# The Effect of Gain and Strong Dissipative Structures on Nonlinear Schrödinger Equations in Optical Fiber

Authors: Xintong Yuan; Chunhua Li
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

We consider the initial value problem for the nonlinear Schrödinger equation satisfying the strong dissipative conditionIλ<0 and Iλ>p-1/2pRλ in one space dimension. Our purpose in this paper is to study how the gain coefficient μ(t) and strong dissipative nonlinearity λvp-1v affect solutions to the nonlinear Schrödinger equation for large initial data. We prove global existence of solutions and present some time decay estimates of solutions for large initial data.


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## Body

## 1. Introduction and Main Results

We consider the Cauchy problem of nonlinear Schrödinger equation:(1)i∂tv+12∂x2v=λvp-1v+iμtv,v0,x=v0x,where v=v(t,x) is a complex valued unknown function, t≥0, x∈R, p>1, the gain coefficient μ(t) is a real valued function, and λ∈C. Equation (1) is applied in problems of dispersion-managed optical fibers and soliton lasers (see [1]). The coefficients λ and μ(t) are, respectively, nonlinearity and amplification. In this work, we study the global existence and investigate time decay estimates of solutions to (1) with the gain coefficient μ(t) and the strong dissipative nonlinearity λvp-1v satisfying Iλ<0 and Iλ>p-1/2pRλ for large initial data, where Iλ and Rλ are the imaginary and real part of λ, respectively.Over the past few decades, the field of fiber optics has made rapid progress. The damped Schrödinger equation(2)i∂tv+12∂x2v=λv2v+iμv,where v=v(t,x) is a complex valued unknown function, t∈R, x∈R, λ, and μ∈C, is one of the simplest nonlinear Schrödinger equations for studying cubic nonlinear effects in optical fibers (see [1]). Equation (2) is applied in several different aspects of optics (see, e.g., [2]). It has been studied extensively in the context of solitons (see [1]). In the case of μ(t)≡0, (1) is reduced to(3)i∂tv+12∂x2v=λvp-1v,v0,x=v0x,where t≥0, x∈R, p>1, and λ∈C. The nonlinearity λvp-1v with Iλ<0 and Iλ>p-1/2pRλ is called strong dissipative. In [3, 4], the large initial problem for (3) with the strong dissipative nonlinearities was investigated. The global solution v(t,x) to (3) decays like t-1/2(log⁡t)-1/2 in the sense of L∞ for t>1, when p=3. Moreover vL∞≤Ct-1/(p-1) for t>1, if 2.686≈5+33/4<p<3 in [3] and 2.586≈19+145/12<p<3 in [4], respectively. To study the time decays of solutions to (3), the estimate of FU(-t)vL∞ is useful. To get a better estimate of FU(-t)vL∞, the contradiction argument was used in [3] and the method of [5] was applied in [4]. In [6], the time decay estimates of global solutions to (3) with 1+2<p<3 in the sense of L2 were considered under the strong dissipative condition. It showed that vL2≤Ct-1/p-1-1/2q for t>1, where 0≤q≤2/3. In the case of Iλ>0, the a priori L2- bound for v is not satisfied, since (4)12vL22-Iλ∫0t∫Rvp+1dxdτ=12v0L22.There are some results about the lifespan of small solutions to (3) with Iλ>0 (see, e.g., [7]). When optical pluses propagate inside a fiber, nonlinearities in (1) affect optical pluses’ shapes. Some related nonlinear Schrödinger equations have been studied (see, e.g., [8, 9]). In the case of μ(t)≡-a<0 and λ=-1, (1) becomes(5)i∂tv+12∂x2v=-vp-1v-iav,v0,x=v0xin