}}{n^{2}-4n+16})\), \((\gamma_{2},\rho_{2})=(\frac{8(\beta+2)}{n\beta},\beta+2)\). For any \(u_{0}\in H^{2}(R^{n})\), there exists δ such that the Cauchy problem (1.1) has a unique solution \(u_{\varepsilon}\) in the space \(L^{\infty}(0,\delta;H^{2}(R^{n}))\cap L^{\gamma^{*}}(0,\delta;W^{2,\rho^{*}}(R^{n})) \cap L^{\gamma_{2}}(0,\delta;W^{2,{\rho_{2}}}(R^{n}))\). Moreover,
$$\|u_{\varepsilon}\|_{L^{\infty}(0,\delta;H^{2})\cap L^{\gamma^{*}}(0,\delta ;W^{2,\rho^{*}}) \cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}})} \leq3\bigl\| U(t)u_{0} \bigr\| _{L^{\gamma^{*}}(0,\delta;W^{2,\rho^{*}})}. $$

3 Main results

Lemma 3.1

Let n, α, β, \((\gamma_{1},\rho_{1})\), \((\gamma _{2},\rho_{2})\) be as in Proposition  2.1. Assume that u is the solution of (1.2), defined on a maximal time interval \([0,T^{*})\), \(0< l< T^{*}\), and \(u_{\varepsilon}\) exists on \([0,l]\). If \(\lim\sup_{\varepsilon\rightarrow 0}\|u_{\varepsilon}\|_{L^{\infty}(0,l;H^{2})\cap L^{\gamma_{1}}(0,l;W^{2,\rho _{1}})\cap L^{\gamma_{2}}(0,l;W^{2,\rho_{2}})}<+\infty\), then we have \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,l;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), where \((q,r)\) is arbitrary admissible pair.

Proof

First, we prove
$$\|u_{\varepsilon}-u\|_{L^{q}(0,l;L^{r})}\rightarrow0 \quad\mbox{as } \varepsilon \rightarrow0. $$
From (2.1) and (2.2), using Strichartz estimates, we have
$$\begin{aligned} &\|u_{\varepsilon}-u\|_{L^{q}(0,l;L^{r})} \\ &\quad\leq\bigl\| J(t) \bigr\| _{L^{q}(0,l;L^{r})}+\bigl\| K(t)\bigr\| _{L^{q}(0,l;L^{r})} \\ &\quad\leq C\bigl\| |u_{\varepsilon}|^{\alpha}u_{\varepsilon}-|u|^{\alpha}u\bigr\| _{L^{\gamma ^{\prime}}(0,l;L^{\rho^{\prime}})} +C\varepsilon\|a\|_{L^{\infty}(0,l)} \bigl\| |u_{\varepsilon}|^{\beta}u_{\varepsilon}\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})}, \end{aligned}$$
(3.1)
where \(J(t)=i\lambda\int_{0}^{t}U(t-s)(|u_{\varepsilon}|^{\alpha}u_{\varepsilon}-|u|^{\alpha}u)(s)\,ds\), \(K(t)=-\varepsilon\int_{0}^{t}U(t-s)a(s)(|u_{\varepsilon}|^{\beta}u_{\varepsilon})(s)\,ds\), \((\gamma,\rho)=(\frac{8(\alpha+2)}{n\alpha},\alpha+2)\).
Since \(\lim\sup_{\varepsilon\rightarrow0}\|u_{\varepsilon}\|_{L^{\infty }(0,l;H^{2})\cap L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})\cap L^{\gamma _{2}}(0,l;W^{2,\rho_{2}})}<+\infty\), there exist \(N_{1}, \varepsilon_{0}\) such that
$$\|u_{\varepsilon}\|_{L^{\infty}(0,l;H^{2})\cap L^{\gamma_{1}}(0,l;W^{2,\rho _{1}})\cap L^{\gamma_{2}}(0,l;W^{2,\rho_{2}})}\leq N_{1} \quad \mbox{for all } \varepsilon< \varepsilon_{0}. $$
Let \(N_{2}=\|u\|_{L^{\infty}(0,l;H^{2})}\), it is obvious that \(N_{2}<+\infty\). Using the Hölder inequality and the Sobolev embedding [12], we have
$$\begin{aligned} \bigl\| |u_{\varepsilon}|^{\alpha}u_{\varepsilon}-|u|^{\alpha}u\bigr\| _{L^{\gamma ^{\prime}}(0,l;L^{\rho^{\prime}})}&\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{a}(0,l;L^{\alpha+2})} +\|u\|^{\alpha}_{L^{a}(0,l;L^{\alpha+2})} \bigr) \|u_{\varepsilon}-u \|_{L^{\gamma}(0,l;L^{\rho})} \\ &\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{\infty}(0,l;H^{2})} +\|u \|^{\alpha}_{L^{\infty}(0,l;H^{2})} \bigr) \|u_{\varepsilon}-u\|_{L^{\gamma}(0,l;L^{\rho})} \\ &\leq C \bigl(N_{1}^{\alpha}+N_{2}^{\alpha}\bigr) \|u_{\varepsilon}-u\|_{L^{\gamma}(0,l;L^{\rho})}, \end{aligned}$$
(3.2)
where \(a=\frac{4\alpha(\alpha+2)}{8-(n-4)\alpha}\).
Similarly, we have
$$\begin{aligned} \bigl\| |u_{\varepsilon}|^{\beta}u_{\varepsilon}\bigr\| _{L^{\gamma_{2}^{\prime}} (0,l;L^{\rho_{2}^{\prime}})} &\leq\|u_{\varepsilon}\|^{\beta} _{L^{b}(0,l;L^{\beta+2})} \|u_{\varepsilon}\|_{L^{\gamma_{2}}(0,l;L^{\rho_{2}})} \\ &\leq\|u_{\varepsilon}\|^{\beta} _{L^{\infty}(0,l;H^{2})} \|u_{\varepsilon}\|_{L^{\gamma_{2}}(0,l;L^{\rho_{2}})} \leq N_{1}^{\beta+1}, \end{aligned}$$
(3.3)
where \(b=\frac{4\beta(\beta+2)}{8-(n-4)\beta}\).
Let \(N_{3}=C\|a\|_{L^{\infty}}N_{1}^{\beta+1}\). Substituting (3.2) and (3.3) into (3.1), we have
$$ \|u_{\varepsilon}-u\|_{L^{q}(0,l;L^{r})}\leq C \bigl(N_{1}^{\alpha}+N_{2}^{\alpha}\bigr) \| u_{\varepsilon}-u\|_{L^{\gamma}(0,l;L^{\rho})}+\varepsilon N_{3}. $$
(3.4)

In the following we will prove that \(\|u_{\varepsilon}-u\|_{L^{\gamma}(0,l;L^{\rho})}\rightarrow0\) as \(\varepsilon\rightarrow 0\).

