
In the present work, we study discontinuous impulsive systems of the type of Cohen-Grossberg Neural Networks (CGNNs) with time-varying delays. The impulsive perturbations are realized not at fixed moments of time, and can be considered as control inputs. The hybrid concept of practical exponential stability with respect to specific manifolds defined by a function is introduced and studied analytically. The established results are applied to the case of Bidirectional Associative Memory (BAM) CGNNs. Lyapunov function method and the Razumikhin technique are the base of the proofs. A numerical example is also presented to demonstrate the applicability and effectiveness of the obtained stability conditions. The proposed results extend and complement some existing stability criteria for impulsive CGNNs with time-varying delays.
Citation: Gani Stamov, Ekaterina Gospodinova, Ivanka Stamova. Practical exponential stability with respect to h−manifolds of discontinuous delayed Cohen–Grossberg neural networks with variable impulsive perturbations[J]. Mathematical Modelling and Control, 2021, 1(1): 26-34. doi: 10.3934/mmc.2021003
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In the present work, we study discontinuous impulsive systems of the type of Cohen-Grossberg Neural Networks (CGNNs) with time-varying delays. The impulsive perturbations are realized not at fixed moments of time, and can be considered as control inputs. The hybrid concept of practical exponential stability with respect to specific manifolds defined by a function is introduced and studied analytically. The established results are applied to the case of Bidirectional Associative Memory (BAM) CGNNs. Lyapunov function method and the Razumikhin technique are the base of the proofs. A numerical example is also presented to demonstrate the applicability and effectiveness of the obtained stability conditions. The proposed results extend and complement some existing stability criteria for impulsive CGNNs with time-varying delays.
The Euler system
{∂∂x(ρu)+∂∂y(ρv)=0,∂∂x(p+ρu2)+∂∂y(ρuv)=0,∂∂x(ρuv)+∂∂y(p+ρv2)=0 | (1) |
is usually used to describe the two-dimensional steady isentropic inviscid compressible flow, where
∂u∂y=∂v∂x. | (2) |
Then the density
ρ(q2)=(1−γ−12q2)1/(γ−1),0<q<√2/(γ−1). | (3) |
The sound speed
div(ρ(|∇φ|2)∇φ)=0, | (4) |
where
Subsonic-sonic flow is one of the most interesting aspects in the mathematical theory of compressible flows. The related problems are usually raised in physical experiments and engineering designs, and there are a lot of numerical simulations and rigorous theory involved in this field (see, e.g., [2,8,15]). Two kinds of subsonic-sonic flows have been intensively studied for decades: the flow past a profile and the flow in a nozzle. The outstanding work [1] by L. Bers proved that there exists a unique two-dimensional subsonic potential flow past a profile provided that the freestream Mach number is less than a critical value and the maximum flow speed tends to the sound speed as the freestream Mach number tends to the critical value. Later, the similar results for multi-dimensional cases were established in [13,9] by G. Dong, R. Finn and D. Gilbarg. These three works did not cover the flow with the critical freestream Mach number. It was shown in [3] based on a compensated compactness framework that the two-dimensional flow with sonic points past a profile may be realized as the weak limit of a sequence of strictly subsonic flows. However, all the subsonic-sonic flows above are obtained in the weak sense and their smoothness and uniqueness are unknown yet, so are the subsonic-sonic flows in an infinitely long nozzle. For a two-dimensional infinitely long nozzle, C. Xie et al. ([22]) proved that there exists a critical value such that a strictly subsonic flow exists uniquely as long as the incoming mass flux is less than the critical value, and a subsonic-sonic flow exists as the weak limit of a sequence of strictly subsonic flows. The multi-dimensional cases were investigated in [24,12,14]. A typical subsonic-sonic flow with precise regularity is a radially symmetric subsonic-sonic flow in a convergent straight nozzle. The structural stability was initially proved in [20] for the case of two-dimensional finitely long nozzle, and some new results can be found in [16,17,18,21,19]. In the recent decade, there are also some studies on rotational subsonic and subsonic-sonic flows, see [4,6,11,7,5,23] and the references therein.
In the present paper, we would like to investigate the subsonic-sonic flow in a class of semi-infinitely long nozzles. Assume precisely that
f′(0)<f(0)=0,(−x)−1/2f″∈L∞((−l0,0]), | (5) |
f(x)>0 for x∈(−∞,0),f′(x)=0 for x∈(−∞,−l0]. | (6) |
The upper and lower wall of the nozzle are described as
Γup:y=fk(x)(x∈(−∞,0]),andΓlow:y=−l1(x∈R), |
respectively, where
fk(x)=kf(x),x∈(−∞,0]. |
The sonic curve of the flow is a free boundary intersecting the upper wall at the origin, which is chosen as the outlet of the nozzle and is denoted by
Γout:x=S(y),y∈[−l1,0],S(0)=0. |
It is assumed further that the subsonic-sonic flow satisfies the slip conditions at
As in [18,21], the subsonic-sonic flow problem can be formulated in the physical plane as
div(ρ(|∇φ|2)∇φ)=0,(x,y)∈Ωk, | (7) |
∂φ∂y(x,−l1)=0,x∈(−∞,S(−l1)), | (8) |
∂φ∂y(x,fk(x))−f′k(x)∂φ∂x(x,fk(x))=0,x∈(−∞,0), | (9) |
|∇φ(S(y),y)|=c∗,φ(S(y),y)=0,y∈(−l1,0), | (10) |
where
The paper is arranged as follows. In Section 2, we formulate the subsonic-sonic flow problem (7)–(10) in the potential plane. Then in Section 3, we solve the fixed boundary problem of a quasilinear degenerate elliptic equation in an unbounded domain. Finally in Section 4, we establish the well-posedness of the subsonic-sonic flow, and prove that the flow is uniformly subsonic at the far fields.