one space dimension, where t≥0, a>0, and p>1. The question (5) has been studied by some mathematicians from the mathematical point (see, e.g., [10, 11]). Let E1(t)=1/2∂xvL22-2/p+1vLp+1p+1. If E1(0)≤0, nonexistence of global solutions to (5) was studied under some assumptions in [10]. It showed that the damping term in (5) cannot prevent blowing up of solutions. In [11], some blow up and global existence of solutions to (5) was investigated. The authors showed that the size of the damping coefficient a affected the solutions. As far as we know, there are not any results about the time decay estimates of solutions to (1) for large initial data. Our question is how the term iμ(t)v and the nonlinearity λvp-1v with strong dissipative condition affect solutions to (1) for large initial data.LetLqR denote the usual Lebesgue space with the norm (6)fLqR=∫Rfxqdx1/qif 1≤q<∞ and (7)fL∞R=ess.supx∈Rfx.For m,s∈R, weighted Sobolev space Hm,s is defined by(8)Hm,s=f∈L2R;fHm,s=1+x2s/21-∂x2m/2fL2R<∞.We write LqR=Lq for 1≤q≤∞ and Hm,0=Hm for simplicity.Let us introduce some notations. We define the dilation operator by(9)Dtϕx=1it1/2ϕxtand define M=ei/2tx2 for t≠0. Evolution operator Ut is written as (10)Ut=MDtFM,where the Fourier transform of f is (11)Ffξ=12π∫Re-ixξfxdx.We also have (12)U-t=M-1F-1Dt-1M-1,where the inverse Fourier transform of f is (13)F-1fx=12π∫Reixξfξdξ.We denote by the same letter C various positive constants. And we write v(t) for the spatial function v(t,·).The standard generator of Galilei transformations is given as(14)Jt=UtxU-t=x+it∂x.We have (15)J2t=Utx2U-t=M-t2∂x2M-1.We also have commutation relations withJβ and L=i∂t+1/2∂x2 such that(16)L,Jβ=0,where β=1,2.Before stating our main theorem, we introduce the function space(17)X1,T=y;U-ty∈C0,T;H1∩H0,1,yX1,T<∞,where yX1,T=sup0≤t<TU(-t)yH1∩H0,1 and T>1. We have the following global existence of solutions to (1) for large initial data.Theorem 1.
Letp>1, Iλ<0, Iλ>p-1/2pRλ, and μ(t)∈C[0,∞). We assume that v0(x)∈H0,1∩H1. Then (1) has a unique global solution v(t,x)∈C[0,∞);H0,1∩H1 satisfying v(t,x)=e∫0tμ(τ)dτu(t,x), where u(t,x)∈X1,∞ is the solution to (19).LetE0(t)=v(t)L22. Multiplying both sides of (1) by v¯, integrating over R, and taking the imaginary parts, we have (18)ddtE0t=2IλvtLp+1p+1+2μtE0t.We could not ensure the sign of d/dtE0(t) by the assumptions in Theorem 1. This case is interesting. We have the equation(19)i∂tu+12∂x2u=λep-1∫0tμτdτup-1u,u0,x=v0xfrom (1) by using the transformation u=e-∫0tμ(τ)dτv. A straightforward calculation shows that v(t,x) solves (1) if and only if u(t,x) solves (19). Thus the transformation u=e-∫0tμ(τ)dτv provides an effective tool to study the global existence of solutions to (1).Remark 2.
Letp>1, Iλ<0, Iλ>p-1/2pRλ, μ(t)∈C[0,∞), and e∫0tμ(τ)dτ≤C0 for all t≥0, where C0>0. We assume that v0(x)∈H0,1∩H1. Then (1) has a unique global solution v(t,x)∈X1,∞.Example 3.
We consider(20)i∂tv+12∂x2v=-ivp-1v+i1+t-2v,v0,x=v0xin one space dimension, where t≥0 and p>1. Since |I(-i)|=1>0=p-1/2p|R(-i)|, and e∫0t(1+τ)-2dτ≤e for t≥0, (20) has a unique global solution v(t,x)∈X1,∞ if v0(x)∈H0,1∩H1.Ifμ(t)≤0 for t≥0 and Iλ<0, we have the result about (1): (21)ddtE0t=2IλvtLp+1p+1+2μtvtL22≤0,where E0(t)=v(t)L22. Time decay estimates of solutions to (1) for large initial data are shown as follows.Theorem 4.