Noting that \(N_{1}, N_{2}<\infty\), we can divide the time interval \([0,l]\) into subintervals \([t_{i},t_{i+1}]\), \(i=0, 1, \ldots, J-1\), where \(t_{0}=0\), \(t_{J-1}=l\) such that in each part \(C(\|u_{\varepsilon}\|^{\alpha}_{L^{a}(t_{i},t_{i+1};L^{\alpha+2})} +\|u\|^{\alpha}_{L^{a}(t_{i},t_{i+1};L^{\alpha+2})})=\frac{1}{2}\).

On \([t_{0},t_{1}]\), since \(u_{\varepsilon}(t_{0})=u(t_{0})=u_{0}\), we have
$$\|u_{\varepsilon}-u\|_{L^{\gamma}(t_{0},t_{1};L^{\rho})}\leq\frac{1}{2}\| u_{\varepsilon}-u\|_{L^{\gamma}(t_{0},t_{1};L^{\rho})}+\varepsilon N_{3}, $$
which means
$$\|u_{\varepsilon}-u\|_{L^{\gamma}(t_{0},t_{1};L^{\rho})}\leq2\varepsilon N_{3}. $$
By (3.4), we have
$$\|u_{\varepsilon}-u\|_{L^{\infty}(t_{0},t_{1};L^{2})}\leq2\varepsilon N_{3}. $$
On \([t_{1},t_{2}]\), we have
$$\begin{aligned} \|u_{\varepsilon}-u\|_{L^{\gamma}(t_{1},t_{2};L^{\rho})}&\leq\bigl\| u_{\varepsilon}(t_{1})-u(t_{1})\bigr\| _{L^{2}}+\frac{1}{2} \|u_{\varepsilon}-u\|_{L^{\gamma}(t_{1},t_{2};L^{\rho})}+\varepsilon N_{3} \\ &\leq3\varepsilon N_{3}+\frac{1}{2}\|u_{\varepsilon}-u \|_{L^{\gamma}(t_{1},t_{2};L^{\rho})}, \end{aligned}$$
from which we can obtain
$$\|u_{\varepsilon}-u\|_{L^{\gamma}(t_{1},t_{2};L^{\rho})}\leq6\varepsilon N_{3}. $$

Especially, we have \(\|u_{\varepsilon}-u\|_{L^{\infty}(t_{1},t_{2};L^{2})}\leq 6\varepsilon N_{3}\).

