# Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity

Authors: Jincheng Shi; Shengzhong Xiao
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. Under suitable initial conditions, the existence of the global classical solutions for the viscosity long-short wave equations is proved. If it does not exist globally, the life span which is the largest time where the solutions exist is also obtained.


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## Body

## 1. Introduction

In this paper, we studied the global well-posedness of the solutions for the long-short wave systems with viscosity which describes the coupling between nonlinear Schrödinger systems and the parabolical systems. In 1977, Benney [2] presented a general theory for deriving nonlinear partial differential equations in which both long and short wave solutions coexist and interact with each other nonlinearly.(1)iut+Cg∇⋅v+△u=Ft,x,u,v,vt+Cl∇⋅v+μ△v=Gt,x,u,v,where ut,x denotes the envelope of the short wave, vt,x is the amplitude of the long wave, the quantity Cl is the long wave speed, and Cg is the group velocity of the short waves. This system arises in the study of surface waves with both gravity and capillary models presented [8] and also in plasma physics [21].In [20], Tsutsumi and Hatano studied the well-posedness of the Cauchy problems for one type of Benney equations(2)iut+△u=vu+u2u,vt=ux2,and they got the well-posedness of the Cauchy problems of (2).In [5, 6], the authors investigated the quasilinear Benney equations with the form,(3)iut+Δu=vu+u2u,vt+fvx=ux2.In [5], they imposed the condition fv=av2−bv3,a>0,b>0 and they established the existence of weak solutions for equation (3). In [6], they supposed that f is a polynomial real function, and they got the existence of local strong solutions for these versions of Benney equations. However, for the existence of the smooth solutions associated with an arbitrary flux-function f and the finite-time blow-up of these equations are still open. There are many papers that deal with the long-short wave equations, for example [1, 4, 14, 15, 17, 22, 23].In the present paper, we extended the results to the viscosity long-short wave equations of the general form,(4)iut+△u=Fu,v,vt−η△v=Dxu2v−Dxfv,u0,x=εϕ;v0,x=εg,where x∈Rn,t≥0, F=Fu,v, and f=fv are nonlinear functions satisfying hypotheses: in a neighborhood of λ^=0, say, for λ^≤1,fλ^=Oλ^1+α, and Fλ^=Oλ^3 where α is an integer ≥1.u=ut,x∈C, v=vt,x∈R,i=−1, and η is the viscosity coefficient.Precisely, we are concerned with the viscosity long-short wave equations with the following form:(5)iut+△u=v2u+u2u+u2v+v3,vt−η△v=Dxu2v−Dxfv,u0,x=εϕ;v0,x=εg.Our purpose is to estimate the life span of solutions which is expressed explicitly in [11, 16].We call the maximal existence time of the solutions u,v in the classical sense the life span (or blow-up time) of u,v. We denote by Tε for ε>0 the life span of the solutions u,v of (5). Under the same assumptions on ϕ and g as above, we will give an upper estimate of Tε in terms of the prescribed conditions and study the best possibility of this estimate. Thus, we can get the existence of the global classical solutions for the viscosity long-short wave equations. If it does not exist globally, we can also get the life span (the largest time in which the solutions exist) with a more general fv in Rn.The studies about the life span are bound in literatures [3, 7, 9, 10, 12, 13, 18, 19]. Recently, some new results about these problems are founded in [24–36].We state our main results as follows.Theorem 1.
Suppose that the nonlinear termf on the right-hand side of (5) satisfies (43) and(6)ϕ∈WS,1Rn∩HS+1Rn,g∈WS,1Rn∩HS+1Rn.Then, for any given integerS≥n+5, there exist positive constants ε0 and c0 with c0ε0≤1 such that, for any ε∈0,ε0, there exists a positive number T=Tε such that Cauchy problem (5) admits on 0,Tε a unique classical solution u∈XS,c0ε,Tε, where Tε can be chosen as follows:(7)Tε=+∞,ifn>2,Tε=eaε−1−1,ifn=2,Tε=bε−2−1,ifn=1,where a,b are positive constants satisfying (148), (149), and (151). Moreover, with eventual modification on a set with zero measure in the variable t, we can obtain the results; for any finite T0 with 0<T0≤T, we have(8)u∈L20,T0,HS+2Rn∩C0,T0;HS+1Rn,(9)v∈L20,T0,HS+2Rn∩C0,T0;HS+1Rn,(10)ut∈L20,T0,HSRn∩C0,T0;HS−1Rn,(11)vt∈L20,T0,HSRn∩C0,T0;HS−1Rn.Moreover, by the Sobolev embedding theorems (observing thatS≥n+5), it easily follows from (8)–(11) that the solutions are classical solutions to the Cauchy problem (5).Remark 1.
In this paper, we took the viscosity coefficientη=1 for simplicity. For the case when η⟶0, we will discuss it in another paper.