Letp>1, μ(t)≤0 for t≥0, μ(t)∈C[0,∞) and the strong dissipative condition Iλ<0, |Iλ|>p-1/2pRλ hold. We assume that v0(x)∈H0,1∩H1. Then (1) has a unique global solution v(t,x)∈X1,∞ satisfying the following time decay estimates: (22)vtL∞≤Ct-1/2e∫0tμτdτand (23)vtH1≤Ce∫0tμτdτfor t>1.From Theorem4, we obtain that the solution v(t,x) to (1) is global and bounded in X1,∞ for large initial data. Moreover, we show that μ(t) determines the time decay rate of the solution, when μ(t) satisfies the assumptions in Theorem 4. Then we consider a special situation of Theorem 4. Let μ(t)≡μ≤0 in (1); we have the following equation:(24)i∂tv+12∂x2v=λvp-1v+iμv,v0,x=v0x,where t≥0, p>1, x∈R, μ≤0, and λ∈C. By Theorem 4, we have the time decay estimates to (24).Corollary 5.
Letμ≤0 and p>1. We assume that v0(x)∈H0,1∩H1 and the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ hold. Then (24) has a unique global solution v(t,x)∈X1,∞ satisfying the following time decay estimates: (25)vtL∞≤Ct-1/2eμtand (26)vtH1≤Ceμtfor t>1.In the last section, we consider the equation(27)i∂tv+12∂x2v=λvp-1v+ia1+tp-1v,v0,x=v0x,where a≤0, p>1, Iλ<0, Iλ>p-1/2pRλ, and t≥0. From (27), we have(28)i∂tu+12∂x2u=λ1+taup-1u,u0,x=v0x,by taking the variable change v(t,x)=(1+t)a/p-1u(t,x). A similar nonlinear equation i∂tu+Δu=λ1+btnα-4/2uαu, where x∈Rn, b∈R,α>0, and λ∈R, was derived to study the rapid decay solutions and scattering properties of the equation i∂tv+Δv=λvαv by letting u(t,x)=(1+bt)-n/2vt/1+bt,x/1+btei(b|x|2/4(1+bt)) in [12].LetIλ<0 and |Iλ|>p-1/2p|Rλ|. Multiplying both sides of (27) by v¯, integrating over R, and taking the imaginary parts, we have(29)ddtE0t=2IλvtLp+1p+1+2ap-11+tvtL22≤0,where E0(t)=v(t)L22. We have global existence and time decay estimates of solutions to (27) for large initial data by Theorem 4. We show a better decay rate of solutions to (27) inspired by the papers [4, 5].Theorem 6.
Leta≤0 and p>1. We assume that v0(x)∈H0,1∩H1 and the strong dissipative condition Iλ<0 and |Iλ|>p-1/2pRλ hold. Then (27) has a unique global solution v(t,x)∈X1,∞ satisfying the following time decay estimates:(30)vtL∞≤Ct-1/2+a/p-1for t>1. Moreover, let p∗(a):=a+5/4+1/416a2+40a+33, and if p∗(a)≤p<2a+3 and -1<a≤0, we have(31)vtL∞≤Ct-1/2+a/p-1-2a+3-p/2p-1for t>1.Whena=0, the exponent p∗(a) coincides with 5+33/4, which is the lower bound given by Kita-Shimomura [3] and Jin-Jin-Li [4]. Let p∗∗(a)≔a+19/12+1/12144a2+264a+145. Since p∗∗(a)<p∗(a) for -1/2<a≤0, the lower bound p∗(a) in Theorem 6 can be improved by p∗∗(a) in Theorem 7. The operator J2 plays an important role in achieving the lower bound p∗∗(a).Theorem 7.
Let-1/2<a≤0, p∗∗(a)≔a+19/12+1/12144a2+264a+145, and p∗∗(a)≤p<2a+3. We assume that v0(x)∈H0,2∩H1 and the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ hold. Then (27) has a unique global solution v(t,x)∈X1,∞ satisfying the following time decay estimates:(32)vtL∞≤Ct-1/2+a/p-1-2a+3-p/2p-1for t>1.Ifa=0, we have p∗∗(a)=19+145/12, which is a lower bound of p shown in [4]. Theorems 6 and 7 say how the strong dissipative nonlinearity and gain coefficient of the nonlinear Schrödinger equation (27) affect decay estimates of solutions under different initial conditions. The rest of this paper is organized as follows. In Section 2, we give proofs of Theorems 1 and 4. Theorems 6 and 7 are proven in Section 3.