By induction, we have
$$\begin{aligned}& \|u_{\varepsilon}-u\|_{L^{\gamma}(t_{i},t_{i+1};L^{\rho})}\leq 2 \bigl(2^{i+1}-1 \bigr) \varepsilon N_{3}, \\& \|u_{\varepsilon}-u\|_{L^{\infty}(t_{i},t_{i+1};L^{2})}\leq 2 \bigl(2^{i+1}-1 \bigr) \varepsilon N_{3}, \quad\mbox{for } i=0, 1, \ldots, J-1. \end{aligned}$$
So we have
$$\|u_{\varepsilon}-u\|_{L^{\gamma}(0,l;L^{\rho})}\leq\sum_{i=0}^{J-1}2 \bigl(2^{i+1}-1 \bigr)\varepsilon N_{3}= \bigl[4 \bigl(2^{J}-1 \bigr)-2J \bigr]\varepsilon N_{3} \rightarrow0. $$
Furthermore, we have
$$\|u_{\varepsilon}-u\|_{L^{q}(0,l;L^{r})}\rightarrow0 \quad\mbox{as } \varepsilon \rightarrow0. $$
Second, we prove
$$\|\nabla u_{\varepsilon}-\nabla u\|_{L^{q}(0,l;L^{r})}\rightarrow0 \quad\mbox{as } \varepsilon\rightarrow0. $$
From (2.1) and (2.2), we have
$$\nabla(u_{\varepsilon}-u)= i\lambda\int_{0}^{t}U(t-s) \nabla \bigl(|u_{\varepsilon}|^{\alpha}u_{\varepsilon}-|u|^{\alpha}u \bigr) (s)\,ds -\varepsilon\int_{0}^{t}U(t-s)a(s) \nabla \bigl(|u_{\varepsilon}|^{\beta}u_{\varepsilon}\bigr) (s)\,ds. $$
Let \(g_{1}(u)=|u|^{\alpha}u\), \(g_{2}(u)=|u|^{\beta}u\). Then, using Strichartz estimates, we have
$$\begin{aligned} &\bigl\| \nabla(u_{\varepsilon}-u)\bigr\| _{L^{q}(0,l;L^{r})} \\ &\quad\leq C\biggl\| \int_{0}^{t}U(t-s)\nabla \bigl(g_{1}(u_{\varepsilon})-g_{1}(u) \bigr) (s)\,ds \biggr\| _{L^{q}(0,l;L^{r})} \\ &\qquad{}+C\varepsilon\biggl\| \int_{0}^{t}U(t-s)a(s) \nabla g_{2}(u_{\varepsilon}) (s)\,ds\biggr\| _{L^{q}(0,l;L^{r})} \\ &\quad\leq C\bigl\| \nabla \bigl(g_{1}(u_{\varepsilon})-g_{1}(u) \bigr)\bigr\| _{L^{{\gamma _{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})}+ C\varepsilon\|a\|_{L^{\infty}(0,l)}\bigl\| \nabla g_{2}(u_{\varepsilon})\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})} \\ &\quad\leq C\bigl\| g^{\prime}_{1}(u_{\varepsilon}) \nabla(u_{\varepsilon}-u)\bigr\| _{L^{{\gamma _{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})}+C\bigl\| \bigl(g^{\prime}_{1}(u_{\varepsilon})-g^{\prime}_{1}(u) \bigr)\nabla u\bigr\| _{L^{{\gamma_{1}}^{\prime}} (0,l;L^{{\rho_{1}}^{\prime}})} \\ &\qquad{}+C\varepsilon\|a\|_{L^{\infty}(0,l)}\bigl\| \nabla g_{2}(u_{\varepsilon}) \bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})}. \end{aligned}$$
(3.5)
Using the Hölder inequality, the Sobolev embedding, and the Young inequality, we obtain
$$\begin{aligned} &\bigl\| g^{\prime}_{1}(u_{\varepsilon}) \nabla(u_{\varepsilon}-u)\bigr\| _{L^{{\gamma _{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;L^{c})} \bigl\| \nabla(u_{\varepsilon}-u)\bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigl\| \nabla(u_{\varepsilon}-u)\bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}\quad \biggl(c=\frac {\rho_{1}\alpha}{\rho_{1}-2} \biggr), \end{aligned}$$
(3.6)
$$\begin{aligned} &\bigl\| \nabla g_{2}(u_{\varepsilon})\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho _{2}^{\prime}})} \leq C\|u_{\varepsilon}\|^{\beta}_{L^{\infty}(0,l;H^{2})} \|u_{\varepsilon}\|_{L^{\gamma_{2}}(0,l;W^{2,\rho_{2}})}, \end{aligned}$$
(3.7)
and
$$\begin{aligned} &\bigl\| \bigl(g^{\prime}_{1}(u_{\varepsilon})-g^{\prime}_{1}(u) \bigr)\nabla u\bigr\| _{L^{{\gamma _{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha-1}_{L^{\gamma_{1}}(0,l;L^{d_{1}})} +\|u \|^{\alpha-1}_{L^{\gamma_{1}}(0,l;L^{d_{1}})} \bigr) \|u_{\varepsilon}-u\|_{L^{\gamma_{1}}(0,l;L^{e_{1}})} \| \nabla u\|_{L^{\gamma_{1}}(0,l;L^{e_{1}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha-1}_{L^{\gamma_{1}}(0,l;W^{2,{\rho_{1}}})} +\|u \|^{\alpha-1}_{L^{\gamma_{1}}(0,l;W^{2,{\rho_{1}}})} \bigr) \bigl\| \nabla(u_{\varepsilon}-u) \bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})} \| u\|_{L^{\gamma_{1}}(0,l;W^{2,{\rho_{1}}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,{\rho_{1}}})} +\|u \|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,{\rho_{1}}})} \bigr) \bigl\| \nabla(u_{\varepsilon}-u) \bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}, \end{aligned}$$
(3.8)
where \(d_{1}=\frac{2n(\alpha+2)(\alpha-1)}{24-(n-4)(\alpha+2)}\), \(e_{1}=\frac {2n(\alpha+2)}{(n-2)(\alpha+2)-8}\).
Substituting (3.6)-(3.8) into (3.5), we have
$$\bigl\| \nabla(u_{\varepsilon}-u)\bigr\| _{L^{q}(0,l;L^{r})}\leq C \bigl(\|u_{\varepsilon}\| ^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,{\rho_{1}}})}+ \|u\|^{\alpha}_{L^{\gamma_{1}} (0,l;W^{2,{\rho_{1}}})} \bigr) \bigl\| \nabla(u_{\varepsilon}-u)\bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}+\varepsilon N_{3}. $$
Similar to the proof in the first step, we have
$$\bigl\| \nabla(u_{\varepsilon}-u)\bigr\| _{L^{q}(0,l;L^{r})}\rightarrow0 \quad \mbox{as } \varepsilon\rightarrow0. $$
At last, we prove
$$\bigl\| D^{2} u_{\varepsilon}-D^{2} u\bigr\| _{L^{q}(0,l;L^{r})} \rightarrow0 \quad\mbox{as } \varepsilon\rightarrow0. $$
By simple computing, we have
$$ D^{2} u_{\varepsilon}-D^{2} u=i(K_{1}+K_{2}+K_{3}), $$
(3.9)
where \(K_{1}=\lambda\int_{0}^{t}U(t-s)A_{1}(u_{\varepsilon},u)(s)\,ds\), \(K_{2}=\lambda\int_{0}^{t}U(t-s)A_{2}(u_{\varepsilon},u)(s)\,ds\), \(K_{3}=-\varepsilon\int_{0}^{t}U(t-s)a(s)A_{3}(u_{\varepsilon})(s)\,ds\). The arrays \(A_{1}(u_{\varepsilon},u)=g_{1}^{\prime}(u_{\varepsilon})D^{2}(u_{\varepsilon}-u)+g_{1}^{\prime\prime}(u_{\varepsilon})D(u_{\varepsilon}-u)\times Du\), \(A_{2}(u_{\varepsilon},u)=Du\times[g_{1}^{\prime\prime}(u_{\varepsilon})Du_{\varepsilon}-g_{1}^{\prime\prime}(u)Du]+ [g_{1}^{\prime}(u_{\varepsilon})-g_{1}^{\prime}(u)]D^{2}u\), \(A_{3}(u_{\varepsilon})= g_{2}^{\prime\prime}(u_{\varepsilon})Du_{\varepsilon}\times Du_{\varepsilon}+ g_{2}^{\prime}(u_{\varepsilon})D^{2}u_{\varepsilon}\).
By the Hölder inequality and the Sobolev embedding, we have
$$ \bigl\| g_{1}^{\prime}(u_{\varepsilon})D^{2}(u_{\varepsilon}-u) \bigr\| _{L^{{\gamma _{1}}^{\prime}}(0,l;L^{\rho_{1}^{\prime}})} \leq\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigl\| D^{2}(u_{\varepsilon}-u)\bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})} $$
