# The Stability of Solutions for the Generalized Degasperis-Procesi Equation with Variable Coefficients

Authors: Jing Chen
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

A generalized Degasperis-Procesi equation with variable coefficients is investigated. TheL1(R) stability of the strong solution for the equation is established under certain assumptions.


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## Body

## 1. Introduction

The Degasperis-Procesi (DP) equation(1)vt-vtxx+4vvx=3vxvxx+vvxxx,t>0,x∈R,was discovered by Degasperis and Procesi [1] in a search for integrable equations similar to the Camassa-Holm equation. Degasperis and Procesi [1] studied a family of third order dispersive nonlinear equations(2)vt+vx+γvxxx-α2vtxx=c1v2+c2vx2+c3vvxxx,where α,γ,c0,c1,c2,c3∈R. It is found in [1] that there are only three equations that satisfy asymptotic integrability conditions within this family. By rescaling and applying a Galilean transformation, the three equations are Korteweg-de Vries equation (3)vt+vxxx+vvx=0,t>0,x∈R,the Camassa-Holm equation(4)vt-vtxx+3vvx=2vxvxx+vvxxx,t>0,x∈R,and the Degasperis-Procesi equation (1). Degasperis et al. [2] proved the formal integrability of (1) and the existence of the nonsmooth solutions by constructing a Lax pair.In recent years, (1) which plays a similar role in water wave theory as the Camassa-Holm equation has caused extensive concern of many scholars (see [1–11]). For example, Coclite and Karlsen [3] established the well-posedness of L1∩BV weak solutions for (1). They proved uniqueness within a class of discontinuous solutions to (1) in [4]. Escher et al. [5] established the precise blow-up rate and proved the existence and uniqueness of global weak solutions to (1) in which the initial data satisfied appropriate conditions. Lai and Wu [7] investigated the local well-posedness of solutions to a generalization of both (1) and (4) in the Sobolev space Hs(R) with s>3/2. Lenells [8] classified all weak traveling wave solutions of the Degasperis-Procesi equation (1). Ai and Gui [9] proved global existence of solutions for the viscous Degasperis-Procesi equation and showed that the blow-up phenomena occurs in finite time. Fu et al. [11] studied the orbital stability of the peakons for the Degasperis-Procesi equation with a strong dispersive term on the line and proved that the shapes of these peakons were stable under small perturbations.As we know, their coefficients play an important role to study the fundamental dynamical properties of the Degasperis-Procesi models. It prompts us to study the following generalized Degasperis-Procesi equation:(5)vt-vtxx+mt,xf′vvx=bt,xf′vvxxx+3f′′vvxvxx+f′′′vvx3+2bxt,xf′vvxx+f′′vvx2+bxxt,xf′vvx,where m(t,x)∈C0(R), b(t,x)∈C03(R), and function f(·) is a polynomial of order n(n≥2). Letting m=4, b=1, f(v)=v2/2, (5) reduces to the Degasperis-Procesi equation (1). We consider the Cauchy problem of (5) with an initial condition v0(x). Namely,(6)v0,x=v0x,x∈R.Assume that (5) possesses a bounded strong solution in its maximum existence time interval [0,T) and v0 lies in L1(R)∩Hs(R)(s>3/2). We use the approaches of Kružkov doubling the variables presented in [12] to prove the L1 stability of the solution for the variable coefficients equation (5). From our knowledge, it has not been acquired in the literature.This paper is organized as follows. Section2 gives several lemmas. The proof of local solution stability is presented in Section 3.