## 2. Proofs of Theorems1 and 4

### 2.1. Proof of Theorem1

We have the equation(33)i∂tu+12∂x2u=λep-1∫0tμτdτup-1u from (1) by using the variable change u=e-∫0tμ(τ)dτv and u(0,x)=v(0,x)=v0(x).Local existence of solutionsu∈X1,T to (33) can be shown by the standard contraction mapping principle (see, e.g., [13]). Therefore, we have local existence of solutions v∈X1,T to (1).First, we consider the equation(34)i∂tu+12∂x2u=λFtup-1u,where F(t)≥0, F(t)∈C1[0,∞), λ∈C, and p>1. The following lemma is useful to study global existence and time decay of solutions to (34).Lemma 8.
Letp>1, F(t)≥0, F(t)∈C1[0,∞), and u(t,x)∈X1,T be the local solution of (34), where T>1. And let the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ be satisfied. Then we have (35)∂t∂xuL2+JuL2+uL2≤0.Proof.
Multiplying both sides of (34) by ∂xu¯ and taking the imaginary parts, we have, by up-1u=up+1/2u¯p-1/2 and the assumptions of F(t),(36)12∂t∂xuL22=I∫R∂xλFtup-1u·∂xu¯dx=Iλp+12Ft∫Rup-1∂xu2dx+Iλp-12Ft∫Rup-3u2∂xu¯2dx≤FtIλp+12+λp-12∫Rup-1∂xu2dx.By (36) and the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ, we obtain(37)12∂t∂xuL22=I∫R∂xλFtup-1u·∂xu¯dx≤0.
We note that(38)Jup-1u=p+12up-1Ju-p-12up-3u2Ju¯,JFtup-1u=x+it∂xFtup-1u=FtJup-1u,and(39)12∂tJuL22=I∫RJλFtup-1u·Ju¯dx.Calculating the right part of (39), we obtain(40)I∫RJλFtup-1u·Ju¯dx=Iλp+12Ft∫Rup-1Ju2dx-p-12IλFt∫Rup-3u2Ju¯2dx≤FtIλp+12+λp-12∫Rup-1Ju2dx≤0under the strong dissipative condition.
Multiplying both sides of (34) by u¯, we obtain(41)12∂tuL22=IλFt∫Rup+1dx≤0 by the assumptions of λ and F(t). Using (37), (40), and (41), we have (42)∂t∂xuL2+JuL2+uL2≤0.By Lemma8, we have(43)uL2+∂xuL2+JuL2≤v0xL2+∂xv0xL2+xv0xL2.From (43), we obtain a unique global solution u∈X1,∞ to (33). Since u=e-∫0tμ(τ)dτv, we have a unique global solution v(t,x)∈C[0,∞);H0,1∩H1 to (1).


### 2.2. Proof of Theorem4

Using the transformu=e-∫0tμ(τ)dτv, we get the equation about u from (1):(44)i∂tu+12∂x2u=λep-1∫0tμτdτup-1u,where u(0,x)=v(0,x)=v0(x). We have local existence of solutions u∈X1,T to (44) and v∈X1,T to (1), respectively (see, e.g., [13]).By Lemma8 and the assumptions in Theorem 4, we have(45)uL2+∂xuL2+JuL2≤v0xL2+∂xv0xL2+xv0xL2.And(46)vL2+∂xvL2+JvL2≤e∫0tμτdτv0xL2+∂xv0xL2+xv0xL2.Thus, we obtain global-in-time existence of solutions u∈X1,∞ to (44) by (45) and v∈X1,∞ to (1) by (46), respectively.We are now in a position to prove time decay estimates of solutions to (1). By the Sobolev inequality (47)uL∞≤CuL21/2∂xuL21/2,the factorization formula Ut=MDtFM, and (45), we have (48)uL∞≤Ct-1/2FMU-tuL∞≤Ct-1/2FMU-tuL21/2∂xFMU-tuL21/2=Ct-1/2uL21/2JuL21/2≤Ct-1/2for t>1. Using the transform u=e-∫0tμ(τ)dτv, we obtain (49)vL∞≤Ct-1/2e∫0tμτdτfor t>1. By (46), we have (50)vH1≤Ce∫0tμτdτfor t>1.