Define a velocity potential
∂φ∂x=u=qcosθ,∂φ∂y=v=qsinθ,∂ψ∂x=−ρv=−ρqsinθ,∂ψ∂y=ρu=ρqcosθ, | (11) |
where
∂θ∂ψ+ρ(q2)+2q2ρ′(q2)qρ2(q2)∂q∂φ=0,1q∂q∂ψ−1ρ(q2)∂θ∂φ=0 | (12) |
in the potential-stream coordinates
∂2A(q)∂φ2+∂2B(q)∂ψ2=0, |
where
A(q)=∫qc∗ρ(s2)+2s2ρ′(s2)sρ2(s2)ds,B(q)=∫qc∗ρ(s2)sds,0<q<√2/(γ−1). |
It is obvious that
N1(c∗−q)≤A′(q)≤N2(c∗−q),N1≤B′(q),−A″(q),−B″(q)≤N2, | (13) |
N1(c∗−q)≤E′(B(q))≤N2(c∗−q),−N2≤E″(B(q)),E‴(B(q))≤−N1, | (14) |
where
Θup(x)=arctanf′k(x),x∈[−l0,0]andΘlow(x)≡0,x∈(−∞,0), |
respectively.
As in [18,21], in order to describe the problem in the potential plane, we denote the flow speed at the upper wall by
Qup(x)=q(x,fk(x)),x∈(−∞,0], |
then the potential function at the upper wall is expressed by
Φup(x)=∫x0Qup(s)(1+(f′k(s))2)1/2ds={∫x0Qup(s)(1+(f′k(s))2)1/2ds,if x∈[−l0,0],ζ0+∫x−l0Qup(s)ds,if x∈(−∞,−l0) | (15) |
with
ζ0=∫−l00Qup(s)(1+(f′k(s))2)1/2ds. |
The inverse function of
∂2A(q)∂φ2(φ,ψ)+∂2B(q)∂ψ2(φ,ψ)=0,(φ,ψ)∈(−∞,0)×(0,m), | (16) |
∂q∂ψ(φ,0)=0,φ∈(−∞,0), | (17) |
∂B(q)∂ψ(φ,m)=f″k(x)(1+(f′k(x))2)3/2Qup(x)|x=Xup(φ),φ∈(−∞,0), | (18) |
q(0,ψ)=c∗,ψ∈(0,m), | (19) |
Qup(x)=q(φ,m)|φ=Φup(x),x∈(−∞,0], | (20) |
where
Definition 2.1. For
0<inf(−∞,0)×(0,m)q≤sup(−∞,0)×(0,m)q≤c∗ |
such that the integral equation
∫0−∞∫m0(A(q(φ,ψ))∂2ξ∂φ2(φ,ψ)+B(q(φ,ψ))∂2ξ∂ψ2(φ,ψ))dψdφ+∫0−∞f″k(x)(1+(f′k(x))2)3/2Qup(x)|x=Xup(φ)ξ(φ,m)dφ=0 |
holds for any
∂ξ∂ψ(⋅,0)|(−∞,0)=∂ξ∂ψ(⋅,m)|(−∞,0)=ξ(0,⋅)|(0,m)=0. |
The existence of solutions to the problem (16)–(20) will be proved by a fixed point argument. Give
δ1≤m≤δ2 | (21) |
with
δ1=c∗ρ(c2∗/4)l12,δ2=c∗ρ(c2∗)(l1+f(−l0)), |
while
max{c∗2,c∗−k1/4}≤Qup(x)≤c∗ for x∈(−∞,0],[Qup]C1/4((−∞,0])≤1. | (22) |
For such
−δ4≤ζ0≤−δ3,c∗2≤Φ′up(x)≤δ5,x∈(−∞,0], | (23) |
|f″k(x)(1+(f′k(x))2)3/2Qup(x)|x=Xup(φ)|≤kδ6(−φ)1/2χ[ζ0,0](φ),φ∈(−∞,0], | (24) |
where
δ3=c∗l02,δ4=c∗l0(1+‖f′‖2L∞((−l0,0)))1/2,δ5=c∗(1+‖f′‖2L∞((−l0,0)))1/2,δ6=‖(−x)1/2f″‖L∞((−l0,0))(2c∗)3/2. |
For
−τ1x≤f(x)≤−τ2x,x∈[−l0,0]. | (25) |
In this section, we deal with the well-posedness of the fixed boundary problem. For the given
The truncated problem is written as
∂2A(qn)∂φ2(φ,ψ)+∂2B(qn)∂ψ2(φ,ψ)=0,(φ,ψ)∈(ζ0−n,0)×(0,m), | (26) |
∂A(qn)∂φ(ζ0−n,ψ)=0,ψ∈(0,m), | (27) |
∂qn∂ψ(φ,0)=0,φ∈(ζ0−n,0), | (28) |
∂B(qn)∂ψ(φ,m)=f″k(x)(1+(f′k(x))2)3/2Qup(x)|x=Xup(φ),φ∈(ζ0−n,0), | (29) |
qn(0,ψ)=c∗,ψ∈(0,m). | (30) |
Note that (26) is degenerate at
qn(0,ψ)=c,ψ∈(0,m), | (31) |
where
The proof can be divided into four steps.