## 2. Preliminaries

Applying the operatorΛ-2=(1-∂x2)-1 to (5), we obtain its equivalent form(7)vt+bt,xf′vvx+Λ-2mt,x-bt,x∂xfv=0,where Λ-2p(t,x)=(1-∂x2)-1p(t,x)=1/2∫R‍e-|x-y|p(t,y)dy.Equations (5) and (6) are equivalent to the problem(8)vt+∂xPt,x,v+Ψt,x,v=0,v0,x=v0x,where P(t,x,v)=b(t,x)f(v) and Ψ(t,x,v)=Λ-2(m(t,x)-b(t,x))∂xf(v)-bx(t,x)f(v). Notice that ∂xP=Px+Pvvx.Remark. According to the statements presented in [7] or [12], we know that problem (8) has a unique local solution in the space C([0,T),Hs(R)) if we assume v0(x)∈Hs(R)(s>3/2).Assume thatv1(t,x) and v2(t,x) are solutions of problem (8) in the domain [0,T)×R with initial functions v10(x) and v20(x)∈L1(R)∩Hs(R)(s>3/2), where T is the maximum existence time of solutions. For simplicity, we denote by c any positive constants. Now we give several lemmas.Lemma 1.
Letv(t,x) be the solution of problem (8) and vL∞(R)≤M. Then (9)Ψt,x,vL∞R≤cMn,where positive constant c depends on v0L∞ and T.Proof.
We have(10)Ψt,x,v=Λ-2mt,x-bt,x∂xfv-bxt,xfv≤12∫Re-x-ymt,y∂yfvdy+12∫Re-x-ybt,y∂yfvdy+bxfv≤c∫Re-x-ysign⁡y-xfvdy+cfv≤cMn,in which we have used ∫R‍e-|x-y|dx=2 to complete the proof.Lemma 2.
Assume thatv1(t,x) and v2(t,x) are solutions of problem (8) in the domain [0,T)×R, v1L∞(R)≤M, and v2L∞(R)≤M. Then(11)∫-∞+∞Ψt,x,v1-Ψt,x,v2dx≤c∫-∞+∞v1-v2dx,where c>0 depends on v10L∞(R),v20L∞(R) and T.Proof.
Using the property of the operatorΛ-2, we get(12)∫-∞+∞Ψt,x,v1-Ψt,x,v2dx≤∫-∞+∞Λ-2m-b∂xfv1-fv2dx+∫-∞+∞fv1-fv2dx≤c∫-∞+∞dx·∫-∞+∞e-x-ysign⁡y-xfv1-fv2dy+c∫-∞+∞v1-v2dx≤c∫-∞+∞fv1-fv2dy·∫-∞+∞e-x-ydx+c∫-∞+∞v1-v2dx≤c∫-∞+∞v1-v2dx,in which we apply the Tonelli Theorem to complete the proof.Letδ(σ)≥0, δ(σ)≡0, for |σ|≥1; ∫-∞+∞‍δ(σ)dσ=1 and δ(σ) is infinitely differential on (-∞,+∞). Set δɛ(σ)=δ(ɛ-1σ)/ɛ, where ɛ is an arbitrary positive constant. It is found that δɛ(σ)∈C0∞(-∞,+∞) and(13)δɛσ≥0,δɛσ=0forσ≥ɛ,δɛσ≤cɛ,∫-∞+∞δɛσdσ=1.Let the functionϕ(x) be defined and locally integrable on (-∞,+∞). Set ϕɛ(x); denote the approximation function of ϕ(x) as(14)ϕɛx=1ɛ∫-∞+∞δx-yɛϕydy.We call x0 a Lebesgue point of the function ϕ(x) if (15)limɛ→0⁡1ɛ∫x0-ɛx0+ɛϕx-ϕx0dx=0.At any Lebesgue point x0, we get (16)limɛ→0⁡ϕɛx0=ϕx0.Since the set of points which are not Lebesgue points of ϕ(x) has measure zero, we have ϕɛ(x)→ϕ(x) as ɛ→0 almost everywhere.For anyT>0, we denote the band {(t,x):[0,T]×R} by ϖT. Let Kr={x:x≤r} and (17)
Π
=
t
,
x
,
τ
,
y
:
t
-
τ
2
≤
ɛ
,
ρ
≤
t
+
τ
2
≤
T
-
ρ
,
x
-
y
2
≤
ɛ
,
x
+
y
2
≤
r
-
ρ
,where r>0, ρ>0.We state the concept of a characteristic cone. LetvL∞(R)≤M, for any R1>0; we define (18)N>maxt,x∈0,T×KR1⁡f′v.Let Ω represent the cone {(t,x):x≤R1-Nt, 0≤t≤T1=min⁡(T,R1N-1)} and let Sτ designate the cross section of the cone Ω by the plane t=τ,τ∈[0,T1].Lemma 3 (see [12]).