(3.10)
and
$$\begin{aligned} &\bigl\| g_{1}^{\prime\prime}(u_{\varepsilon})D(u_{\varepsilon}-u)^{\bot} \times Du_{\varepsilon}\bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\quad\leq\|u_{\varepsilon}\|^{\alpha-1}_{L^{\gamma_{1}}(0,l;L^{d_{1}})} \bigl\| D(u_{\varepsilon}-u)\bigr\| _{L^{\gamma_{1}}(0,l;L^{e_{1}})} \|Du_{\varepsilon}\|_{L^{\gamma_{1}}(0,l;L^{e_{1}})} \\ &\quad\leq\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigl\| D^{2}(u_{\varepsilon}-u) \bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}. \end{aligned}$$
(3.11)
Thus we have from (3.10) and (3.11)
$$\begin{aligned} \|K_{1}\|_{L^{q}(0,l;L^{r})} &\leq\bigl\| g_{1}^{\prime}(u_{\varepsilon})D^{2}(u_{\varepsilon}-u) \bigr\| _{L^{\gamma _{1}^{\prime}}(0,l;L^{\rho_{1}^{\prime}})} +\bigl\| g_{1}^{\prime\prime}(u_{\varepsilon})D(u_{\varepsilon}-u)^{\bot} \times Du_{\varepsilon}\bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\leq C\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigl\| D^{2}(u_{\varepsilon}-u)\bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}. \end{aligned}$$
(3.12)
Similar to the proof of (3.11), we obtain
$$\begin{aligned} &\bigl\| g_{1}^{\prime\prime}(u_{\varepsilon})D(u_{\varepsilon}-u)^{\bot} \times Du\bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\quad\leq \bigl(\|u\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} +\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigr) \bigl\| D^{2}(u_{\varepsilon}-u) \bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}. \end{aligned}$$
(3.13)
Noting that \(\alpha\geq2\), we have
$$\begin{aligned} &\bigl\| \bigl(g_{1}^{\prime\prime}(u_{\varepsilon})-g_{1}^{\prime\prime}(u) \bigr)Du^{\bot}\times Du\bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\quad\leq C\bigl\| \bigl(|u_{\varepsilon}|^{\alpha-2}+|u|^{\alpha-2} \bigr) (u_{\varepsilon}-u)Du^{\bot}\times Du\bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha-2}_{L^{\gamma_{1}}(0,l;L^{c})} +\|u \|^{\alpha-2}_{L^{\gamma_{1}}(0,l;L^{c})} \bigr) \|u_{\varepsilon}-u\|_{L^{\gamma_{1}}(0,l;L^{d_{2}})} \|Du\|^{2}_{L^{\gamma _{1}}(0,l;L^{e_{2}})} \\ &\quad\leq \bigl(\|u\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} +\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigr) \bigl\| D^{2}(u_{\varepsilon}-u) \bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}, \end{aligned}$$
(3.14)
where \(e_{2}=\frac{2n(\alpha+2)}{(n-2)(\alpha+2)-8}\), \(\frac{1}{{\rho_{1}}^{\prime}}=\frac{(\rho_{1}-2)(\alpha-2)}{\rho_{1}\alpha}+\frac {1}{d_{2}}+\frac{2}{e_{2}}\).
Similarly, using the Hölder inequality and the Sobolev embedding, we obtain
$$\begin{aligned} &\bigl\| \bigl(g_{1}^{\prime}(u_{\varepsilon})-g_{1}^{\prime}(u) \bigr)D^{2}u\bigr\| _{L^{{\gamma _{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha-1}_{L^{\gamma_{1}}(0,l;L^{c})}+\|u\| ^{\alpha-1}_{L^{\gamma_{1}}(0,l;L^{c})} \bigr) \|u_{\varepsilon}-u\|_{L^{\gamma_{1}}(0,l;L^{c})} \bigl\| D^{2}u\bigr\| _{L^{\gamma _{1}}(0,l;L^{\rho_{1}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})}+\|u\| ^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigr) \bigl\| D^{2}(u_{\varepsilon}-u) \bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}. \end{aligned}$$
(3.15)
Thus we have from (3.13) and (3.15)
$$\begin{aligned} \|K_{2}\|_{L^{q}(0,l;L^{r})} \leq{}& \bigl\| A_{2}(u_{\varepsilon},u) \bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ \leq{}&\bigl\| g_{1}^{\prime\prime}(u_{\varepsilon})D(u_{\varepsilon}-u)^{\bot} \times Du\bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &{}+\bigl\| \bigl(g_{1}^{\prime\prime}(u_{\varepsilon})-g_{1}^{\prime\prime}(u) \bigr)Du^{\bot}\times Du\bigr\| _{L^{{\gamma_{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ &{}+\bigl\| \bigl(g_{1}^{\prime}(u_{\varepsilon})-g_{1}^{\prime}(u) \bigr)D^{2}u\bigr\| _{L^{{\gamma _{1}}^{\prime}}(0,l;L^{{\rho_{1}}^{\prime}})} \\ \leq{}& C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} +\|u \|^{\alpha}_{L^{\gamma_{1}}(0,l;W^{2,\rho_{1}})} \bigr) \bigl\| D^{2}(u_{\varepsilon}-u) \bigr\| _{L^{\gamma_{1}}(0,l;L^{\rho_{1}})}. \end{aligned}$$
(3.16)
Similar to the proof of (3.3), we obtain
$$\begin{aligned} \bigl\| g_{2}^{\prime}(u_{\varepsilon})D^{2}u_{\varepsilon}\bigr\| _{L^{\gamma _{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})} &\leq\|u_{\varepsilon}\|^{\beta}_{L^{b}(0,l;L^{\beta+2})} \bigl\| D^{2}u_{\varepsilon}\bigr\| _{L^{\gamma_{2}}(0,l;L^{\rho_{2}})}\leq N_{1}^{\beta+1} \\ &\leq\|u_{\varepsilon}\|^{\beta}_{L^{\infty}(0,l;H^{2})} \|u_{\varepsilon}\|_{L^{\gamma_{2}}(0,l;W^{2,\rho_{2}})}\leq N_{1}^{\beta+1} \end{aligned}$$
(3.17)
and
$$\begin{aligned} \bigl\| g_{2}^{\prime\prime}(u_{\varepsilon})Du_{\varepsilon}^{\bot} \times Du_{\varepsilon}\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})} &\leq\|u_{\varepsilon}\|^{\beta-1}_{L^{b}(0,l;L^{\beta+2})} \|Du_{\varepsilon}\|^{2}_{L^{\gamma_{2}}(0,l;L^{\rho^{2}})} \\ &\leq \|u_{\varepsilon}\|^{\beta+1}_{L^{\gamma_{2}}(0,l;W^{2,\rho_{2}})}\leq N_{1}^{\beta+1}. \end{aligned}$$
(3.18)
From (3.17) and (3.18), we immediately obtain
$$\begin{aligned} \|K_{3}\|_{L^{q}(0,l;L^{r})} &\leq\varepsilon\|a \|_{L^{\infty}(0,l)} \bigl\| A_{3}(u_{\varepsilon})\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})} \\ &\leq\varepsilon\|a\|_{L^{\infty}(0,l)} \bigl[\bigl\| g_{2}^{\prime} (u_{\varepsilon})D^{2}u_{\varepsilon}\bigr\| _{L^{\gamma_{2}^{\prime}} (0,l;L^{\rho_{2}^{\prime}})} + \bigl\| g_{2}^{\prime\prime}(u_{\varepsilon})Du_{\varepsilon}^{\bot} \times Du_{\varepsilon}\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})} \bigr] \\ &\leq\varepsilon N_{3}. \end{aligned}$$
(3.19)
Taking, respectively, \((q,r)=(\gamma,\rho)\) and \((q,r)=(\gamma_{1},\rho_{1})\) in (3.9), (3.12), (3.16), and (3.19), similar to the method of the first step, we can obtain
$$\bigl\| D^{2} u_{\varepsilon}-D^{2} u\bigr\| _{L^{q}(0,l;L^{r})} \rightarrow0 \quad\mbox{as } \varepsilon\rightarrow0. $$
 □