Step 1. Well-posedness of the problem (26)–(29), (31) for
Lemma 3.1. Assume that
c∗/6≤qn,c(φ,ψ)<c∗,(φ,ψ)∈[ζ0−n,0]×[0,m], | (32) |
qn,c(ζ0−n,ψ)≤c∗−k3/4,ψ∈[0,m]. | (33) |
Proof. The uniqueness result follows from Proposition 3.2 in [20]. Set
k1=min{(c∗6)4/3,(c∗48δ22δ34)2,(c∗96δ34)4,(A(c∗/4)−A(c∗/6)8δ22δ34)2,(A(c∗/3)−A(c∗/4)16δ34)4,(2δ1δ5/24B′(5c∗/6)δ6)2,(2δ2δ5/24B′(c∗/6)δ6A′(c∗/6))2,(1δ22e2δ4)4,(3A′(5c∗/6)4δ24e2δ4B′(5c∗/6))2}. |
For
¯qn,c(φ,ψ)=23c∗+(k1/2ψ2+k1/4(φ−2)eφ)Λ(φ),(φ,ψ)∈[ζ0−n,0]×[0,m],q_n,c(φ,ψ)=A−1(A(c∗/4)−(k1/2ψ2+k1/4(φ−2)eφ)Λ(φ)),(φ,ψ)∈[ζ0−n,0]×[0,m], |
where
Λ(φ)=max{0,(φ+2δ4)3},φ∈(−∞,0]. |
Thanks to (13), (14), (23) and (24), direct calculations show that
c∗2≤¯qn,c(φ,ψ)≤5c∗6,c∗6≤q_n,c(φ,ψ)≤c∗3,(φ,ψ)∈[ζ0−n,0]×[0,m],∂A(ˉqn,c)∂φ(ζ0−n,ψ)=∂A(q_n,c)∂φ(ζ0−n,ψ)=0,ψ∈(0,m),∂ˉqn,c∂ψ(φ,0)=∂q_n,c∂ψ(φ,0)=0,φ∈(ζ0−n,0), |
∂B(ˉqn,c)∂ψ(φ,m)=2k1/2mB′(¯qn,c(φ,m))Λ(φ)≥2k1/2δ2δ34B′(5c∗/6)χ[ζ0,0](φ)≥kδ6(−φ)1/2χ[ζ0,0](φ),φ∈(ζ0−n,0),∂B(ˉqn,c)∂ψ(φ,m)=−2k1/2mB′(q_n,c(φ,m))A′(q_n,c(φ,m))Λ(φ)≤−2k1/2δ2δ34B′(c∗/6)A′(c∗/6)χ[ζ0,0](φ)≤−kδ6(−φ)1/2χ[ζ0,0](φ),φ∈(ζ0−n,0), |
∂2A(ˉqn,c)∂φ2(φ,ψ)+∂2B(ˉqn,c)∂ψ2(φ,ψ)≤B′(¯qn,c(φ,ψ))(A′(¯qn,c(φ,ψ))B′(¯qn,c(φ,ψ))∂2ˉqn,c∂φ2(φ,ψ)+∂2ˉqn,c∂ψ2(φ,ψ))≤2k1/4B′(¯qn,c(φ,ψ))(φ+2δ4)×(A′(5c∗/6)B′(5c∗/6)(3k1/4δ22−6e−2δ4)+4k1/4δ24)χ[−2δ4,0](φ)≤2k1/4B′(¯qn,c(φ,ψ))(φ+2δ4)×(−3e−2δ4A′(5c∗/6)B′(5c∗/6)+4k1/4δ24)χ[−2δ4,0](φ)≤0,(φ,ψ)∈(ζ0−n,0)×(0,m), |
and
∂2A(q_n,c)∂φ2(φ,ψ)+∂2B(q_n,c)∂ψ2(φ,ψ)≥∂2A(q_n,c)∂φ2(φ,ψ)+B′(q_n,c(φ,ψ))A′(q_n,c(φ,ψ))∂2A(q_n,c)∂ψ2(φ,ψ)≥2k1/4(φ+2δ4)(6e−2δ4−3k1/4δ22−4k1/4δ24B′(c∗/3)A′(c∗/3))χ[−2δ4,0](φ)≥2k1/4(φ+2δ4)(3e−2δ4−4k1/4δ24B′(c∗/3)A′(c∗/3))χ[−2δ4,0](φ)≥0,(φ,ψ)∈(ζ0−n,0)×(0,m), |
where
Step 2. A priori estimates of the average of solutions to the problem (26)–(29), (31).