Let the functionϕ(t,x) be bounded and measurable in cylinder [0,T1]×Kr. For any ρ∈(0,min⁡[r,T1]) and any ɛ∈(0,ρ), the function (19)Vɛ=1ɛ2⨌Π⁡ϕt,x-ϕτ,ydtdxdτdysatisfies (20)limɛ→0⁡Vɛ=0.Lemma 4 (see [12]).
If∂F(ϕ)/∂ϕ is bounded, the function H(ϕ,ψ)=sign⁡(ϕ-ψ)(F(ϕ)-F(ψ)) satisfies the Lipschitz condition in ϕ and ψ.Lemma 5.
Ifv(t,x) is the solution of problem (11) on ϖT, η(t,x)∈C0∞(ϖT), it holds that(21)
∬
ϖ
T
⁡
v
-
k
η
t
+
sign
⁡
v
-
k
P
t
,
x
,
v
-
P
t
,
x
,
k
η
x
-
sign
⁡
v
-
k
P
x
t
,
x
,
k
+
Ψ
t
,
x
,
v
η
d
t
d
x
=
0
,where k is an arbitrary constant.Proof.
Suppose thatΦ(v) is a twice differential function. Multiplying the first equation of problem (8) by Φ′(v)η(t,x) and integrating over ϖT, we get(22)
∬
ϖ
T
⁡
Φ
′
v
η
v
t
+
Φ
′
v
η
P
x
+
Φ
′
v
η
P
v
v
x
+
Φ
′
v
η
Ψ
t
,
x
,
v
d
t
d
x
=
0
.Using the method of integration by parts, we get(23)∬ϖT⁡Φ′vηvtdtdx=-∬ϖT⁡Φvηtdtdx.Notice that (24)∫kvΦ′zPzt,x,zdzx′=Φ′vPvvvx+∫kvΦ′′zPzzx+Φ′zPzxdz.So (25)∫-∞+∞∫kvΦ′zPzdzηxdx=-∫-∞+∞Φ′vPvvxηdx-∫-∞+∞∫kvΦ′′zPzzx+Φ′zPzxdzηdx.Then we have(26)∬ϖT⁡Φ′vPvvxηdtdx=-∬ϖT⁡∫kvv′zPzdz·ηxdtdx+∬ϖT⁡∫kvΦ′′zPzzx+Φ′zPzxdzηdtdx.Substituting (23) and (26) into (22), we get(27)
∬
ϖ
T
⁡
Φ
′
v
η
t
+
∫
k
v
Φ
′
z
P
z
d
z
η
x
+
∫
k
v
Φ
′
′
z
P
z
z
x
d
z
η
+
∫
k
v
Φ
′
z
P
z
x
d
z
η
-
Φ
′
v
η
P
x
-
Φ
′
v
η
Ψ
d
t
d
x
=
0
.Let Φɛ(v) be an approximation of the function v-k. When ɛ→0, Φɛ(v)→Φ(v). Setting Φ(v)=v-k, then Φ′(v)=sign⁡(v-k), Φ′′(v)=0. Hence,(28)∫kvΦ′zPzdz=sign⁡v-kPt,x,v-Pt,x,k,(29)∫kvΦ′zPzxdz=sign⁡v-kPxt,x,v-Pxt,x,k,combining with (27), we complete the proof.


## 3. Main Result

Set functionη(t,x)∈C0∞(ϖT), η(t,x)≡0, outside the cylinder {(t,x)}=[ρ,T-2ρ]×Kr-2ρ, where Kr-2ρ={x:x≤r-2ρ}, r>0, 0<2ρ<min⁡(T,r). Now we give the main result of this work.Theorem 6.