Noting that if \(\alpha=\frac{8}{n-4}\), a in (3.2) will be meaningless. So we will need the following lemma for the critical case.

Lemma 3.2

Let n, α, β, \((\gamma^{*},\rho^{*})\), \((\gamma _{2},\rho_{2})\) be as in Proposition  2.2. Assume that u is the solution of (1.2), defined on a maximal time interval \([0,T^{*})\), \(0< l< T^{*}\), and \(u_{\varepsilon}\) exists on \([0,l]\). If \(\lim\sup_{\varepsilon\rightarrow 0}\|u_{\varepsilon}\|_{L^{\infty}(0,l;H^{2})\cap L^{\gamma ^{*}}(0,l;W^{2,\rho^{*}})\cap L^{\gamma_{2}}(0,l;W^{2,\rho_{2}})}<+\infty\), then we have \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,l;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), where \((q,r)\) is arbitrary admissible pair.

Proof

Using the Hölder inequality and a Sobolev embedding, we have
$$\begin{aligned} &\bigl\| |u_{\varepsilon}|^{\alpha}u_{\varepsilon}-|u|^{\alpha}u\bigr\| _{L^{\gamma ^{*'}}(0,l;L^{\rho^{*'}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma^{*}}(0,l;L^{\frac{\rho ^{*}\alpha}{\rho^{*}-2}})} +\|u\|^{\alpha}_{L^{\gamma^{*}}(0,l;L^{\frac{\rho^{*}\alpha}{\rho^{*}-2}})} \bigr) \|u_{\varepsilon}-u \|_{L^{\gamma^{*}}(0,l;L^{\rho^{*}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma^{*}}(0,l;W^{2,\rho^{*}})} +\|u \|^{\alpha}_{L^{\gamma^{*}}(0,l;W^{2,\rho^{*}})} \bigr) \|u_{\varepsilon}-u\|_{L^{\gamma^{*}}(0,l;L^{\rho^{*}})}. \end{aligned}$$
(3.20)
From (2.1) and (2.2), using Strichartz estimates, we have
$$\begin{aligned} \|u_{\varepsilon}-u\|_{L^{q}(0,l;L^{r})} &\leq C \bigl\| |u_{\varepsilon}|^{\alpha}u_{\varepsilon}-|u|^{\alpha}u \bigr\| _{L^{\gamma ^{*'}}(0,l;L^{\rho^{*'}})} +C\varepsilon\|a\|_{L^{\infty}(0,l)}\bigl\| |u_{\varepsilon}|^{\beta}u_{\varepsilon}\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})} \\ &\quad\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{L^{\gamma^{*}}(0,l;W^{2,\rho^{*}})} +\|u \|^{\alpha}_{L^{\gamma^{*}}(0,l;W^{2,\rho^{*}})} \bigr) \|u_{\varepsilon}-u\|_{L^{\gamma^{*}}(0,l;L^{\rho^{*}})} \\ &\qquad{}+C\varepsilon\|a\|_{L^{\infty}(0,l)}\bigl\| |u_{\varepsilon}|^{\beta}u_{\varepsilon}\bigr\| _{L^{\gamma_{2}^{\prime}}(0,l;L^{\rho_{2}^{\prime}})}; \end{aligned}$$
(3.21)
similarly as in Lemma 3.1, we can obtain
$$\|u_{\varepsilon}-u\|_{L^{q}(0,l;L^{r})}\rightarrow0 \quad\mbox{as } \varepsilon \rightarrow0. $$
Noting that for \((\gamma_{1},\rho_{1})\) in Lemma 3.1 in the case \(\alpha =\frac{8}{n-4}\), \(2\leq\beta<\frac{8}{n-4}\), we have
$$(\gamma_{1},\rho_{1})= \bigl(\gamma^{*}, \rho^{*} \bigr), $$
thus obviously
$$\bigl\| \nabla(u_{\varepsilon}-u)\bigr\| _{L^{q}(0,l;L^{r})}\rightarrow0 \quad \mbox{as } \varepsilon\rightarrow0 $$
and
$$\bigl\| D^{2} u_{\varepsilon}-D^{2} u\bigr\| _{L^{q}(0,l;L^{r})} \rightarrow0 \quad\mbox{as } \varepsilon\rightarrow0, $$
for all admissible pairs \((q,r)\). □

Remark 3.1

For the critical case \(2\leq\alpha<\frac{8}{n-4}\), \(\beta=\frac{8}{n-4}\), we only take the working space as \(L^{\infty}(0,\delta;H^{2}(R^{n})) \cap L^{\gamma_{1}}(0,\delta;W^{2,\rho_{1}}(R^{n})) \cap L^{\gamma^{*}}(0,\delta;W^{2,\rho^{*}}(R^{n}))\).

For the case \(\alpha=\beta=\frac{8}{n-4}\), we take the working space as \(L^{\infty}(0,\delta;H^{2}(R^{n})) \cap L^{\gamma^{*}}(0,\delta; W^{2,\rho^{*}}(R^{n}))\).

Theorem 3.1

Assume that \(n>4\), \(a\in L^{\infty}(0,\infty)\), \(2\leq\alpha\leq\frac{8}{n-4}\), and \(2\leq\beta\leq\frac{8}{n-4}\). Assume that u is the solution of (1.2) with initial value \(u_{0}\in H^{2}(R^{n})\), defined on a maximal time interval \([0,T^{*})\). Then we have:
  1. (1)

    For any given \(0< T< T^{*}\), there is a solution \(u_{\varepsilon}\) on \([0,T]\).

     
  2. (2)

    \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,T;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), where \((q,r)\) is an arbitrary admissible pair.

     

Proof

(1) The case \(2\leq\alpha<\frac{8}{n-4}\) and \(2\leq\beta<\frac{8}{n-4}\).

From Proposition 2.1, we find that there exists \(u_{\varepsilon}\) on \([0,\delta]\) such that
$$\|u_{\varepsilon}\|_{L^{\infty}(0,\delta;H^{2})\cap L^{\gamma_{1}}(0,\delta ;W^{2,\rho_{1}}) \cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}})}\leq2\|u_{0}\|_{H^{2}}. $$
So for small ε, we have
$$\lim\sup_{\varepsilon\rightarrow 0}\|u_{\varepsilon}\|_{L^{\infty}(0,\delta;H^{2})\cap L^{\gamma_{1}}(0,\delta ;W^{2,\rho_{1}})\cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}})}< +\infty. $$

Using Lemma 3.1, we have \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,\delta ;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), for any arbitrary admissible pair \((q,r)\).

Especially, we have \(\|u_{\varepsilon}(\delta)\|_{H^{2}}\leq2\|u_{0}\|_{H^{2}}\). Again using Proposition 2.1, there exists \(u_{\varepsilon}\) on \([\delta ,2\delta]\) such that
$$\|u_{\varepsilon}\|_{L^{\infty}(\delta,2\delta;H^{2}) \cap L^{\gamma_{1}}(\delta,2\delta;W^{2,\rho_{1}}) \cap L^{\gamma_{2}}(\delta,2\delta;W^{2,\rho_{2}})} \leq2\bigl\| u_{\varepsilon}(\delta) \bigr\| _{H^{2}}\leq2\|u_{0}\|_{H^{2}}. $$
By a continuation extension method, we obtain the solution \(u_{\varepsilon}\) on \([0,T] \) (\(0< T< T^{*}\)) such that
$$\|u_{\varepsilon}\|_{L^{\infty}(0,T;H^{2})\cap L^{\gamma}(0,T;W^{2,\rho}) \cap L^{\gamma_{1}}(0,T;W^{2,\rho_{1}})}\leq2\|u_{0}\|_{H^{2}}. $$
So
$$\lim\sup_{\varepsilon\rightarrow0}\|u_{\varepsilon}\|_{L^{\infty}(0,T;H^{2}) \cap L^{\gamma_{1}}(0,T;W^{2,\rho_{1}}) \cap L^{\gamma_{2}}(0,T;W^{2,\rho_{2}})}< +\infty, $$
using Lemma 3.1, we immediately have \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,T;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), for any arbitrary admissible pair \((q,r)\).