## 2.1. Proof of Theorem1

We have the equation(33)i∂tu+12∂x2u=λep-1∫0tμτdτup-1u from (1) by using the variable change u=e-∫0tμ(τ)dτv and u(0,x)=v(0,x)=v0(x).Local existence of solutionsu∈X1,T to (33) can be shown by the standard contraction mapping principle (see, e.g., [13]). Therefore, we have local existence of solutions v∈X1,T to (1).First, we consider the equation(34)i∂tu+12∂x2u=λFtup-1u,where F(t)≥0, F(t)∈C1[0,∞), λ∈C, and p>1. The following lemma is useful to study global existence and time decay of solutions to (34).Lemma 8.
Letp>1, F(t)≥0, F(t)∈C1[0,∞), and u(t,x)∈X1,T be the local solution of (34), where T>1. And let the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ be satisfied. Then we have (35)∂t∂xuL2+JuL2+uL2≤0.Proof.
Multiplying both sides of (34) by ∂xu¯ and taking the imaginary parts, we have, by up-1u=up+1/2u¯p-1/2 and the assumptions of F(t),(36)12∂t∂xuL22=I∫R∂xλFtup-1u·∂xu¯dx=Iλp+12Ft∫Rup-1∂xu2dx+Iλp-12Ft∫Rup-3u2∂xu¯2dx≤FtIλp+12+λp-12∫Rup-1∂xu2dx.By (36) and the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ, we obtain(37)12∂t∂xuL22=I∫R∂xλFtup-1u·∂xu¯dx≤0.
We note that(38)Jup-1u=p+12up-1Ju-p-12up-3u2Ju¯,JFtup-1u=x+it∂xFtup-1u=FtJup-1u,and(39)12∂tJuL22=I∫RJλFtup-1u·Ju¯dx.Calculating the right part of (39), we obtain(40)I∫RJλFtup-1u·Ju¯dx=Iλp+12Ft∫Rup-1Ju2dx-p-12IλFt∫Rup-3u2Ju¯2dx≤FtIλp+12+λp-12∫Rup-1Ju2dx≤0under the strong dissipative condition.
Multiplying both sides of (34) by u¯, we obtain(41)12∂tuL22=IλFt∫Rup+1dx≤0 by the assumptions of λ and F(t). Using (37), (40), and (41), we have (42)∂t∂xuL2+JuL2+uL2≤0.By Lemma8, we have(43)uL2+∂xuL2+JuL2≤v0xL2+∂xv0xL2+xv0xL2.From (43), we obtain a unique global solution u∈X1,∞ to (33). Since u=e-∫0tμ(τ)dτv, we have a unique global solution v(t,x)∈C[0,∞);H0,1∩H1 to (1).


## 2.2. Proof of Theorem4

Using the transformu=e-∫0tμ(τ)dτv, we get the equation about u from (1):(44)i∂tu+12∂x2u=λep-1∫0tμτdτup-1u,where u(0,x)=v(0,x)=v0(x). We have local existence of solutions u∈X1,T to (44) and v∈X1,T to (1), respectively (see, e.g., [13]).By Lemma8 and the assumptions in Theorem 4, we have(45)uL2+∂xuL2+JuL2≤v0xL2+∂xv0xL2+xv0xL2.And(46)vL2+∂xvL2+JvL2≤e∫0tμτdτv0xL2+∂xv0xL2+xv0xL2.Thus, we obtain global-in-time existence of solutions u∈X1,∞ to (44) by (45) and v∈X1,∞ to (1) by (46), respectively.We are now in a position to prove time decay estimates of solutions to (1). By the Sobolev inequality (47)uL∞≤CuL21/2∂xuL21/2,the factorization formula Ut=MDtFM, and (45), we have (48)uL∞≤Ct-1/2FMU-tuL∞≤Ct-1/2FMU-tuL21/2∂xFMU-tuL21/2=Ct-1/2uL21/2JuL21/2≤Ct-1/2for t>1. Using the transform u=e-∫0tμ(τ)dτv, we obtain (49)vL∞≤Ct-1/2e∫0tμτdτfor t>1. By (46), we have (50)vH1≤Ce∫0tμτdτfor t>1.