Lemma 3.2. Assume that
1m∫m0A(qn,c(φ,ψ))dψ=1m∫m0A(qn,c(ζ0,ψ))dψ,φ∈[ζ0−n,ζ0]. | (34) |
Furthermore, there exist three constants
A(c)−kσ2min{−φ,−ζ0}≤1m∫m0A(qn,c(φ,ψ))dψ≤A(c)−kσ1min{−φ,−ζ0},φ∈[ζ0−n,0]. | (35) |
Proof. The proof is similar to the proof of Lemma 3.2 in [21]. Integrating (26) over
d2dφ2∫m0A(qn,c(φ,ψ))dψ=−f″k(x)(1+(f′k(x))2)3/2Qup(x)|x=Xup(φ),φ∈(ζ0−n,0). | (36) |
And (27) yields that
ddφ∫m0A(qn,c(ζ0−n,ψ))dψ=0. | (37) |
One gets from (6), (36) and (37) that
ddφ∫m0A(qn,c(φ,ψ))dψ=0,φ∈[ζ0−n,ζ0], | (38) |
and
ddφ∫m0A(qn,c(φ,ψ))dψ=−∫φζ0f″k(x)(1+(f′k(x))2)3/2Qup(x)|x=Xup(s)ds=−∫Xup(φ)−l0f″k(x)Φ′up(x)(1+(f′k(x))2)3/2Qup(x)dx=−∫Xup(φ)−l0(arctanf′k(x))′dx=−arctanf′k(Xup(φ)),φ∈[ζ0,0]. | (39) |
Thus (34) follows from (38). As in the proof of Lemma 3.2 in [21], it follows from (15) and (39) that
1m∫m0A(qn,c(φ,ψ))dψ=1m∫m0A(qn,c(0,ψ))dψ+1m∫0φarctanf′k(Xup(˜φ))d˜φ=A(c)+1m∫0φarctanf′k(Xup(˜φ))d˜φ=A(c)−kc∗f(Xup(φ))+O(k5/4),φ∈[ζ0,0], | (40) |
where
Step 3. A priori derivative estimates of solutions to the problem (26)–(29), (31).
Lemma 3.3. Assume that
|∂qn,c∂ψ(φ,ψ)|≤kσ3(min{−φ,−ζ0})1/2,(φ,ψ)∈(ζ0−n,0)×(0,m), | (41) |
|A(qn,c(φ1,ψ1))−A(qn,c(φ2,ψ2))|≤kσ4(|φ1−φ2|1/2+|ψ1−ψ2|),(φ1,ψ1),(φ2,ψ2)∈[ζ0−n,0]×[0,m], | (42) |
where
Proof. The proof is similar to Proposition 3.2 in [20]. Set
z(φ,ψ)=∂B(qn,c)∂ψ(φ,ψ),(φ,ψ)∈[ζ0−n,0]×[0,m]. |
Then
j1(φ,ψ)∂2z∂φ2+∂2z∂ψ2+j2(φ,ψ)∂z∂φ+j3(φ,ψ)∂z∂ψ+j4(φ,ψ)z=0,(φ,ψ)∈(ζ0−n,0)×(0,m), | (43) |
∂z∂φ(ζ0−n,ψ)=0,ψ∈(0,m), | (44) |
z(φ,0)=0,φ∈(ζ0−n,0), | (45) |
z(φ,m)=f″k(x)(1+(f′k(x))2)3/2Qup(x)|x=Xup(φ),φ∈(ζ0−n,0), | (46) |
z(0,ψ)=0,ψ∈(0,m), | (47) |
where
j1=E′(B(qn,c))>0,j2=E″(B(qn,c))E′(B(qn,c))∂A(qn,c)∂φ,j3=−E″(B(qn,c))E′(B(qn,c))∂B(qn,c)∂ψ,j4=(E‴(B(qn,c))(E′(B(qn,c)))2−(E″(B(qn,c)))2(E′(B(qn,c)))3)(∂A(qn,c)∂φ)2≤−(E″(B(qn,c)))2(E′(B(qn,c)))3(∂A(qn,c)∂φ)2≤0 |
and
14j1(φ,ψ)(−φ)−3/2−j4(φ,ψ)(−φ)1/2≥√−j1(φ,ψ)j4(φ,ψ)(−φ)−1/2≥−12j2(φ,ψ)(−φ)−1/2,(φ,ψ)∈(ζ0−n,0)×(0,m). |
Due to (24), one can show that
z±(φ,ψ)=±kδ6(−φ)1/2,(φ,ψ)∈[ζ0−n,0]×[0,m] |
are a supersolution and a subsolution to the problem (43)–(47), respectively. The comparison principle (Proposition 3.2 in [20]) implies that
|z(φ,ψ)|≤kδ6(−φ)1/2,(φ,ψ)∈[ζ0−n,0]×[0,m]. | (48) |
Define
˜z±(φ,ψ)=±kδ6(−ζ0)1/2,(φ,ψ)∈[ζ0−n,ζ0]×[0,m]. |
It is easy to verify that
j1(φ,ψ)∂2z∂φ2+∂2z∂ψ2+j2(φ,ψ)∂z∂φ+j3(φ,ψ)∂z∂ψ+j4(φ,ψ)z=0,(φ,ψ)∈(ζ0−n,ζ0)×(0,m),∂z∂φ(ζ0−n,ψ)=0,ψ∈(0,m),z(φ,0)=0,φ∈(ζ0−n,ζ0),z(φ,m)=0,φ∈(ζ0−n,ζ0),z(ζ0,ψ)=z(ζ0,ψ),ψ∈(0,m), |
respectively. The comparison principle shows that
|z(φ,ψ)|≤kδ6(−ζ0)1/2,(φ,ψ)∈[ζ0−n,ζ0]×[0,m], |
which, together with (48), leads to (41). Finally, (42) can be proved in the same way as the proof of Proposition 3.2 in [20].
Step 4. Well-posedness of the truncated problem (26)–(30).