(2) Case 1: \(\alpha=\frac{8}{n-4}\), \(2\leq\beta<\frac{8}{n-4}\).

From Proposition 2.2, we find that there exists \(u_{\varepsilon}\) on \([0,\delta]\) such that
$$\|u_{\varepsilon}\|_{L^{\infty}(0,\delta;H^{2})\cap L^{\gamma^{*}}(0,\delta ;W^{2,\rho^{*}}) \cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}})}\leq3\bigl\| U(t)u_{0} \bigr\| _{L^{\gamma ^{*}}(0,\delta;W^{2,\rho^{*}})}. $$
So for small ε, we have
$$\lim\sup_{\varepsilon\rightarrow 0}\|u_{\varepsilon}\|_{L^{\infty}(0,\delta;H^{2})\cap L^{\gamma ^{*}}(0,\delta;W^{2,\rho^{*}})\cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}})}< +\infty. $$

Using Lemma 3.2, we have \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,\delta ;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), for any arbitrary admissible pair \((q,r)\).

Noting that
$$\bigl\| U(t)u_{\varepsilon}(\delta)\bigr\| _{L^{\gamma^{*}}(0,\delta;W^{2,\rho ^{*}})}\leq C\bigl\| u_{\varepsilon}( \delta)\bigr\| _{H^{2}} \leq3C\bigl\| U(t)u_{0}\bigr\| _{L^{\gamma^{*}}(0,\delta;W^{2,\rho^{*}})}, $$
so, again using Proposition 2.2, there exists \(u_{\varepsilon}\) on \([\delta,2\delta]\) such that
$$\begin{aligned} \|u_{\varepsilon}\|_{L^{\infty}(\delta,2\delta;H^{2})\cap L^{\gamma ^{*}}(\delta,2\delta;W^{2,\rho^{*}}) \cap L^{\gamma_{2}}(\delta,2\delta;W^{2,\rho_{2}})}&\leq3C\bigl\| U(t)u_{\varepsilon}(\delta) \bigr\| _{L^{\gamma^{*}}(0,\delta;W^{2,\rho ^{*}})} \\ &\leq(3C)^{2}\bigl\| U(t)u_{0}\bigr\| _{L^{\gamma^{*}}(0,\delta;W^{2,\rho^{*}})}. \end{aligned}$$
By continuation extension method, we obtain the solution \(u_{\varepsilon }\) on \([0,T]\) (\(0< T< T^{*}\)) such that
$$\|u_{\varepsilon}\|_{L^{\infty}(0,T;H^{2})\cap L^{\gamma^{*}}(0,T;W^{2,\rho^{*}}) \cap L^{\gamma_{2}}(0,T;W^{2,\rho_{2}})}\leq C(T)\bigl\| U(t)u_{0} \bigr\| _{L^{\gamma ^{*}}(0,\delta;W^{2,\rho^{*}})}. $$
So
$$\lim\sup_{\varepsilon\rightarrow0}\|u_{\varepsilon}\|_{L^{\infty }(0,T;H^{2})\cap L^{\gamma^{*}}(0,T;W^{2,\rho^{*}}) \cap L^{\gamma_{2}}(0,T;W^{2,\rho_{2}})}\leq C(T) \bigl\| U(t)u_{0}\bigr\| _{L^{\gamma ^{*}}(0,\delta;W^{2,\rho^{*}})}< +\infty, $$
using Lemma 3.2, we immediately have \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,T;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), for any arbitrary admissible pair \((q,r)\).

Case 2: \(\beta=\frac{8}{n-4}\), \(2\leq\alpha<\frac{8}{n-4}\) or \(\alpha =\beta=\frac{8}{n-4}\).

See Remark 3.1, the proof is similar; here we omit it. □

Lemma 3.3

Assume that u is the global solution of (1.2) with the initial valve \(u_{0}\in H^{2}(R^{n})\) and \(u\in L^{q}_{loc}(0,\infty ;W^{2,r}(R^{n}))\). Then we have:
  1. (1)

    The solution \(u_{\varepsilon}\) of (1.1) with the initial valve \(u_{0}\) is global for sufficiently small ε.

     
  2. (2)

    \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,\infty;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), where \((q,r)\) is an arbitrary admissible pair.

     

Proof

(1) We will prove that \(u_{\varepsilon}\) is also global for small ε if u is global.

From Theorem 3.1, we can see
$$\bigl\| u_{\varepsilon}(T)-u(T)\bigr\| _{H^{2}}\rightarrow0 \quad\mbox{as } \varepsilon \rightarrow0 $$
for all \(T<\infty\).
Since u is global, for any \(\eta>0\), there exists sufficient large T such that
$$\|u\|_{L^{\gamma_{1}}(T,\infty;W^{2,\rho_{1}})}\leq\frac{\eta}{4}, $$
\((\gamma_{1},\rho_{1})\) is the same as in Theorem 3.1.

Case 1: \(2\leq\alpha<\frac{8}{n-4}\), \(2\leq\beta\leq\frac{8}{n-4}\).