Assume thatv1(t,x) and v2(t,x) are two strong solutions of problem (8) with initial data v10(x),v20(x)∈L1(R)∩Hs(R)(s>3/2). Let T>0 be the maximum existence time of v1(t,x) and v2(t,x). If v1(t,x)L∞≤M and v2(t,x)L∞≤M, for any t∈[0,T], it holds that(30)v1t,x-v2t,xL1≤ect∫-∞+∞v10x-v20xdx,where c is a positive constant depending on v10L∞(R) and v20L∞(R).Proof.
Setk=ψ(τ,y), η(t,x)=h(t,x,τ,y). Using the Kružkov device of doubling the variables in [12] and Lemma 5, we get(31)
⨌
ϖ
T
×
ϖ
T
⁡
v
1
t
,
x
-
v
2
τ
,
y
h
t
+
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
·
P
t
,
x
,
v
1
t
,
x
-
P
τ
,
y
,
v
2
τ
,
y
h
x
-
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
·
P
x
t
,
x
,
v
2
τ
,
y
+
Ψ
t
,
x
,
v
1
t
,
x
h
d
t
d
x
d
τ
d
y
=
0
.Similarly, we have(32)
⨌
ϖ
T
×
ϖ
T
⁡
v
2
τ
,
y
-
v
1
t
,
x
h
τ
+
sign
⁡
v
2
τ
,
y
-
v
1
t
,
x
·
P
τ
,
y
,
v
2
τ
,
y
-
P
t
,
x
,
v
1
t
,
x
h
y
-
sign
⁡
v
2
τ
,
y
-
v
1
t
,
x
·
P
y
τ
,
y
,
v
1
t
,
x
+
Ψ
τ
,
y
,
v
2
τ
,
y
h
d
t
d
x
d
τ
d
y
=
0
.Adding (31) and (32), we obtain(33)
⨌
ϖ
T
×
ϖ
T
⁡
v
1
t
,
x
-
v
2
τ
,
y
h
t
+
h
τ
+
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
P
t
,
x
,
v
1
t
,
x
-
P
t
,
x
,
v
2
τ
,
y
h
x
+
h
y
+
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
·
P
τ
,
y
,
v
2
τ
,
y
-
P
t
,
x
,
v
2
τ
,
y
h
x
+
P
τ
,
y
,
v
1
t
,
x
-
P
t
,
x
,
v
1
t
,
x
h
y
+
P
y
τ
,
y
,
v
1
t
,
x
-
P
x
t
,
x
,
v
2
τ
,
y
h
-
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
Ψ
t
,
x
,
v
1
t
,
x
-
Ψ
τ
,
y
,
v
2
τ
,
y
h
d
t
d
x
d
τ
d
y
=
⨌
ϖ
T
×
ϖ
T
⁡
A
1
+
A
2
+
A
3
+
A
4
d
t
d
x
d
τ
d
y
=
0
.
Set function(34)ht,x,τ,y=ηt+τ2,x+y2δɛt-τ2δɛx-y2=η⋯λɛ⋮,in which (⋯)=t+τ/2,x+y/2 and ⋮=(t-τ/2,x-y/2). Thus, we obtain(35)ht+hτ=ηt⋯λɛ⋮,hx+hy=ηx⋯λɛ⋮.