## 3. Proofs of Theorems6 and 7

### 3.1. Proof of Theorem6

From Theorem4, we have global existence of solutions v∈X1,∞ to (27) and the time decay estimates (30). To get a better decay estimates of solutions to (27), we use the method of [4, 5]. Changing a variable such as u=(1+t)-a/p-1v, we have(51)i∂tu+12∂x2u=λ1+taup-1u,and u(0,x)=v(0,x)=v0(x).Multiplying both sides of (51) by FU-t, we get (52)FU-tλ1+taup-1u=λ1+taFM-1F-1Dt-1M-1up-1u=λ1+tat-p-1/2FU-tup-1FU-tu+λRt,where(53)Rt=1+tat-p-1/2×FMU-tup-1FMU-tu-FU-tup-1FU-tu+1+tat-p-1/2FM-1-1F-1FMU-tup-1FMU-tu.Therefore, we have (54)i∂tFU-tu=λ1+tat-p-1/2FU-tup-1FU-tu+λRt,where the remainder term R(t) is given in (53). Substituting FU-tu by f, we have the following equation about f:(55)i∂tft=λ1+tat-p-1/2ftp-1ft+λRt.We have the following estimates ofR(t). Since the proof of these estimates is similar to that in [4, 6], we omit the proof.Lemma 9.
Letu(t,x) be a solution of (51) in the function space X1,∞. Then, we have(56)RtL∞≤Ct-p/2-1/4-aFMU-tuL∞p-1+FU-tuL∞p-1JuL2and(57)RtL2≤Ct-p/2+aFU-tuL2p-1/2JuL2p+1/2for t>1.From (55), we obtain(58)∂tft2=2Iλ1+tat-p-1/2ftp+1+2IλRtft¯.Hence, we get (59)∂tft≤2aIλt-p-1/2+aftp+CRtL∞for t>1. By Lemmas 8 and 9, we obtain (60)∂tft≤2aIλt-p/2+1/2+aftp+Ct-p/2+1/4+afor t>1.Multiplying both sides of the above bytδ, where δ>0, we have (61)tδ∂tft≤2aIλtδ-p/2+1/2+aftp+Ctδ-p/2+1/4+a.By the Young inequality, we have(62)∂ttδft≤δtδ-1ft-2aIλtδ-p/2+1/2+aftp+Ctδ-p/2+1/4+a≤p-1pp/p-1·12aIλ1/p-1·δp/p-1tδ-1+2a+p/2p-1+Ctδ-p/2+1/4+afor t>1. Let δ=2a+3-p/2(p-1) for 1<p<2a+3 and -1<a≤0. Integrating in time from 1 to t, we get(63)ft≤Ct-2a+3-p/2p-1+Ct-p/2-5/4-afor 1<p<2a+3, p>5/2+2a, -1<a≤0, and t>1. To guarantee that(64)0<2a+3-p2p-1≤p2-54-a,we need (65)52+2a<p,1<p<2a+3,1≤p2p-4a-5.Let p∗(a)=a+5/4+1/416a2+40a+33. Then, from (63), we get(66)ft≤Ct-2a+3-p/2p-1for p∗(a)≤p<2a+3, -1<a≤0, and t>1.Sincep∗(a)≤p<2a+3 and -1<a≤0, we have 2a+3-p/2(p-1)<1/4. By (66), we have (67)uL∞≤Ct-1/2FU-tuL∞+ct-3/4xU-tuL∞≤Ct-1/2-2a+3-p/2p-1for t>1. Using the transform v(t,x)=(1+t)a/p-1u(t,x), we have our desired result.