The remainder of this paper is as follows: Section2 is devoted to giving some useful estimates which will be used in Section 3. In Section 3, we consider the Cauchy problem for n-dimensional nonlinear evolution equations and we get our main results. In Section 4, conclusion is given.


## 2. Preliminary

First, we considered the following homogeneous equations:(12)iut+△u=0,ux,0=ϕx,(13)vt−△v=0,vx,0=gx.It is well known that, by means of the Fourier transformation, the solutions to the Cauchy problem of equations (12) and (13) can be expressed in the following explicit form:(14)ux,t=14πitn/2∫Rneix−y2/4tϕydy,and(15)vx,t=14πtn/2∫Rne−x−y2/4tgydy,where y=y1,y2,…,yn and x−y2=∑i=1nxi−yi2.For simplicity, we wrote (14) in the form,(16)ux,t=Stϕ,and (15) in the form,(17)vx,t=Rtg,where(18)St:ϕ⟶ut,⋅,Rt:g⟶vt,⋅.Then, by Duhamel’s principle, the solutions to the Cauchy problem for inhomogeneous heat equations(19)iut+△u=Ft,x,vt−△v=Gt,x,ux,0=ϕx,vx,0=gx,can be denoted as(20)u=Stϕ+∫0tSt−τFτ,⋅dτ,and(21)v=Rtg+∫0tRt−τGτ,⋅dτ,or precisely,(22)ux,t=14πitn/2∫Rneix−y2/4tϕydy+14πn/2∫0t∫Rn1it−τn/2eix−y2/4t−τFτ,ydydτ,vx,t=14πtn/2∫Rne−x−y2/4tgydy+14πn/2∫0t∫Rn1t−τn/2e−x−y2/4t−τGτ,ydydτ.Now, we used the explicit expressions (14) and (15) to establish some decay estimates t⟶∞ for solutions to Cauchy problem (12) and (13) for the n-dimensional homogeneous equations.The following lemmas were employed by Li and Chen [16], but for completeness, we included them here without proving.Lemma 1.
Suppose that all norms appearing on the right-hand side below are bounded. For any integerN≥0, solutions (14) and (15) to the Cauchy problem (12) and (13) satisfy the following estimates:(23)RtgWN,∞≤c01+t−n/2gWN+n+1,1Rn,(24)RtgWN,1≤gWN,1Rn,∀t≥0,(25)StϕWN,∞≤c01+t−n/2ϕWN+n+1,1Rn,and(26)StϕHN=ϕHNRn.Lemma 2.
Suppose thatϕ∈HSRn,F∈L20,T;HS−1, where S≥1 is an integer, then the Cauchy problem(27)ut−Δu=Ft,x,t=0,u=ϕx,admits the following estimate:(28)∫0t∑k=2Dxkuτ,⋅HS−12dτ≤c0ϕHS2+∫0tFτ,⋅HS−12dτ,where c0 is a positive constant independent of t,k=k1,k2,…,kn is a multi-index,k=k1+k2+⋯+kn, and Dxk=∂k/∂x1k1,∂x2k2,…,∂xnkn.Lemma 3.
Under assumption1/r=1/p+1/q, if all norms appearing on the right-hand side below are bounded, then for any given integer S≥0, we have(29)fgWS,rRn≤cfLpRngWS,qRn+fWS,qRnfLpRn.Furthermore, suppose thatF=Fw is a sufficiently smooth function of w=w1,w2,…,wN, satisfying that if(30)w≤υ0,then(31)Fw=Ow1+α,α≥1integer.For any given integerS≥0, if a vector function w=wx satisfies(32)wL∞≤υ0,and such that all norms appearing on the right-hand side below are bounded, then(33)FwWS,rRn≤CSwWS,qRn∏i=1αwLpiRn,where CS is a positive constant (depending on υ0) and(34)1r=∑i=1α1pi+1q,1≤pii=1,…,α,q,r≤+∞.In particular, we have(35)FwWS,rRn≤CSwWS,qRnwLpRnwL∞rnα−1,where CS is a positive constant (depending on υ0) and(36)1r=1p+1q,1≤p,q,r≤+∞.Lemma 4.
Suppose thatF=Fw satisfies the same assumptions as in Lemma 3. If vector functions w=w¯x and w=w¯¯x satisfy (30), respectively, and such that all norms appearing on the right-hand side below are bounded, then for any given integer S≥0, let(37)w∗=w¯−w¯¯.We have(38)Fw¯−Fw¯¯WS,rRn≤CSw∗LpRnw¯WS,qRn+w¯¯WS,qRn+w∗WS,qRnw¯LpRn+w¯¯LpRnw¯L∞Rn+w¯¯L∞Rnα−1,where p,q and r satisfy 1/r=1/p+1/q and CS is a positive constant (depending on υ0).


