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

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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 $$\varepsilon\rightarrow0$$.

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  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  and , Pausader Benoit gives the global well-posedness and scattering for $$n\geq5$$ and radial initial data. In , Miao et al. study the defocusing case and obtain the global existence for $$n\geq9$$. In , Zhang and Zheng obtain the global solution and scattering for $$n=8$$. Pausader Benoit also discusses the mass-critical case in .

For the case $$\varepsilon\neq0$$, $$a(t)$$ is the gain (loss) if $$a(t)<0$$ ($$a(t)>0$$). In , 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  and . 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 $$\varepsilon\rightarrow0$$.

The preliminary estimates

First, we denote by $$U(t)$$ ($$t\in R$$) the fundamental solution operator of the fourth-order Schrödinger equation , 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  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 ): 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 , 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,

$$\|u_{\varepsilon}\|_{L^{\infty}(0,\delta;H^{2})\cap L^{\gamma^{*}}(0,\delta ;W^{2,\rho^{*}}) \cap L^{\gamma_{2}}(0,\delta;W^{2,\rho_{2}})} \leq3\bigl\| U(t)u_{0} \bigr\| _{L^{\gamma^{*}}(0,\delta;W^{2,\rho^{*}})}.$$

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 , 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 , 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. □

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.

References

1. 1.

Cui, SB, Guo, CH: Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $$H^{s}(R^{n})$$ and applications. Nonlinear Anal. 67(3), 687-707 (2007)

2. 2.

Pausader, B: The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete Contin. Dyn. Syst. 24(4), 1275-1292 (2009)

3. 3.

Pausader, B: Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ. 4(3), 197-225 (2007)

4. 4.

Miao, C, Xu, G, Zhao, L: Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth order in dimensions $$n\geq 9$$. J. Differ. Equ. 251(12), 3381-3402 (2011)

5. 5.

Zhang, J, Zheng, J: Energy critical fourth-order Schrödinger equations with subcritical perturbations. Nonlinear Anal. 73(4), 1004-1014 (2010)

6. 6.

Pausader, B: The mass-critical fourth-order Schrödinger equation in high dimensions. J. Hyperbolic Differ. Equ. 7(4), 651-705 (2010)

7. 7.

Allayarov, IM, Tsoy, EN: Dynamics of fronts in optical media with linear gain and nonlinear losses. Phys. Lett. A 377(7), 550-554 (2013)

8. 8.

Feng, B, Zhao, D, Sun, C: The limit behavior of solutions for the nonlinear Schrödinger equation including nonlinear loss/gain with variable coefficient. J. Math. Anal. Appl. 405(1), 240-251 (2013)

9. 9.

Feng, B, Zhao, D, Sun, C: On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain. J. Math. Anal. Appl. 416(2), 901-923 (2014)

10. 10.

Ben-Artzi, M, Koch, H, Saut, J-C: Dispersion estimates for fourth order Schrödinger equations. C. R. Math. Acad. Sci. Paris 330(2), 87-92 (2000)

11. 11.

Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

12. 12.

Adams, RA, Fournier, JJF: Sobolev Spaces. Academic Press, Singapore (2009)

13. 13.

Bergh, J, Löfstöm, T: Interpolation Spaces. Springer, New York (1976)

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).

Author information

Correspondence to Cuihua Guo. 