### 3.2. Proof of Theorem7

We have global existence of solutionsv∈X1,∞ to (27) by Theorem 4. We consider the decay estimates of solutions to (27) by using the method of [4, 5] in the following steps.Since[J2,i∂t+1/2∂x2]=0 holds, from (51), we have(68)12∂tJ2uL22=Iλ∫RJ21+taup-1u·J2u¯dx.By the strong dissipative condition Iλ<0 and Iλ>p-1/2p|Rλ| and using uL∞≤Ct-1/2uL21/2JuL21/2 for t>0, we have(69)Iλ∫RJ21+taup-1u·J2u¯dx≤Iλp+12∫R1+taup-1J2u2dx+Iλp-12∫R1+taup-3u2J2u¯2dx+C∫R1+taup-2Ju2J2udx≤1+taIλp+12+Iλp-12∫Rup-1J2u2dx+C1+tauL∞p-2JuL∞J2uL2JuL2≤Ct-p-1/2+auL2p-2/2JuL23/2+p-2/2J2uL23/2for t>0. Thus, we obtain(70)J2uL2≤v0H0,2+Ct-p+3+2afor t>0.By using a similar method to that in [4], we have the estimate of R(t) as follows. Here we omit the proof.Lemma 10.
Letu be a solution of (51) in the function space X1,∞. Then, we have (71)RtL∞≤Ct-p/2+1/4-av0H0,1p-1J2uL2for t>1.(58) shows(72)∂tft2=2Iλ1+tat-p-1/2ftp+1+2IλRtft¯,where f=FU(-t)u. Let 3-p+2a>0. By Lemma 10 and (58), we get (73)∂tft≤2aIλt-p-1/2+aftp+Ct-3p/2+11/4+3afor t>1. Multiplying both sides of the above by tρ, where ρ>0, we have (74)tρ∂tft≤2aIλtρ-p/2+1/2+aftp+Ctρ-3p/2+11/4+3a.By the Young inequality, we have (75)∂ttρft≤ρtρ-1ft-2aIλtρ-p/2+1/2+aftp+Ctρ-3p/2+11/4+3a≤p-1pp/p-1·12aIλ1/p-1·ρp/p-1tρ-1+2a+p/2p-1+Ctρ-3p/2+11/4+3afor t>1. Let ρ=2a+3-p/2(p-1) for 1<p<2a+3 and -1<a≤0. Integrating in time from 1 to t, we get(76)ft≤Ct-2a+3-p/2p-1+Ct-3p/2-15/4-3afor 1<p<2a+3, p>2a+13/6, -1<a≤0, and t>1. To obtain(77)0<2a+3-p2p-1≤3p2-154-3a,we need (78)1<p<2a+3,p>2a+136,6p2-12a+19p+8a+9≥0.Let p∗∗(a)≕a+19/12+1/12144a2+264a+145. Then, from (76), we get(79)ft≤Ct-2a+3-p/2p-1for p∗∗(a)≤p<2a+3, -1<a≤0, and t>1.Whenp∗∗(a)≤p<2a+3 and -1/2<a≤0, we have 2a+3-p/2(p-1)<1/4. By (79), we have (80)uL∞≤Ct-1/2FU-tuL∞+ct-3/4xU-tuL∞≤Ct-1/2-2a+3-p/2p-1for t>1. By using the relation v(t,x)=(1+t)a/p-1u(t,x), we have our desired result.


## 3.1. Proof of Theorem6

From Theorem4, we have global existence of solutions v∈X1,∞ to (27) and the time decay estimates (30). To get a better decay estimates of solutions to (27), we use the method of [4, 5]. Changing a variable such as u=(1+t)-a/p-1v, we have(51)i∂tu+12∂x2u=λ1+taup-1u,and u(0,x)=v(0,x)=v0(x).Multiplying both sides of (51) by FU-t, we get (52)FU-tλ1+taup-1u=λ1+taFM-1F-1Dt-1M-1up-1u=λ1+tat-p-1/2FU-tup-1FU-tu+λRt,where(53)Rt=1+tat-p-1/2×FMU-tup-1FMU-tu-FU-tup-1FU-tu+1+tat-p-1/2FM-1-1F-1FMU-tup-1FMU-tu.Therefore, we have (54)i∂tFU-tu=λ1+tat-p-1/2FU-tup-1FU-tu+λRt,where the remainder term R(t) is given in (53). Substituting FU-tu by f, we have the following equation about f:(55)i∂tft=λ1+tat-p-1/2ftp-1ft+λRt.We have the following estimates ofR(t). Since the proof of these estimates is similar to that in [4, 6], we omit the proof.Lemma 9.
Letu(t,x) be a solution of (51) in the function space X1,∞. Then, we have(56)RtL∞≤Ct-p/2-1/4-aFMU-tuL∞p-1+FU-tuL∞p-1JuL2and(57)RtL2≤Ct-p/2+aFU-tuL2p-1/2JuL2p+1/2for t>1.From (55), we obtain(58)∂tft2=2Iλ1+tat-p-1/2ftp+1+2IλRtft¯.Hence, we get (59)∂tft≤2aIλt-p-1/2+aftp+CRtL∞for t>1. By Lemmas 8 and 9, we obtain (60)∂tft≤2aIλt-p/2+1/2+aftp+Ct-p/2+1/4+afor t>1.Multiplying both sides of the above bytδ, where δ>0, we have (61)tδ∂tft≤2aIλtδ-p/2+1/2+aftp+Ctδ-p/2+1/4+a.By the Young inequality, we have(62)∂ttδft≤δtδ-1ft-2aIλtδ-p/2+1/2+aftp+Ctδ-p/2+1/4+a≤p-1pp/p-1·12aIλ1/p-1·δp/p-1tδ-1+2a+p/2p-1+Ctδ-p/2+1/4+afor t>1. Let δ=2a+3-p/2(p-1) for 1<p<2a+3 and -1<a≤0. Integrating in time from 1 to t, we get(63)ft≤Ct-2a+3-p/2p-1+Ct-p/2-5/4-afor 1<p<2a+3, p>5/2+2a, -1<a≤0, and t>1. To guarantee that(64)0<2a+3-p2p-1≤p2-54-a,we need (65)52+2a<p,1<p<2a+3,1≤p2p-4a-5.Let p∗(a)=a+5/4+1/416a2+40a+33. Then, from (63), we get(66)ft≤Ct-2a+3-p/2p-1for p∗(a)≤p<2a+3, -1<a≤0, and t>1.Sincep∗(a)≤p<2a+3 and -1<a≤0, we have 2a+3-p/2(p-1)<1/4. By (66), we have (67)uL∞≤Ct-1/2FU-tuL∞+ct-3/4xU-tuL∞≤Ct-1/2-2a+3-p/2p-1for t>1. Using the transform v(t,x)=(1+t)a/p-1u(t,x), we have our desired result.