## 3. The Proof of Theorem1

In this section, we shall discuss the Cauchy problem for n-dimensional nonlinear evolution equations:(39)iut+Δu=v2u+u2u+u2v+v3,Δ=∂2∂x12+⋯+∂2∂xn2,vt−Δv=Dxu2v−Dxfv,(40)t=0:u=εϕx,x=x1,x2,…,xn,(41)t=0:v=εgx,x=x1,x2,…,xn,where Dxu=ux1,…,uxn=uxi;i=1,…,n, and ε>0 is a small parameter.Let(42)λ^=λ;λi,i=1,…,n.For the nonlinear termf in (39), we give the following hypotheses: in a neighborhood of λ^=0, say, for λ^≤1,f=fλ^ is sufficiently smooth and(43)fλ^=Oλ^1+α,where α is an integer ≥1.For any given integerS such that S≥n+5, and any positive numbers E,T0<T≤+∞, we introduce the following set of functions:(44)XS,E,T=u=ut,x;v=vt,x|DS,Tu,v≤E,where(45)DS,Tu,v=sup0<t≤T1+tn/2vWS−n−3,∞Rn+sup0<t≤TvWS−1,1Rn+∫0T∑|k|=2DxkvHS−12dt1/2+sup0<t≤T1+tn/2uWS−n−3,∞Rn+sup0<t≤TuHSRn.We now define a map(46)u˜,v˜⟶Mu˜,v˜=u,v,where u,v are the solutions of the linear equations(47)iut+Δu=v˜2u˜+u˜2u˜+u˜2v˜+v˜3,vt−Δv=Dxu˜2v˜−Dxfv˜,t=0:u=εϕx,t=0:v=εgx.The next step is to prove that the mapM is a strict contraction in the complete metric space XS,E,T.Lemma 5.
For anyu˜,v˜∈XS,E,T (in which S≥n+5,E≤1),u,v=Mu˜,v˜ satisfies(48)DS,Tu,v≤c1˜ε+E21+∫0t1+τ−n/2dτ.Proof. From (21), we have(49)v=εRtg+∫0tRt−τDxfv˜dτ+∫0tRt−τDxu˜2v˜dτ.By (23), we have(50)vt,⋅WS−n−3,∞≤cε1+t−n/2gWS−2,1+c∫0t1+t−τ−n/2Dxfv˜WS−2,1dτ+c∫0t1+t−τ−n/2Dxu˜2v˜WS−2,1dτ.Using (29) and (33), we get(51)Dxfv˜WS−2,1≤cv˜WS−1,1v˜W1,∞α,(52)Dxu˜2v˜WS−2,1≤cv˜WS−1,1u˜W1,∞2+cu˜HS−12v˜W1,∞.A combination of (50)–(52) leads to(53)vt,⋅WS−n−3,∞≤cε1+t−n/2gWS−2,1+c∫0t1+t−τ−n/2v˜WS−2,1v˜W1,∞αdτ+c∫0t1+t−τ−n/2u˜HS−12v˜W1,∞+u˜W1,∞2v˜WS−1,1dτ.Noting thatS≥n+5, by the definition of XS,E,T, we have(54)v˜W1,∞≤v˜WS−n−3,∞≤E1+t−n/2,(55)v˜WS−2,1≤E,(56)u˜HS−1≤u˜HS≤E,(57)u˜W1,∞≤u˜WS−n−3,∞≤E1+t−n/2.On combining (53)–(57), we get(58)vt,⋅WS−n−3,∞≤cε1+t−n/2+cE1+α∫0t1+t−τ−n/21+τ−nα/2dτ+c∫0t1+t−τ−n/2E31+τ−n/2+1+τ−ndτ.We point out the fact that, ifb≥a≥0, then we have(59)∫0t1+t−τ−a1+τ−bdτ≤c1+t−a∫0t1+τ−bdτ.The proof of (59) was shown in [16], but to complete it, we include here. Let(60)I1=∫0t/21+t−τ−a1+t−bdτ≤1+t2−a∫0t/21+t−bdτ,≤c1+t−a∫0t1+t−bdτ,(61)I2=∫t/2t1+t−τ−a1+τ−bdτ≤1+t2−b∫tt1+t−τ−adτ,≤c1+τ−b∫0t1+t−τ−adτ≤c1+τ−b∫0t/21+τ−adτ≤c1+τ−a∫0t1+τ−bdτ.The combination of (60) and (61) gives (59).Thus, from (58) and (59), we obtain(62)sup0<t≤T1+tn/2vt,⋅WS−n−3,∞≤cε+E1+α∫0t1+τ−nα/2dτ+E3∫0t1+τ−n/2dτ.By (49) and (24), we also have(63)vWS−1,1≤cεgWS−1,1+∫0tDxfv˜WS−1,1dτ+∫0tDxu˜2v˜WS−1,1dτ.From (29) and (33), we have(64)Dxfv˜WS−1,1≤cv˜HS2v˜W1,∞α−1,(65)Dxu˜2v˜WS−1,1≤cu˜HS2v˜W1,∞+cu˜W1,∞2v˜WS,1.Moreover, in fact,(66)v˜HS+12=v˜H12+∑|k|=2Dxkv˜HS−12,v˜H12≤cv˜W1,∞v˜W1,1≤cv˜WS−n−3,∞v˜WS−1,1,Thus, we have(67)v˜HS2≤v˜HS+12≤cE21+t−n/2+∑|k|=2Dxkv˜HS−12.On combining (63)–(65) and (67) and the definition of XS,E,T, we obtain(68)sup0<t≤TvWS−1,1≤cε+E1+α1+∫0t1+τ−nα/2dτ+E3∫0t1+τ−n/2+1+τ−ndτ.By (28), we have(69)∫0T∑|k|=2DxkvHS−12dτ≤cε2gHS−12+∫0TDxfv˜τ,⋅HS−12dτ+∫0TDxu˜2v˜HS−12dτ.Noting the hypotheses of (43), it follows from (33) that(70)Dxfv˜HS−1≤cv˜HSv˜W1,∞α,Dxu˜2v˜HS−1≤cu˜HSu˜W1,∞v˜W1,∞+cv˜HSu˜W1,∞2.Using (67), ∫0∞1+τ−1+2αn/2dτ≤c, and the definition of XS,E,T, we obtain(71)∫0T∑k=2DxkvHS−12dτ1/2≤cε+E1+α+E3∫0T1+τ−ndτ+1.We can also get(72)u=εStϕ+∫0tSt−τv˜2u˜+u˜2u˜+u˜2v˜+v˜3dτ.By (25), we have(73)uWS−n−3,∞≤cε1+t−n/2ϕWS−2,1+c∫0t1+t−τ−n/2v˜2u˜WS−2,1dτ+c∫0t1+t−τ−n/2u˜2u˜WS−2,1dτ+c∫0t1+t−τ−n/2u˜2v˜WS−2,1dτ+c∫0t1+t−τ−n/2v˜3WS−2,1dτ.By (29) and (33), we have(74)v˜2u˜WS−2,1≤v˜L∞v˜u˜WS−2,1+v˜WS−2,1v˜u˜L∞,≤v˜L∞v˜HS−2u˜HS−2+v˜WS−2,1v˜L∞u˜L∞≤E21+t−n/2v˜HS+E31+t−n.From (67), we have(75)v˜HS≤cE1+t−n/4+c∑k=2Dxkv˜HS−121/2.Thus,(76)∫0t1+t−τ−n/21+τ−n/2v˜HSdτ≤cE∫0t1+t−τ−n/21+τ−3n/4dτ+c∫0t1+t−τ−n/21+τ−n/2∑|k|=2Dxkv˜HS−121/2dτ,≤cE∫0t1+t−τ−n/21+τ−3n/4dτ+1+τ−n/2∫0t1+τ−ndτ1/2E≤cE1+τ−n/2∫0t1+τ−3n/4dτ+∫0t1+τ−ndτ1/2.n deriving (76), we have used ∫0tabdτ≤∫0ta2dτ1/2∫0tb2dτ1/2.Using (29), we have(77)u˜2v˜WS−2,1≤cu˜2WS−2,1v˜L∞+cu˜2L∞v˜WS−2,1,≤cu˜HS−22v˜L∞+cu˜L∞2v˜WS−2,1≤cE31+t−n/2+cE31+t−n,and(78)v˜3WS−2,1≤cv˜WS−2,1v˜L∞2≤cE31+t−n.On combining (73)–(78), we obtain(79)uWS−n−3,∞≤cε1+t−n/2ϕWS−2,1+cE31+t−n/2∫0t1+τ−3n/4dτ+∫0t1+τ−n/2d+τ∫0t1+τ−ndτ1/2+∫0t1+τ−ndτ,Thus,(80)sup0<t≤T1+tn/2uWS−n−3,∞≤cε+E31+∫0t1+τ−n/2dτ.Finally, from (26) and (72), we have(81)uHS≤cεϕHS+c∫0tv˜2u˜HSdτ+c∫0tu˜2u˜HSdτ+c∫0tu˜2v˜HSdτ+c∫0tv˜3HSdτ≤cε+c∫0tv˜HSu˜L∞v˜L∞+u˜HSv˜L∞2+u˜HSu˜L∞v˜L∞dτ+c∫0tu˜HSu˜L∞2+v˜HSv˜L∞2+u˜L∞2v˜HSdτ,On combining (75) and (81), we obtain(82)uHS≤cε+E31+∫0t1+τ−n/2dτ.On combining the above discussions, we obtain(83)DS,Tu,v≤c1˜ε+E21+∫0t1+τ−n/2dτ.Lemma 6.