We will prove that the formA3 in (33) approaches zero as ɛ→0. In fact, the coefficients of ηx and ηy in A3 vanish for t-τ+x-y=0. Thus the integrals of A3 can be rewritten as the following concrete form:(36)
⨌
ϖ
T
×
ϖ
T
⁡
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
·
P
τ
,
y
,
v
2
τ
,
y
-
P
t
,
x
,
v
2
τ
,
y
λ
ɛ
x
-
P
x
t
,
x
,
v
2
τ
,
y
λ
ɛ
+
P
τ
,
y
,
v
1
t
,
x
-
P
t
,
x
,
v
1
t
,
x
λ
ɛ
y
+
P
y
τ
,
y
,
v
1
t
,
x
λ
ɛ
η
⋯
d
t
d
x
d
τ
d
y
=
A
ɛ
.In the following computations, we omit the index ɛ of function λ. Applying the Taylor formula, we have the relations(37)Pτ,y,v2τ,y-Pt,x,v2τ,yλx-Pxt,x,v2τ,yλ=Pτ,y,v2τ,y-Pt,x,v2τ,yλx=Pττ,y,v2τ,yτ-tλx+Pyτ,y,v2τ,yy-xλx-Pyτ,y,v2τ,yλ+α1λx+α0λ=Pττ,y,v2τ,yτ-tλx+Pyτ,y,v2τ,yy-xλx+α1λx+α0λ.It is seen that the identity λy=-λx. In a similar way, we obtain(38)Pτ,y,v1t,x-Pt,x,v1t,xλy+Pyτ,y,v1t,xλ=Pττ,y,v1t,xτ-tλy+Pyτ,y,v1t,xy-xλy+Pyτ,y,v1t,xλ+β1λy=Pττ,y,v1t,xt-τλx+Pyτ,y,v1t,xx-yλx+β1λy.The functions α0, α1, and β1 in (37) and (38) satisfy (39)α0+α1+β1≤dɛd,where d=|t-τ|+x-y and ɛ(d)→0 as d→0. There are λ=λɛ≡0 for |t-τ|≥2ɛ or |x-y|≥2ɛ and (40)λx+λy≤cɛ3,η⋯-ητ,y≤ct-τ+x-y.Hence, we get(41)
A
ɛ
=
⨌
ϖ
T
×
ϖ
T
⁡
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
·
P
τ
τ
,
y
,
v
2
τ
,
y
-
P
τ
t
,
x
,
v
2
τ
,
y
·
τ
-
t
λ
x
+
P
y
τ
,
y
,
v
2
t
,
x
-
P
y
τ
,
y
,
v
1
t
,
x
·
y
-
x
λ
x
η
τ
,
y
d
t
d
x
d
τ
d
y
+
I
ɛ
,where I(ɛ)→0 as ɛ→0.
We denote the integrand in (41) as(42)
B
ɛ
=
K
1
τ
,
y
,
v
1
t
,
x
,
v
2
τ
,
y
τ
-
t
η
τ
,
y
λ
x
+
K
2
τ
,
y
,
v
1
t
,
x
,
v
2
τ
,
y
y
-
x
η
τ
,
y
λ
x
=
K
1
τ
,
y
,
v
1
t
,
x
,
v
2
τ
,
y
-
K
1
τ
,
y
,
v
1
τ
,
y
,
v
2
τ
,
y
τ
-
t
η
τ
,
y
λ
x
+
K
2
τ
,
y
,
v
1
t
,
x
,
v
2
τ
,
y
-
K
2
τ
,
y
,
v
1
τ
,
y
,
v
2
τ
,
y
·
y
-
x
η
τ
,
y
λ
x
+
K
1
τ
,
y
,
v
1
τ
,
y
,
v
2
τ
,
y
τ
-
t
η
τ
,
y
λ
x
+
K
2
τ
,
y
,
v
1
τ
,
y
,
v
2
τ
,
y
y
-
x
η
τ
,
y
λ
x
,where K1 and K2 satisfy the Lipschitz condition in v. Applying the property of the function η(τ,y)λɛ and the method of integration by parts, we have (43)
⨌
ϖ
T
×
ϖ
T
⁡
K
1
τ
,
y
,
v
1
τ
,
y
,
v
2
τ
,
y
·
τ
-
t
η
τ
,
y
λ
x
+
K
2
τ
,
y
,
v
1
τ
,
y
,
v
2
τ
,
y
·
y
-
x
η
τ
,
y
λ
x
d
t
d
x
d
τ
d
y
=
0
.Hence (44)Aɛ-Iɛ=⨌ϖT×ϖT⁡Bɛdtdxdτdy≤c·⨌ϖT×ϖT⁡v1t,x-v2τ,y·τ-t+y-xητ,yλx+λɛdtdxdτdy≤cɛ2⨌Π⁡v1t,x-v2τ,ydtdxdτdy.Using Lemma 3, Aɛ-I(ɛ)→0 as ɛ→0. Therefore, we have(45)limɛ→0⁡⨌ϖT×ϖT⁡A3dtdxdτdy=0.It follows from (33) to (45) that(46)
⨌
ϖ
T
×
ϖ
T
⁡
v
1
t
,
x
-
v
2
τ
,
y
η
t
+
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
·
P
t
,
x
,
v
1
t
,
x
-
P
t
,
x
,
v
2
τ
,
y
η
x
·
λ
ɛ
d
t
d
x
d
τ
d
y
+
⨌
ϖ
T
×
ϖ
T
⁡
sign
⁡
v
1
t
,
x
-
v
2
τ
,
y
·
Ψ
t
,
x
,
v
1
t
,
x
-
Ψ
τ
,
y
,
v
2
τ
,
y
·
η
λ
ɛ
d
t
d
x
d
τ
d
y
≥
0
.