Lemma 3.4. Assume that
|∂qn∂ψ(φ,ψ)|≤kσ3(min{−φ,−ζ0})1/2,(φ,ψ)∈(ζ0−n,0)×(0,m), | (49) |
|A(qn(φ1,ψ1))−A(qn(φ2,ψ2))|≤kσ4(|φ1−φ2|1/2+|ψ1−ψ2|),(φ1,ψ1),(φ2,ψ2)∈[ζ0−n,0]×[0,m], | (50) |
c∗−σ6k1/2(min{−φ,−ζ0})1/2≤qn(φ,ψ)≤c∗−σ5k1/2(min{−φ,−ζ0})1/2,(φ,ψ)∈[ζ0−n,0]×[0,m], | (51) |
where
Proof. The uniqueness result follows from Proposition 3.2 in [20]. For
Ck={c∈[c∗/3,c∗):the problem (26)–(29), (31) admits a solutionqn,c∈C∞((ζ0−n,0)×(0,m))∩C1([ζ0−n,0]×[0,m])with (32) and (33)}. |
It follows from Lemma 3.1 and the comparison principle (Proposition 3.2 in [20]) that
qn,c(φ,ψφ)≤c∗−(kσ1N2)1/2(min{−φ,−ζ0})1/2, |
which, together with (41), yields
qn,c(φ,ψ)=qn,c(φ,ψφ)+∫ψψφ∂qn,c∂ψ(φ,˜ψ)d˜ψ≤c∗−((σ1N2)1/2−k1/2σ3δ2)k1/2(min{−φ,−ζ0})1/2,(φ,ψ)∈[ζ0−n,0]×[0,m]. | (52) |
Choose
σ5=(σ14N2)1/2,k3=min{k1,k2,σ14σ23δ22N2,σ45δ2416}. |
For
c∗/4≤qn,c(φ,ψ)≤c∗−σ5k1/2(min{−φ,−ζ0})1/2,(φ,ψ)∈[ζ0−n,0]×[0,m], | (53) |
qn,c(ζ0−n,ψ)≤c∗−2k3/4,ψ∈[0,m]. | (54) |
It follows from
|A(qn,c(φ,ψ))−A(c)|≤kσ4(−φ)1/2,ψ∈[0,m]. | (55) |
Thanks to (53)–(55), one can prove from the comparison principle (Proposition 3.2 in [20]) and the continuous dependence of solutions to the problem (26)–(29), (31) that
Let
qn,c1(φ,ψ)≤qn,c2(φ,ψ),(φ,ψ)∈[ζ0−n,0]×[0,m]. |
Set
qn(φ,ψ)=limc→c−∗qn,c(φ,ψ),(φ,ψ)∈[ζ0−n,0]×[0,m]. |
Due to (41), (42) and (53), it is clear that
qn(φ,˜ψφ)≥c∗−(kσ2N1)1/2(min{−φ,−ζ0})1/2. |
This estimate above and (49) yield
qn(φ,ψ)=qn(φ,˜ψφ)+∫ψ˜ψφ∂qn∂ψ(φ,˜ψ)d˜ψ≥c∗−((σ2N1)1/2+k1/2σ3δ2)k1/2(min{−φ,−ζ0})1/2,(φ,ψ)∈[ζ0−n,0]×[0,m]. |
Hence the first inequality in (51) holds for
Let us establish the existence of the solution to the problem (16)–(19).
Proposition 1. Assume that
|∂q∂ψ(φ,ψ)|≤kσ3(min{−φ,−ζ0})1/2,(φ,ψ)∈(−∞,0)×(0,m), | (56) |
|A(q(φ1,ψ1))−A(q(φ2,ψ2))|≤kσ4(|φ1−φ2|1/2+|ψ1−ψ2|),(φ1,ψ1),(φ2,ψ2)∈(−∞,0]×[0,m], | (57) |
\begin{gather} c_*-\sigma_6k^{1/2}(\min\{-\varphi,-\zeta_0\})^{1/2} \leq q(\varphi,\psi)\leq c_*-\sigma_5k^{1/2}(\min\{-\varphi,-\zeta_0\})^{1/2}, \\ \quad (\varphi,\psi)\in(-\infty,0]\times[0,m], \end{gather} | (58) |
where
\begin{align} \frac{1}{m}\int_0^m A(q(\varphi,\psi)){\rm d}\psi = A(q_\infty),\quad\varphi\in(-\infty,\zeta_0], \end{align} | (59) |
where
\begin{align} q_\infty = A^{-1}\bigg(\frac{1}{m}\int_0^m A(q(\zeta_0,\psi)){\rm d}\psi\bigg) \in[c_*-\sigma_6k^{1/2}(-\zeta_0)^{1/2}, c_*-\sigma_5k^{1/2}(-\zeta_0)^{1/2}]. \end{align} | (60) |
Proof. For any
q_n\in C^\infty((\zeta_0-n,0)\times(0,m)) \cap C^1([\zeta_0-n,0)\times[0,m]) \cap C^{1/2}([\zeta_0-n,0]\times[0,m]) |
satisfying (49)–(51). Therefore, there exists a subsequence of
q\in C^\infty((-\infty,0)\times(0,m))\cap C^1((-\infty,0)\times[0,m])\cap C((-\infty,0]\times[0,m]). |
Integrating (16) over
\frac{{\rm d}^2}{{\rm d}\varphi^2}\int_0^m A(q(\varphi,\psi)){\rm d}\psi = 0, \quad\varphi\in(-\infty,\zeta_0), |
and then there exists some constant
\begin{align} \frac{{\rm d}}{{\rm d}\varphi}\int_0^m A(q(\varphi,\psi)){\rm d}\psi = C, \quad\varphi\in(-\infty,\zeta_0), \end{align} | (61) |
which implies that
\begin{align} \int_0^m A(q(\varphi,\psi)){\rm d}\psi = \int_0^m A(q(\zeta_0,\psi)){\rm d}\psi+C(\varphi-\zeta_0), \quad\varphi\in(-\infty,\zeta_0). \end{align} | (62) |
It follows from (57) and (62) that
\begin{align*} |C||\varphi-\zeta_0| &\leq \int_0^m|A(q(\varphi,\psi))-A(q(\zeta_0,\psi))| {\rm d}\psi \\ &\leq k\sigma_4\delta_2|\varphi-\zeta_0|^{1/2}, \quad\varphi\in(-\infty,\zeta_0), \end{align*} |
that is,
\begin{align} |C|\leq k\sigma_4\delta_2|\varphi-\zeta_0|^{-1/2}, \quad\varphi\in(-\infty,\zeta_0). \end{align} | (63) |
One can get
\frac{1}{m}\int_0^m A(q(\varphi,\psi)){\rm d}\psi = \frac{1}{m}\int_0^m A(q(\zeta_0,\psi)){\rm d}\psi,\quad\varphi\in(-\infty,\zeta_0]. |
Therefore, (59) holds.