For anyu˜,v˜, u˜˜,v˜˜∈XS,E,T(in which S≥n+5,E≤1 and 0<T≤+∞, if u¯,v¯=Mu˜,v˜,u¯¯,v¯¯=Mu˜˜,v˜˜ also satisfy u¯,v¯,u¯¯,v¯¯∈XS,E,T, then(84)DS,Tu∗,v∗≤c˜2EDS,Th,p1+∫0t1+τ−n/2dτ,where u∗=u¯−u¯¯,v∗=v¯−v¯¯,h=u˜−u˜˜,p=v˜−v˜˜.Proof. By the definition ofM, we have(85)iut∗+△u∗=v˜2u˜−v˜˜2u˜˜+u˜2u˜−u˜˜2u˜˜+u˜2v˜−u˜˜2v˜˜+v˜3−v˜˜3,vt∗−△v∗=Dxu˜2v˜−Dxu˜˜2v˜˜−Dxfv˜+Dxfv˜˜,(86)u∗x,0=v∗x,0=0.Similar to (53), we have(87)v∗WS−n−3,∞≤c∫0t1+t−τ−n/2Dxfv˜−Dxfv˜˜WS−2,1dτ+c∫0t1+t−τ−n/2Dxu˜2v˜−Dxu˜˜2v˜˜WS−2,1dτ.Noting thatS≥n+5, and the definition of XS,E,T and using the hypotheses (43) and (38) (in which we take r=1,p=+∞,q=1), we get(88)Dxfv˜−Dxfv˜˜WS−2,1≤cpWS−1,1v˜W1,∞+v˜˜W1,∞α+cpW1,∞v˜WS−1,1+v˜˜WS−1,1v˜W1,∞+v˜˜W1,∞α−1≤cpWS−1,1v˜WS−n−3,∞+v˜˜WS−n−3,∞α+cpWS−n−3,∞v˜WS−1,1+v˜˜WS−1,1v˜WS−n−3,∞+v˜˜WS−n−3,∞α−1,Dxu˜2v˜−Dxu˜˜2v˜˜WS−2,1=Dxu˜2−u˜˜2v˜+u˜˜2v˜−v˜˜WS−2,1.For we have(89)Dxu˜˜2v˜−v˜˜WS−2,1≤cu˜˜WS−1,12pW1,∞+u˜˜W1,∞2pWS−1,1,≤cu˜˜Hs2pW1,∞+u˜˜W1,∞2pWS−1,1≤c1+t−n/2E2DS,Th,p+cE21+t−nDS,Th,p,we also have(90)Dxu˜2−u˜˜2v˜WS−2,1≤cDxu˜+u˜˜hv˜WS−2,1,≤cu˜+u˜˜hWS−1,1v˜W1,∞+cu˜+u˜˜hW1,∞v˜WS−1,1≤cu˜HShHS+u˜˜HShHSv˜W1,∞+cu˜W1,∞hW1,∞+u˜˜W1,∞hW1,∞v˜WS−1,1≤c1+t−n/2E2DS,Th,p+cE21+t−nDS,Th,p.Thus, we can get(91)Dxu˜2v˜−Dxu˜˜2v˜˜WS−2,1≤c1+t−n/2E2DS,Th,p+cE21+t−nDS,Th,p.Putting (88) and (91) into (87), and using the definition of XS,E,T and (59), we obtain(92)sup0<t≤T1+tn/2v∗WS−n−3,∞≤cEαDS,Th,p∫0t1+τ−nα/2dτ+cE2DS,Th,p∫0t1+τ−n/2dτ.Next, similar to (63), we have(93)v∗WS−1,1≤∫0tDxfv˜−Dxfv˜˜WS−1,1dτ+∫0tDxu˜2v˜−Dxu˜˜2v˜˜WS−1,1dτ.Using (38) (in which we take r = 1, p = q = 2), we have(94)Dxfv˜−Dxfv˜˜WS−1,1≤cpHSv˜H1+v˜˜H1+pH1v˜HS+v˜˜HSv˜W1,∞+v˜˜W1,∞α−1.By the definition of⋅HS and the definition of DS,Th,p, we get(95)pHS2≤pHS+12=pH12+∑|k|=2DxkpHS−12≤cpWS−n−3,∞pWS−1,1+∑|k|=2DxkpHS−12,≤c1+t−n/2DS,T2h,p+∑k=2DxkpHS−12.Moreover, we have(96)pH12≤cpW1,∞pW1,1≤c1+t−n/2DS,T2h,p.Thus, we see(97)pHS≤c1+t−n/4DS,Th,p+c∑k=2DxkpHS−121/2,(98)pH1≤c1+t−n/4DS,Th,p.Similarly, we have(99)v˜HS+v˜˜HS≤c1+t−n/4E+c∑k=2Dxkv˜HS−12+∑k=2Dxkv˜˜HS−121/2,(100)v˜H1+v˜˜H1≤c1+t−n/4E.Besides, we have(101)v˜W1,∞+v˜˜W1,∞≤cE1+t−n/2.On combining (94)–(101), we obtain(102)∫0tDxfv˜−Dxfv˜˜WS−1,1dτ≤cEα∫0t1+τ−nα/2dτDS,Th,p+cEα∫0t1+τ−nα−1/2/2∑k=2DxkpHS−121/2dτ+cEα−1DS,Th,p∫0t1+τ−nα−1/2/2∑k=2Dxkv˜HS−12+∑k=2Dxkv˜˜HS−121/2dτ.Noting thatα is an integer ≥1, we have(103)α−12≥α2.It follows from (102) that(104)∫0tDxfv˜−Dxfv˜˜WS−1,1dτ≤cEαDS,Th,p1+∫0t1+τ−nα/2dτ.Similar to (91), we can get(105)Dxu˜2v˜−Dxu˜˜2v˜˜WS−1,1≤c1+t−n/2E2DS,Th,p+cE21+t−nDS,Th,p.On combining (93), (104), and (105), we get(106)sup0<t≤Tv∗WS−1,1≤cEαDS,Th,p∫0t1+τ−nα/2dτ+1+cE2DS,Th,p∫0t1+τ−n/2dτ.By means of (29), it comes from (85) that(107)∫0T∑|k|=2Dxkv∗HS−12dτ≤c∫0TDxfv˜−Dxfv˜˜HS−12dτ+c∫0TDxu˜2v˜−Dxu˜˜2v˜˜HS−12dτ.By (38) (in which we take r = q = 2 and p=+∞), we get(108)Dxfv˜−Dxfv˜˜HS−1≤cpHSv˜W1,∞+v˜˜W1,∞+pW1,∞v˜HS+v˜˜HSv˜W1,∞+v˜˜W1,∞α−1.Still using (97), (99), and (101) and notifying that(109)pW1,∞≤cDS,Th,p1+t−n/2.it follows that(110)∫0TDxfv˜−Dxfv˜˜HS−12dτ≤cE2α∫0t1+τ−1+2α/2ndτDS,T2h,p+cE2α∫0T1+τ−nα∑k=2DxkpHS−12dτ+cE2α−1DS,Th,p2∫0T1+τ−nα∑k=2Dxkv˜HS−12+∑k=2Dxkv˜˜HS−12dτ.For we know that(111)Dxu˜2v˜−Dxu˜˜2v˜˜HS−1=Dxu˜2−u˜˜2v˜˜+u˜˜2v˜−v˜˜HS−1,≤cu˜+u˜˜hv˜HS+cu˜˜2pHS≤cu˜HS+u˜˜HShL∞v˜L∞+cu˜L∞+u˜˜L∞hHSv˜L∞+cu˜L∞+u˜˜L∞hL∞v˜HS+cu˜HSu˜L∞pL∞+cu˜L∞2pHS≤cE21+t−nDS,Th,p+cE1+t−nDS,Th,pv˜Hs+cE21+t−npHS.From (67) and (97), we obtain(112)Dxu˜2v˜−Dxu˜˜2v˜˜HS−1≤cE21+t−nDS,Th,p+cE21+t−5n/4DS,Th,p+cE1+t−nDS,Th,p∑k=2Dxkv˜HS−121/2+cE21+t−n∑k=2DxkpHS−121/2.Thus,(113)∫0tDxu˜2v˜−Dxu˜˜2v˜˜HS−12dτ≤cE4∫0T1+τ−2ndτ+cE2DS,T2h,p∫0T∑|k|=2Dxkv˜HS−12dτ+cE4∫0T∑|k|=2DxkpHS−12dτ,≤cE4DS,T2h,p,in deriving (111), we have used ∫0T1+τ−2ndτ≤c and 1+t−1≤1.On combining (107), (111), and (112) and using 1+t−1≤1, we obtain(114)∫0T∑k=2Dxkv∗HS−12dτ1/2≤cEαDS,Th,p+cE2DS,Th,p.Similar to (73), we obtain(115)u∗WS−n−3,∞≤c∫0t1+t−τ−n/2u˜2u˜−u˜˜2u˜˜WS−2,1+u˜2v˜−u˜˜2v˜˜WS−2,1dτ+c∫0t1+t−τ−n/2v˜2u˜−v˜˜2u˜˜WS−2,1+v˜3−v˜˜3WS−2,1dτ.We have from (38) that(116)u˜2u˜−u˜˜2u˜˜WS−2,1≤chHS−2u˜HS−2+u˜˜HS−2u˜L∞+u˜˜L∞,≤cE2DS,Th,p1+t−n/2,(117)u˜2v˜−u˜˜2v˜˜WS−2,1≤u˜2pWS−2,1+u˜+u˜˜hv˜˜WS−2,1,≤cu˜2WS−2,1pL∞+cu˜2L∞pWS−2,1+cu˜hWS−2,1v˜˜L∞+cu˜hL∞v˜˜WS−2,1+cu˜˜hWS−2,1v˜˜L∞+cu˜˜hL∞v˜˜WS−2,1≤cu˜HS2pL∞+cu˜L∞2pWS−2,1+cu˜HShHSv˜˜L∞+cu˜L∞hL∞v˜˜WS−2,1+cu˜˜HShHSv˜˜L∞+cu˜˜L∞hL∞v˜˜WS−2,1.We can obtain(118)u˜2v˜−u˜˜2v˜˜WS−2,1≤cE21+t−n/2DS,Th,p+cE21+t−nDS,Th,p.and we also have(119)v˜2u˜−v˜˜2u˜˜WS−2,1=v˜2u˜−v˜˜2u˜+v˜˜2u˜−v˜˜2u˜˜WS−2,1≤v˜+v˜˜pu˜WS−2,1+v˜˜2hWS−2,1.For we have(120)v˜pu˜WS−2,1≤cv˜HS−2u˜HS−2pL∞+pHS−2u˜HS−2v˜L∞+pHS−2v˜HS−2u˜L∞.Using (99), we obtain(121)v˜HS−2u˜HS−2pL∞≤Ev˜HS1+t−n/2DS,Th,p≤cE21+t−3n/4DS,Th,p+cE1+t−n/2∑k=2Dxkv˜˜HS−12DS,Th,p,So, we have(122)∫0t1+t−τ−n/2v˜HS−2u˜HS−2pL∞dτ≤cE21+t−n/2∫0t1+τ−3n/4dτDS,Th,p+cE∫0t1+t−τ−n1+τ−ndτ1/2∫0t∑k=2Dxkv˜˜HS−12dτ1/2DS,Th,p,≤cE21+t−n/2∫0t1+t−3n/4dτDS,Th,p+cE21+t−n/21+∫0t1+t−ndτ.Following