We note that the first two terms of the integrand of (46) have the form(47)Fɛ=Ft,x,τ,y,v1t,x,v2τ,yλɛ⋮,where F satisfies the Lipschitz condition in all its variables. Then(48)
⨌
ϖ
T
×
ϖ
T
⁡
F
ɛ
d
t
d
x
d
τ
d
y
=
⨌
ϖ
T
×
ϖ
T
⁡
F
t
,
x
,
τ
,
y
,
v
1
t
,
x
,
v
2
τ
,
y
·
λ
ɛ
d
t
d
x
d
τ
d
y
=
⨌
ϖ
T
×
ϖ
T
⁡
F
t
,
x
,
τ
,
y
,
v
1
t
,
x
,
v
2
τ
,
y
-
F
t
,
x
,
t
,
x
,
v
1
t
,
x
,
v
2
t
,
x
λ
ɛ
d
t
d
x
d
τ
d
y
+
⨌
ϖ
T
×
ϖ
T
⁡
F
t
,
x
,
t
,
x
,
v
1
t
,
x
,
v
2
t
,
x
·
λ
ɛ
d
t
d
x
d
τ
d
y
=
J
11
ɛ
+
J
12
.
Note thatFɛ=0 outside the region Π. Applying the estimate λɛ⋮≤c/ɛ2 and Lemma 4, we get(49)J11ɛ≤cɛ+1ɛ2⨌Π⁡v1t,x-v2τ,ydtdxdτdy,where c is a positive constant independent of ɛ. Using Lemma 3, we know J11(ɛ)→0 as ɛ→0.
For the termJ12, we substitute t=α, t-τ/2=β, x=ζ, x-y/2=γ. Combining with the identity (50)∫-ɛɛ∫-∞+∞λɛβ,γdβdγ=1,we obtain(51)J12=22∬ϖT×ϖT⁡Fα,ζ,α,ζ,ϕα,ζ,ψα,ζ·∫-ɛɛ∫-∞+∞λɛβ,γdβdγdαdζ=4·∬ϖT⁡Ft,x,t,x,v1t,x,v2t,xdtdx.Thus, we have(52)limɛ→0⁡⨌ϖT×ϖT⁡Fɛ=4∬ϖT⁡Ft,x,t,x,v1t,x,v2t,xdtdx.