The solution to the problem (16)–(19) has the following regularity and asymptotic behavior.
Proposition 2. Assume that
\begin{align} \Big|\frac{\partial q}{\partial \varphi}(\varphi,\psi)\Big| \leq\sigma_7k^{1/4}(-\varphi)^{-1/2},\quad (\varphi,\psi)\in[2\zeta_0,0)\times(0,m), \end{align} | (64) |
where
\begin{align} \begin{split} \Big|\frac{\partial q}{\partial \varphi}(\varphi,\psi)\Big| \leq\sigma_8k^{1/2}(-\varphi)^{-2},\quad &\Big|\frac{\partial q}{\partial \psi}(\varphi,\psi)\Big| \leq\sigma_8k(-\varphi)^{-2}, \\ &(\varphi,\psi)\in(-\infty,2\zeta_0)\times(0,m), \end{split} \end{align} | (65) |
and hence
\begin{align} \|q(\varphi,\psi)-q_\infty\|_ {L^\infty((-\infty,\zeta)\times(0,m))} \leq\sigma_9k(-\zeta)^{-2}, \quad\zeta\in(-\infty,2\zeta_0), \end{align} | (66) |
where
Proof. Similarly to the proof of Proposition 4.1 in [18], one can prove that
In the remaining of the proof, we use
q(\varphi,\psi_\varphi) = q_\infty, |
which, together with (56), yields
\begin{align} \|q(\varphi,\psi)-q_\infty\|_{L^\infty((-\infty,\zeta_0)\times(0,m))} \leq\int_0^m\Big\|\frac{\partial q}{\partial \psi}\Big\| _{L^\infty((-\infty,\zeta_0)\times(0,m))}{\rm d}\psi \leq\mu_1k. \end{align} | (67) |
Note that
\begin{align*} &\frac{\partial}{\partial \varphi}\Big(a(\varphi,\psi)\frac{\partial q}{\partial \varphi}\Big) +\frac{\partial}{\partial \psi}\Big(b(\varphi,\psi)\frac{\partial q}{\partial \psi}\Big) = 0, &&(\varphi,\psi)\in(-\infty,\zeta_0)\times(0,m), \\ &\frac{\partial q}{\partial \psi}(\varphi,0) = 0, &&\varphi\in(-\infty,\zeta_0), \\ &\frac{\partial q}{\partial \psi}(\varphi,m) = 0, &&\varphi\in(-\infty,\zeta_0), \end{align*} |
where
a(\varphi,\psi) = A'(q(\varphi,\psi)),\quad b(\varphi,\psi) = B'(q(\varphi,\psi)),\quad (\varphi,\psi)\in(-\infty,\zeta_0)\times(0,m). |
Fix integer
\left\{\begin{array}{ll} \hat{\varphi} = k^{-1/4}(\varphi-n\zeta_0)/n, &\quad\varphi\in[4n\zeta_0,n\zeta_0/2], \\ \hat{\psi} = \psi/n, &\quad\psi\in[0,m], \end{array}\right. |
and setting
\hat{q}(\hat{\varphi},\hat{\psi}) = q(n\zeta_0+k^{1/4}n\hat{\varphi},n\hat{\psi})-q_\infty, \quad(\hat{\varphi},\hat{\psi})\in [3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2]\times[0,m/n]. |
One can verify that
\hat{q}\in C^\infty((3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2)\times(0,m/n))\cap C^1([3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2]\times[0,m/n]) |
solves
\begin{align} &\frac{\partial}{\partial \hat{\varphi}} \Big(k^{-1/2}\hat{a}(\hat{\varphi},\hat{\psi}) \frac{\partial \hat{q}}{\partial \hat{\varphi}}\Big) +\frac{\partial}{\partial \hat{\psi}} \Big(\hat{b}(\hat{\varphi},\hat{\psi}) \frac{\partial \hat{q}}{\partial \hat{\psi}}\Big) = 0, && \\ & &&(\hat{\varphi},\hat{\psi}) \in(3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2)\times(0,m/n), \end{align} | (68) |
\begin{align} &\frac{\partial \hat{q}}{\partial \hat{\psi}}(\hat{\varphi},0) = 0, &&\hat{\varphi}\in (3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2), \end{align} | (69) |
\begin{align} &\frac{\partial \hat{q}}{\partial \hat{\psi}}(\hat{\varphi},m/n) = 0, &&\hat{\varphi} \in(3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2), \end{align} | (70) |
where
\begin{align*} \hat{a}(\hat{\varphi},\hat{\psi}) = a(n\zeta_0+k^{1/4}n\hat{\varphi},n\hat{\psi}),\quad &\hat{b}(\hat{\varphi},\hat{\psi}) = b(n\zeta_0+k^{1/4}n\hat{\varphi},n\hat{\psi}), \\ &(\hat{\varphi},\hat{\psi})\in [3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2]\times[0,m/n]. \end{align*} |
Extending the problem (68)–(70) into the domain