the same procedure as (122), we obtain(123)∫0t1+t−τ−n/2pHS−2u˜HS−2v˜L∞dτ≤cE21+t−n/2∫0t1+t−3n/4dτDS,Th,p+cE21+t−n/21+∫0t1+t−ndτ.We now begin to deal with∫0t1+t−τ−n/2v˜˜2hWS−2,1dτ.We can easily get(124)v˜˜HS−2≤v˜˜HS≤c1+t−n/4E+c∑k=2Dxkv˜˜HS−121/2.Thus, we have(125)v˜˜2hWS−2,1≤cv˜˜HS−2hHS−2v˜˜L∞+cv˜˜HS−22hL∞,≤cE1+t−n/2DS,Th,pv˜˜HS−2+c1+t−n/2DS,Th,pv˜˜HS−22≤cE21+t−n/2DS,Th,p+c1+t−n/2DS,Th,pE21+t−n/2+∑k=2Dxkv˜HS−12.Now, we need to seek a new method to give a bound for(126)∫0t1+t−τ−n/21+τ−n/2DS,Th,p∑k=2Dxkv˜HS−12dτ.For we know that,(127)mτ=1+t−τ−n/21+τ−n/2,is a continuous differentiable function in 0,t, thus we have(128)m′τ=n21+t−τ−n/2−11+τ−n/2−12τ−t.We can easily get whenτ=t/2, we have m′τ=0.For we have(129)m0=1+t−n/2,mt=1+t−n/2,mt2=1+t2−n=1+t+t24−n/2.Thus, we obtain(130)mτ≤1+t−n/2,in0,t.So, we can easily see that(131)∫0t1+t−τ−n/21+τ−n/2DS,Th,p∑k=2Dxkv˜HS−12dτ≤E21+t−n/2DS,Th,p.On combining (125)–(132), we obtain(132)∫0t1+t−τ−n/2v˜˜2hWS−2,1dτ≤cE21+t−n/2∫0t1+τ−n/2dτDS,Th,p+cE21+t−n/2∫0t1+τ−ndτDS,Th,p.Using (38), we obtain(133)v˜3−v˜˜3WS−2,1≤cpL∞v˜WS,1+v˜˜WS,1+cpWS,1v˜L∞+v˜˜L∞v˜L∞+v˜˜L∞,≤c1+t−nE2DS,Th,p.On combining (116)–(134), we obtain(134)sup0<t≤T1+tn/2u∗WS−n−3,∞≤cE2DS,Th,p1+∫0t1+τ−n/2dτ.Finally, applying (26), we have(135)u∗HS≤c∫0tv˜2u˜−v˜˜2u˜˜HSdτ+∫0tu˜2u˜−u˜˜2u˜˜HSdτ+∫0tu˜2v˜−u˜˜2v˜˜HSdτ+∫0tv˜3−v˜˜3HSdτ.From (38) (taking r=2,p=∞,q=2), we get(136)u˜2u˜−u˜˜2u˜˜HS≤chL∞u˜HS+u˜˜HS+hHSu˜L∞+u˜˜L∞u˜L∞+u˜˜L∞,Thus, we can easily get(137)∫0tu˜2u˜−u˜˜2u˜˜HSdτ≤c∫0t1+τ−nE2dτDS,Th,p.Similar to (119), we also have(138)v˜2u˜−v˜˜2u˜˜HS≤v˜+v˜˜pu˜HS+v˜˜2hHS,(139)v˜pu˜HS≤cv˜L∞pL∞u˜HS+cv˜L∞u˜L∞pHS+cv˜HSpL∞u˜L∞,≤c1+t−n/2E21+t−n/2DS,Th,p+c1+t−n/2E21+t−n/21+t−n/4DS,Th,p+∑|k|=2‖Dxkp‖HS−121/2+cEDS,Th,p1+t−n/21+t−n/4E+∑|k|=2Dxkv˜HS−121/2.We can get(140)∫0tv˜pu˜HSdτ≤c∫0t1+τ−nE2DS,Th,pdτ+c∫0t1+τ−3n/4E2DS,Th,pdτ+cE2DS,Th,p.We need only to give a bound for∫0tv˜˜2hHSdτ.(141)v˜˜2hHS≤cv˜HSv˜L∞hL∞+v˜L∞2hHS≤c1+t−n/2E+∑k=2DxkvHS−121/2E1+t−nDS,Th,p+E21+t−nDS,Th,p.We obtain(142)∫0tv˜˜2hHSdτ≤cE2DS,Th,p+∫0t1+τ−ndτE2DS,Th,p,(143)u˜2v˜−u˜˜2v˜˜HS≤u˜2pHS+u˜+u˜˜hv˜˜HS≤cu˜HSu˜L∞pL∞+cu˜L∞2pHS+cu˜HShL∞v˜˜L∞+cu˜L∞hL∞v˜˜HS+cu˜˜HSu˜L∞pL∞+cu˜˜L∞2pHS+cu˜˜HShL∞v˜˜L∞+cu˜˜L∞hL∞v˜˜HS,≤c1+t−nE2DS,Th,p+cE1+t−nv˜˜HS+c1+t−nE2pHS.On combining (135)–(144), (97), and (99), we can conclude that(144)u∗HS≤cE2DS,Th,p1+∫0t1+τ−ndτ.On combining (92), (106), (115), (135), and (145), we obtain (84).Lemma 7.