Similarly, the integrand of the third term in (46) can be represented as(53)F¯ɛ=sign⁡v1t,x-v2τ,y·Ψt,x,v1t,x-Ψτ,y,v2τ,ygλɛ=F¯t,x,τ,y,v1t,x,v2τ,yλɛ⋮.Then(54)
⨌
ϖ
T
×
ϖ
T
⁡
F
¯
ɛ
d
t
d
x
d
τ
d
y
=
⨌
ϖ
T
×
ϖ
T
⁡
F
¯
t
,
x
,
τ
,
y
,
v
1
t
,
x
,
v
2
τ
,
y
-
F
¯
t
,
x
,
t
,
x
,
v
1
t
,
x
,
v
2
t
,
x
λ
ɛ
d
t
d
x
d
τ
d
y
+
⨌
ϖ
T
×
ϖ
T
⁡
F
¯
t
,
x
,
t
,
x
,
v
1
t
,
x
,
v
2
t
,
x
·
λ
ɛ
d
t
d
x
d
τ
d
y
=
J
21
ɛ
+
J
22
.It holds that (55)J21ɛ≤cɛ+1ɛ2⨌Π⁡v1t,x-v2τ,ydtdxdτdy.Using Lemma 3, it yields J21(ɛ)→0 as ɛ→0. Repeating the steps as before, we have (56)J22=4∬ϖT⁡F¯t,x,t,x,v1t,x,v2t,xdtdx.From (46) to (56), we get(57)
∬
ϖ
T
⁡
v
1
t
,
x
-
v
2
t
,
x
η
t
+
sign
⁡
v
1
t
,
x
-
v
2
t
,
x
·
P
t
,
x
,
v
1
t
,
x
-
P
τ
,
y
,
v
2
τ
,
y
η
x
d
t
d
x
+
∬
ϖ
T
⁡
sign
⁡
v
1
t
,
x
-
v
2
t
,
x
·
Ψ
t
,
x
,
v
1
-
Ψ
t
,
x
,
v
2
η
d
t
d
x
≥
0
.
We set(58)gt=∫-∞+∞v1t,x-v2t,xdx,μɛσ=∫-∞σδɛσdσ.Take two numbers ρ, τ∈(0,T1) and ρ<τ. In (57), we set(59)ηt,x=μɛt-ρ-μɛt-τχt,x,ɛ<min⁡ρ,T1-τ,in which(60)χt,x=χθt,x=1-μθx+Nt-R1+θ,where θ is a small positive constant and χ(t,x)=0 outside the cone Ω. When θ→0,R1→+∞, we observe that χθ→1. By the definition of the number N, we have (61)0=χt+Nχx≥χt+Nχx,t,x∈Ω.
Applying (57)–(60), we get(62)
∬
ϖ
T
1
⁡
v
1
t
,
x
-
v
2
t
,
x
δ
ɛ
t
-
ρ
-
δ
ɛ
t
-
τ
·
χ
θ
t
,
x
d
t
d
x
+
∫
0
T
1
d
t
·
∫
-
∞
+
∞
Ψ
t
,
x
,
v
1
-
Ψ
t
,
x
,
v
2
·
μ
ɛ
t
-
ρ
-
μ
ɛ
t
-
τ
χ
θ
t
,
x
d
x
≥
0
.In (62), sending θ→0,R1→+∞ and using Lemma 2, we obtain(63)∫0T1δɛt-ρ-δɛt-τhtdt+c∫0T1μɛt-ρ-μɛt-τgtdt≥0,where c is independent of ɛ.
Applying the properties of the functionδɛ for ɛ≤min⁡(ρ,T1-ρ), we get (64)∫0T1δɛt-ρgt-gρdt=∫0T1δɛt-ρgt-gρdt≤cɛ∫ρ-ɛρ+ɛgt-gρdt.Then(65)∫0T0δɛt-ρgtdt⟶gρasɛ⟶0.Let (66)Wρ=∫0T1μɛt-ρgtdt=∫0T1dt∫-∞t-ρδɛσgtdσ.We observe that (67)W′ρ=-∫0T1δɛt-ρgtdt.Letting ɛ→0, it derives that (68)W′ρ⟶-gρ,Wρ⟶W0-∫0ρgtdt,Wτ⟶W0-∫0τgtdt.Therefore, we have(69)Wρ-Wτ⟶∫ρτgtdtasɛ⟶0.
From (63)–(69), we obtain inequality (70)gρ+c∫ρτgtdt≥gτ.Choosing ρ→0,τ→t, we get (71)∫-∞+∞ϕ0,x-ψ0,xdx+c∫0tdt∫-∞+∞ϕt,x-ψt,xdx≥∫-∞+∞ϕt,x-ψt,xdx.Applying the Gronwall inequality, we complete the proof of Theorem 6.



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*Source: 207427-2015-10-12.xml*