\begin{align*} &\frac{\partial}{\partial \check{\varphi}} \Big(k^{-1/2}\check{a}(\check{\varphi},\check{\psi}) \frac{\partial \check{q}}{\partial \check{\varphi}}\Big) +\frac{\partial}{\partial \check{\psi}} \Big(\check{b}(\check{\varphi},\check{\psi}) \frac{\partial \check{q}}{\partial \check{\psi}}\Big) = 0, && \\ & &&(\check{\varphi},\check{\psi}) \in(3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2)\times(0,2m), \\ &\frac{\partial \check{q}}{\partial \check{\psi}}(\check{\varphi},0) = 0, &&\check{\varphi}\in(3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2), \\ &\frac{\partial \check{q}}{\partial \check{\psi}}(\check{\varphi},2m) = 0, &&\check{\varphi}\in(3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2), \end{align*} |
where for
\begin{align*} \check{a}(\check{\varphi},\check{\psi}) & = \left\{\begin{array}{ll} \hat{a}(\check{\varphi},\check{\psi}-(i-1)m/n), &\quad\hbox{if $i$ is odd}, \\ \hat{a}(\check{\varphi},im/n-\check{\psi}), &\quad\hbox{if $i$ is even}, \end{array}\right. \\ \check{b}(\check{\varphi},\check{\psi}) & = \left\{\begin{array}{ll} \hat{b}(\check{\varphi},\check{\psi}-(i-1)m/n), &\quad\hbox{if $i$ is odd}, \\ \hat{b}(\check{\varphi},im/n-\check{\psi}), &\quad\hbox{if $i$ is even}. \end{array}\right. \end{align*} |
Duo to (13), (51) and (67), one gets that
\begin{align*} \mu_2k^{1/2}\leq\check{a}(\check{\varphi},\check\psi)\leq\mu_3k^{1/2},\quad &\mu_2\leq\check{b}(\check{\varphi},\check\psi)\leq\mu_3,\quad \\ &(\check{\varphi},\check{\psi})\in [-4k^{-1/4},3k^{-1/4}\varepsilon/(4n)]\times[0,2m], \end{align*} |
and
\|\check{q}\|_{L^\infty ((3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2) \times(0,2m))}\leq\mu_1k. |
It follows from the Hölder continuity estimates for uniformly elliptic equations that there exists a number
[\check{q}]_{\beta;(5k^{-1/4}\zeta_0/2,-k^{-1/4}\zeta_0/4) \times(0,2m)} \leq\mu_4\|\check{q}\|_{L^\infty ((3k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/2) \times(0,2m))} \leq\mu_5k, |
which implies
\begin{align*} [\check{a}]_{\beta;(5k^{-1/4}\zeta_0/2,-k^{-1/4}\zeta_0/4)\times(0,2m)} &\leq\mu_6k, \\ [\check{b}]_{\beta;(5k^{-1/4}\zeta_0/2,-k^{-1/4}\zeta_0/4)\times(0,2m)} &\leq\mu_6k. \end{align*} |
The Schauder estimates on uniformly elliptic equations imply that
\begin{align} \|\check{q}\|_{C^{1,\beta}((2k^{-1/4}\zeta_0,-k^{-1/4}\zeta_0/8) \times(0,2m))} &\leq\mu_7\|\check{q}\|_{L^\infty ((5k^{-1/4}\zeta_0/2,-k^{-1/4}\zeta_0/4)\times(0,2m))} \\ &\leq\mu_8k. \end{align} | (71) |
Transforming (71) into the
\begin{align} \begin{split} \Big\|\frac{\partial q}{\partial \varphi}\Big\|_{L^\infty((3n\zeta_0,3n\zeta_0/4)\times(0,m))} &\leq\mu_9k^{3/4}n^{-1}, \\ \Big\|\frac{\partial q}{\partial \psi}\Big\|_{L^\infty((3n\zeta_0,3n\zeta_0/4)\times(0,m))} &\leq\mu_9kn^{-1}. \end{split} \end{align} | (72) |
Similar to (67), we have from (72) that
\begin{align} \|q(\varphi,\psi)-q_\infty\|_{L^\infty((3n\zeta_0,3n\zeta_0/4)\times(0,m))} &\leq\int_0^m\Big\|\frac{\partial q_{n}}{\partial \psi}\Big\| _{L^\infty((3n\zeta_0,3n\zeta_0/4)\times(0,m))}{\rm d}\psi \\ &\leq\mu_{10}kn^{-1}. \end{align} | (73) |
Using (73) and the same operation on
\begin{align*} \begin{split} \Big\|\frac{\partial q}{\partial \varphi}\Big\|_{L^\infty((2n\zeta_0,n\zeta_0)\times(0,m))} &\leq\mu_{11}k^{1/2}n^{-2}, \\ \Big\|\frac{\partial q}{\partial \psi}\Big\|_{L^\infty((2n\zeta_0,n\zeta_0)\times(0,m))} &\leq\mu_{11}kn^{-2}, \end{split} \end{align*} |
Then the arbitrariness of
Remark 1. Through the similar process of the proof of Proposition 2, one can show that for any positive integer