From (2.2), (2.3)-(2.4), using a continuity argument we can obtain
$$\begin{aligned} &\bigl\| U(t)u(T)\bigr\| _{L^{\gamma_{1}}(0,\infty;W^{2,\rho_{1}})} \\ &\quad\leq C\bigl\| u(T)\bigr\| _{L^{\gamma_{1}}(0,\infty;W^{2,\rho_{1}})} +C\biggl\| \int _{T}^{t}U(t-\tau)|u|^{\alpha}u(\tau)\,d\tau \biggr\| _{L^{\gamma_{1}}(0,\infty ;W^{2,\rho_{1}})} \\ &\quad\leq C\bigl\| u(t)\bigr\| _{L^{\gamma_{1}}(T,\infty;W^{2,\rho_{1}})} +C\bigl\| |u|^{\alpha+1}\bigr\| _{L^{\gamma_{1}}(T,\infty;W^{2,\rho_{1}})} \\ &\quad\leq\frac{\eta}{2}. \end{aligned}$$
Thus we have
$$\begin{aligned} \bigl\| U(t)u_{\varepsilon}(T)\bigr\| _{L^{\gamma_{1}}(0,\infty;W^{2,\rho_{1}})} \leq{}&\bigl\| U(t) \bigl(u_{\varepsilon}(T)-u(T) \bigr)\bigr\| _{L^{\gamma_{1}}(0,\infty;W^{2,\rho_{1}})} \\ &{}+ \bigl\| U(t)u(T)\bigr\| _{L^{\gamma_{1}}(0,\infty;W^{2,\rho_{1}})} \leq\eta. \end{aligned}$$
Obviously \(\|U(t)u_{\varepsilon}(T)\|_{L^{q}(0,\infty;W^{2,r})}\leq\eta\) for suitable T and any admissible pair \((q,r)\).
Furthermore we define the working space as follows:
$$\begin{aligned} X(0,t)={}&L^{\frac{2(n+4)}{n}} \bigl(0,t;L^{\frac{2(n+4)}{n}} \bigl(R^{n} \bigr) \bigr) \cap L^{\frac{2(n+4)}{n-4}} \bigl(0,t;W^{2,\frac{2n(n+4)}{n^{2}+16}} \bigl(R^{n} \bigr) \bigr) \\ &{} \cap L^{\frac{2(n+4)}{n-2}} \bigl(0,t;W^{2,\frac{2n(n+4)}{n^{2}+8}} \bigl(R^{n} \bigr) \bigr)\cap L^{\gamma_{2}} \bigl(0,t;W^{2,\rho_{2}} \bigl(R^{n} \bigr) \bigr) \cap L^{\infty} \bigl(0,t;H^{2} \bigl(R^{n} \bigr) \bigr), \end{aligned}$$
where \((\gamma_{2},\rho_{2})\) is the same as in Theorem 3.1.
Using the Hölder inequality, the interpolation inequality [13], and the Sobolev embedding, we have
$$\begin{aligned} &\bigl\| |u_{\varepsilon}|^{\alpha}u_{\varepsilon}\bigr\| _{L^{\frac {2(n+4)}{n+8}}(0,t;L^{\frac{2(n+4)}{n+8}})} \\ &\quad\leq\|u_{\varepsilon}\|^{\alpha}_{L^{\frac{(n+4)\alpha}{4}}(0,t;L^{\frac {(n+4)\alpha}{4}})} \|u_{\varepsilon}\|_{L^{\frac{2(n+4)}{n}}(0,t;L^{\frac{2(n+4)}{n}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{2-\frac{(n-4)\alpha}{4}}_{L^{\frac {2(n+4)}{n}}(0,t;L^{\frac{2(n+4)}{n}})} \|u_{\varepsilon}\|^{\frac{n\alpha}{4}-2}_{L^{\frac {2(n+4)}{n-4}}(0,t;L^{\frac{2(n+4)}{n-4}})} \|u_{\varepsilon}\|_{L^{\frac{2(n+4)}{n}}(0,t;L^{\frac{2(n+4)}{n}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{3-\frac{(n-4)\alpha}{4}}_{L^{\frac {2(n+4)}{n}}(0,t;L^{\frac{2(n+4)}{n}})} \|u_{\varepsilon}\|^{\frac{n\alpha}{4}-2}_{L^{\frac {2(n+4)}{n-4}}(0,t;W^{2,\frac{2n(n+4)}{n^{2}+16}})} \\ &\quad \leq C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)}. \end{aligned}$$
Similarly, we can obtain
$$\begin{aligned}& \bigl\| \nabla \bigl(|u_{\varepsilon}|^{\alpha}u_{\varepsilon}\bigr)\bigr\| _{L^{\frac{2(n+4)}{n+8}}(0,t;L^{\frac{2(n+4)}{n+8}})} \leq C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)}; \\& \bigl\| |u_{\varepsilon}|^{\alpha-1} D^{2}u_{\varepsilon}\bigr\| _{L^{\frac{2(n+4)}{n+8}}(0,t;L^{\frac{2(n+4)}{n+8}})} \leq C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)}. \end{aligned}$$
For the case \(4< n<8\), we have
$$\begin{aligned} &\bigl\| |u_{\varepsilon}|^{\alpha-1} \nabla u_{\varepsilon}\cdot\nabla u_{\varepsilon}\bigr\| _{L^{\frac{2(n+4)}{n+8}}(0,t;L^{\frac{2(n+4)}{n+8}})} \\ &\quad\leq\bigl\| |u_{\varepsilon}|^{\alpha-1}\bigr\| _{L^{\frac {2(n+4)}{8-n}}(0,t;L^{\frac{2(n+4)}{8-n}})} \|\nabla u_{\varepsilon}\|^{2}_{L^{\frac{2(n+4)}{n}}(0,t;L^{\frac {2(n+4)}{n}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{\frac{8-n}{4}-\frac{(n-4)(\alpha-1)}{4}} _{L^{\frac{2(n+4)}{n}}(0,t;L^{\frac{2(n+4)}{n}})} \|u_{\varepsilon}\|^{\frac{n(\alpha-1)}{4}-\frac{8-n}{4}} _{L^{\frac{2(n+4)}{n-4}}(0,t;W^{2,\frac{2(n+4)}{n^{2}+16}})} \|u_{\varepsilon}\|^{2}_{L^{\frac{2(n+4)}{n}}(0,t;W^{2,\frac{2(n+4)}{n}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)}. \end{aligned}$$
For the case \(8\leq n<12\), we have
$$\begin{aligned} &\bigl\| |u_{\varepsilon}|^{\alpha-1} \nabla u_{\varepsilon}\cdot\nabla u_{\varepsilon}\bigr\| _{L^{\frac{2(n+4)}{n+8}}(0,t;L^{\frac{2(n+4)}{n+8}})} \\ &\quad\leq\bigl\| |u_{\varepsilon}|^{\alpha-1}\bigr\| _{L^{\frac {2(n+4)}{12-n}}(0,t;L^{\frac{2(n+4)}{12-n}})} \|\nabla u_{\varepsilon}\|^{2}_{L^{\frac{2(n+4)}{n-2}}(0,t;L^{\frac {2(n+4)}{n-2}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{\frac{12-n}{4}-\frac{(n-4)(\alpha-1)}{4}} _{L^{\frac{2(n+4)}{n}}(0,t;L^{\frac{2(n+4)}{n}})} \|u_{\varepsilon}\|^{\frac{n(\alpha-1)}{4}-\frac{12-n}{4}} _{L^{\frac{2(n+4)}{n-2}}(0,t;W^{2,\frac{2(n+4)}{n^{2}+8}})} \|u_{\varepsilon}\|^{2}_{L^{\frac{2(n+4)}{n}}(0,t;W^{2,\frac{2(n+4)}{n}})} \\ &\quad\leq C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)}; \end{aligned}$$
thus we have
$$\bigl\| D^{2} \bigl(|u_{\varepsilon}|^{\alpha}u_{\varepsilon}\bigr)\bigr\| _{L^{\frac{2(n+4)}{n+8}}(0,t;L^{\frac{2(n+4)}{n+8}})} \leq C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)}. $$
Noting that \((\frac{2(n+4)}{n+8},\frac{2(n+4)}{n+8})\) is an admissible pair, using Strichartz estimates, we can obtain
$$\biggl\| \int_{T}^{t}U(t-\tau)|u_{\varepsilon}|^{\alpha}u_{\varepsilon}\biggr\| _{X(0,t)} \leq C\bigl\| |u_{\varepsilon}|^{\alpha}u_{\varepsilon}\bigr\| _{L^{\frac {2(n+4)}{n-8}}(0,t;W^{2,\frac{2(n+4)}{n-8}})} \leq C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)}. $$
Using (2.1), we have
$$\bigl\| u_{\varepsilon}(T)\bigr\| _{X(0,t)} \leq C\bigl\| U(t)u_{\varepsilon}(T) \bigr\| _{X(0,t)}+C\|u_{\varepsilon}\|^{\alpha+1}_{X(0,t)} +C \|u_{\varepsilon}\|^{\beta+1}_{X(0,t)}. $$
Using a continuity argument, we immediately have
$$\bigl\| u_{\varepsilon}(T)\bigr\| _{X(0,t)}\leq3\eta \quad \mbox{for sufficiently small } \varepsilon, $$
which means that \(\|u_{\varepsilon}\|_{X(T,\infty)}\leq M\), where M is a constant.