## 3.2. Proof of Theorem7

We have global existence of solutionsv∈X1,∞ to (27) by Theorem 4. We consider the decay estimates of solutions to (27) by using the method of [4, 5] in the following steps.Since[J2,i∂t+1/2∂x2]=0 holds, from (51), we have(68)12∂tJ2uL22=Iλ∫RJ21+taup-1u·J2u¯dx.By the strong dissipative condition Iλ<0 and Iλ>p-1/2p|Rλ| and using uL∞≤Ct-1/2uL21/2JuL21/2 for t>0, we have(69)Iλ∫RJ21+taup-1u·J2u¯dx≤Iλp+12∫R1+taup-1J2u2dx+Iλp-12∫R1+taup-3u2J2u¯2dx+C∫R1+taup-2Ju2J2udx≤1+taIλp+12+Iλp-12∫Rup-1J2u2dx+C1+tauL∞p-2JuL∞J2uL2JuL2≤Ct-p-1/2+auL2p-2/2JuL23/2+p-2/2J2uL23/2for t>0. Thus, we obtain(70)J2uL2≤v0H0,2+Ct-p+3+2afor t>0.By using a similar method to that in [4], we have the estimate of R(t) as follows. Here we omit the proof.Lemma 10.
Letu be a solution of (51) in the function space X1,∞. Then, we have (71)RtL∞≤Ct-p/2+1/4-av0H0,1p-1J2uL2for t>1.(58) shows(72)∂tft2=2Iλ1+tat-p-1/2ftp+1+2IλRtft¯,where f=FU(-t)u. Let 3-p+2a>0. By Lemma 10 and (58), we get (73)∂tft≤2aIλt-p-1/2+aftp+Ct-3p/2+11/4+3afor t>1. Multiplying both sides of the above by tρ, where ρ>0, we have (74)tρ∂tft≤2aIλtρ-p/2+1/2+aftp+Ctρ-3p/2+11/4+3a.By the Young inequality, we have (75)∂ttρft≤ρtρ-1ft-2aIλtρ-p/2+1/2+aftp+Ctρ-3p/2+11/4+3a≤p-1pp/p-1·12aIλ1/p-1·ρp/p-1tρ-1+2a+p/2p-1+Ctρ-3p/2+11/4+3afor t>1. Let ρ=2a+3-p/2(p-1) for 1<p<2a+3 and -1<a≤0. Integrating in time from 1 to t, we get(76)ft≤Ct-2a+3-p/2p-1+Ct-3p/2-15/4-3afor 1<p<2a+3, p>2a+13/6, -1<a≤0, and t>1. To obtain(77)0<2a+3-p2p-1≤3p2-154-3a,we need (78)1<p<2a+3,p>2a+136,6p2-12a+19p+8a+9≥0.Let p∗∗(a)≕a+19/12+1/12144a2+264a+145. Then, from (76), we get(79)ft≤Ct-2a+3-p/2p-1for p∗∗(a)≤p<2a+3, -1<a≤0, and t>1.Whenp∗∗(a)≤p<2a+3 and -1/2<a≤0, we have 2a+3-p/2(p-1)<1/4. By (79), we have (80)uL∞≤Ct-1/2FU-tuL∞+ct-3/4xU-tuL∞≤Ct-1/2-2a+3-p/2p-1for t>1. By using the relation v(t,x)=(1+t)a/p-1u(t,x), we have our desired result.



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*Source: 7297090-2019-03-03.xml*