Letc0=2maxc1˜,c2˜, where c1˜,c2˜ are the constants given in Lemma 5 and Lemma 6, respectively. If there exists a positive number ε0 with c0ε0≤1 such that for any ε∈0,ε0, if E=Eε=c0ε0 and Eε>0 allow(145)Eε1+∫0t1+τ−n/2dτ≤1c0,then the map M has a unique fixed point in XS,E,T.Proof.
By (146) and Lemmas 5 and 6, it is easy to see that for any u˜,v˜∈XS,E,T,u,v=Mu˜,v˜ satisfies(146)DS,Tu,v≤Eε,and that for any u˜,v˜,u˜˜,v˜˜∈XS,E,T,(147)DS,Tu∗,v∗≤12DS,Th,p.
Namely,M maps XS,E,T into itself. Moreover, M is a contraction with respect to the metric XS,E,T. Thus, it follows from the standard contracting mapping principle that the map M possesses a fixed point u,v∈XS,E,T. The proof of Lemma 7 is finished.
The fixed point obtained in Lemma7 is obviously the classical solution to the Cauchy problem (39)–(41) on 0<t≤Tε.
Now, we determineε0 and Tε for any ε∈0,ε0 such that (145) holds. In what follows, we always take ε0 so small that c0ε0≤1.(1)
In the case thatn>2, since(148)∫0T1+t−n/2dt≤c,∀T>0.(i)
We can chooseTε=+∞, and let ε0 be so small that for any 0<ε≤ε0, (145) holds. In this case, we get the global solution.(2)
In the case thatn=2, since(149)∫0T1+t−1dt=ln1+T,(i)
we can chooseTε=eaε−1−1, where a is a positive constant that satisfies(150)2ac02≤1.(ii)
Thus, ifε0 is so small that(151)2c02ε0≤1,(145) holds. In that case, we get the so-called almost global solutions.(3)
In the case thatn=1, since(152)∫0T1+t−1/2dt=11−n/21+T1/2−1≤21+T1/2,we can choose Tε=bε−2−1, where b is a positive constant that satisfies(153)4b1/2c02≤1.
Thus, we still get (145), provided (153) holds.
On combining the above discussions, we get the desired Theorem1.


## 4. Conclusion

In the present paper, the global existence and blow-up for the classical solutions of the long-short wave equations with viscosity are studied. Using the method proposed in this paper, similar results can be obtained for other wave equations with different nonlinear terms. Our method is also valid for the weak solution. For the case whenη⟶0, our method is no longer applicable. There may be difficulty in obtaining the energy estimates. We must seek new method to overcome this difficulty. We think it is interesting. We will discuss it in another paper.



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