Furthermore, we have \(\|u_{\varepsilon}\|_{L^{q}(T,\infty;W^{2,r})}\leq M\), for any admissible pair \((q,r)\). Thus \(u_{\varepsilon}\) is global.

Case 2: \(\alpha=\frac{8}{n-4}\), \(2\leq\beta\leq\frac{8}{n-4}\).

We need the following working space:
$$Y(0,t)=L^{\frac{2n}{n-4}} \bigl(0,t;W^{2,\frac{2n^{2}}{n^{2}-4n+16}}\bigl(R^{n} \bigr) \bigr)\cap L^{\gamma_{2}} \bigl(0,t;W^{2,\rho_{2}} \bigl(R^{n} \bigr) \bigr) \cap L^{\infty} \bigl(0,t;H^{2} \bigl(R^{n} \bigr) \bigr). $$
The process of proof is similar to the case 1, so here we omit the detailed proof.

(2) In the sequel, we prove \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,\infty;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), for any admissible pair \((q,r)\).

Using (2.1) and (2.2), we have
$$\begin{aligned}& \begin{aligned}[b] &u_{\varepsilon}(T+t)-u(T+t) \\ &\quad=U(t) \bigl(u_{\varepsilon}(T)-u(T) \bigr)+i\int_{0}^{t}U(t- \tau) \bigl(|u_{\varepsilon}|^{\alpha}u_{\varepsilon}-|u|^{\alpha}u \bigr) (T+\tau)\,d\tau \\ &\qquad{}+\varepsilon\int_{0}^{t}U(t-\tau)a(T+\tau)|u|^{\beta}u\bigl((T+\tau)\bigr)\,d\tau \\ &\quad=a(t)+b(t)+c(t); \end{aligned} \\& \bigl\| a(t)\bigr\| _{L^{q}(0,\infty;W^{2,r})}\leq C\bigl\| u_{\varepsilon}(T)-u(T)\bigr\| _{H^{2}} \rightarrow0 \quad\mbox{as } \varepsilon\rightarrow0; \\& \begin{aligned}[b] \bigl\| b(t)\bigr\| _{L^{q}(0,\infty;W^{2,r})} &\leq C\bigl\| |u_{\varepsilon}|^{\alpha}u_{\varepsilon}- |u|^{\alpha}u \bigr\| _{L^{\frac{2(n+4)}{n+8}}(0,\infty;W^{2,\frac{2(n+4)}{n+8}})} \\ &\leq C \bigl(\|u_{\varepsilon}\|^{\alpha}_{X(0,\infty)}+\|u \|^{\alpha }_{X(0,\infty)} \bigr) \|u_{\varepsilon}-u\|_{X(0,\infty)} \rightarrow0; \end{aligned} \\& \bigl\| c(t)\bigr\| _{L^{q}(0,\infty;W^{2,r})}\leq C\varepsilon\|u_{\varepsilon}\| ^{\beta+1}_{X(0,\infty)}. \end{aligned}$$
Thus we have
$$\bigl\| u_{\varepsilon}(T+t)-u(T+t)\bigr\| _{L^{q}(0,\infty;W^{2,r})}\rightarrow0 \quad\mbox{as } \varepsilon\rightarrow0. $$
 □

Theorem 3.2

Assume that \(n>4\), \(a\in L^{\infty}(0,\infty)\), \(2\leq\alpha\leq\frac{8}{n-4}\), and \(2\leq\beta\leq\frac{8}{n-4}\). One of the following conditions holds:
  1. (i)

    \(\lambda<0\),

     
  2. (ii)

    \(\lambda>0\), \(\|u_{0}\|_{H^{2}}\) is small.

     
Then we have
  1. (1)

    The solution \(u_{\varepsilon}\) of (1.1) is global for small ε.

     
  2. (2)

    \(u_{\varepsilon}\rightarrow u\) in \(L^{q}(0,\infty;W^{2,r}(R^{n}))\) as \(\varepsilon\rightarrow0\), where \((q,r)\) is arbitrary admissible pair.

     

Proof

Note that the solution u of (1.2) is global provided the conditions (i) \(\lambda<0\) or (ii) \(\lambda>0\), \(\|u_{0}\| _{H^{2}}\) is small hold. Combing Lemma 3.3, the proof of Theorem 3.2 immediately is complete. □

4 Conclusions

The appearance of gain/loss does not affect the local well-posedness of the solution. Moreover, the solution \(u_{\varepsilon}\) will converge to u in the space \(L^{q}(0,T;W^{2,r}(R^{n}))\) as ε converges to 0. Furthermore, if (i) \(\lambda<0\), or (ii) \(\lambda>0\), \(\|u_{0}\|_{H^{2}}\) is small, then we have found that the global solution \(u_{\varepsilon}\) will converge to u in the space \(L^{q}(0,\infty;W^{2,r}(R^{n}))\) as ε converges to 0.

Declarations

Acknowledgements

This work is supported by Natural Science of the Shanxi province (No. 2013011003-2) and the Natural Science Foundation of China (Nos. 61473180, 11571209, 61503230).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Science, Shanxi University, Taiyuan, Shanxi, 030006, P.R. China

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