\begin{align*} \Big|\frac{\partial q}{\partial \varphi}(\varphi,\psi)\Big| \leq\sigma_8'k^{1-\lambda/4}(-\varphi)^{-\lambda},\quad &\Big|\frac{\partial q}{\partial \psi}(\varphi,\psi)\Big| \leq\sigma_8'k(-\varphi)^{-\lambda}, \\ &(\varphi,\psi)\in(-\infty,2\zeta_0)\times(0,m) \end{align*} |
and
\|q(\varphi,\psi)-q_\infty\|_ {L^\infty((-\infty,\zeta)\times(0,m))} \leq\sigma_9'k(-\zeta)^{-\lambda}, \quad\zeta\in(-\infty,2\zeta_0), |
where
The solution to the problem (16)–(19) is also unique for small
Proposition 3. There exists a constant
Proof. In the proof, we use
w_i(\varphi,\psi) = A(q^{(i)}(\varphi,\psi)),\quad (\varphi,\psi)\in(-\infty,0]\times[0,m],\quad i = 1,\,2. |
Then
\begin{align*} &\frac{\partial^{2} w_{i}}{\partial \varphi^{2}} +\frac{\partial^{2} B\left(A^{-1}\left(w_{i}\right)\right)}{\partial \psi^{2}} = 0, &&(\varphi,\psi)\in(-\infty,0)\times(0,m), \\ &\frac{\partial w_{i}}{\partial \psi}(\varphi,0) = 0, &&\varphi\in(-\infty,0), \\ &\frac{\partial B\left(A^{-1}\left(w_{i}\right)\right)}{\partial \psi}(\varphi,m) = \frac{f''_k(x)}{(1+(f'_k(x))^2)^{3/2}Q_{\rm up}(x)}\Big|_{x = X_{\rm up}(\varphi)}, && \\ &&&\varphi\in(-\infty,0), \\ &w_i(0,\psi) = 0, &&\psi\in(0,m). \end{align*} |
Set
w(\varphi,\psi) = w_1(\varphi,\psi)-w_2(\varphi,\psi),\quad (\varphi,\psi)\in(-\infty,0]\times[0,m]. |
It is easy to show that
\begin{align} &\frac{\partial^{2} w}{\partial \varphi^{2}} +\frac{\partial^{2}}{\partial \psi^{2}}(h(\varphi,\psi)w) = 0, &&(\varphi,\psi)\in(-\infty,0)\times(0,m), \end{align} | (74) |
\begin{align} &\frac{\partial w}{\partial \psi}(\varphi,0) = 0, &&\varphi\in(-\infty,0), \end{align} | (75) |
\begin{align} &\frac{\partial(h w)}{\partial \psi}(\varphi,m) = 0, &&\varphi\in(-\infty,0), \end{align} | (76) |
\begin{align} &w(0,\psi) = 0, &&\psi\in(0,m), \end{align} | (77) |
where
\begin{align*} h(\varphi,\psi) & = \int_0^1 \frac{B'(A^{-1}(\eta w_1(\varphi,\psi) +(1-\eta)w_2((\varphi,\psi))))} {A'(A^{-1}(\eta w_1(\varphi,\psi) +(1-\eta)w_2((\varphi,\psi))))}{\rm d}\eta, \\ & \quad (\varphi,\psi)\in(-\infty,0)\times(0,m). \end{align*} |
Thanks to (56), (58), (64) and (65), direct calculations yield
\begin{gather} \nu_1k^{1/2}\langle-\varphi\rangle^{1/2} \leq h(\varphi,\psi)\leq \nu_1k^{1/2}\langle-\varphi\rangle^{1/2}, \quad(\varphi,\psi)\in(-\infty,0)\times(0,m), \end{gather} | (78) |
\begin{gather} \Big|\frac{\partial h}{\partial \psi}(\varphi,\psi)\Big| \leq\left\{\begin{array}{ll} \nu_2(-\varphi)^{-1/2}, \quad(\varphi,\psi)\in[2\zeta_0,0)\times(0,m), \\ \nu_2(-\varphi)^{-2}, \quad(\varphi,\psi)\in(-\infty,2\zeta_0)\times(0,m), \end{array}\right. \end{gather} | (79) |
\begin{gather} \Big|\frac{\partial w}{\partial \varphi}(\varphi,\psi)\Big| \leq\nu_2k(-\varphi)^{-2}, \quad(\varphi,\psi)\in(-\infty,2\zeta_0)\times(0,m), \end{gather} | (80) |
where
\begin{align*} &\int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +\int_\zeta^0\int_0^mh(\varphi,\psi)\Big(\frac{\partial w}{\partial \psi}\Big)^2 {\rm d}\psi{\rm d}\varphi \\ = \,&-\int_\zeta^0\int_0^m\frac{\partial h}{\partial \psi}(\varphi,\psi) w\frac{\partial w}{\partial \psi}{\rm d}\psi{\rm d}\varphi -\int_0^mw(\zeta,\psi)\frac{\partial w}{\partial \varphi}(\zeta,\psi) {\rm d}\psi, \end{align*} |
which, together with (78)–(80), yields
\begin{align*} &\int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +k^{-1/2}\int_\zeta^0\int_0^m \langle-\varphi\rangle^{1/2}\Big(\frac{\partial w}{\partial \psi}\Big)^2 {\rm d}\psi{\rm d}\varphi \\ \leq\,&\nu_3\int_{2\zeta_0}^0\int_0^m(-\varphi)^{-1/2} \Big|w\frac{\partial w}{\partial \psi}\Big|{\rm d}\psi{\rm d}\varphi +\nu_3\int_\zeta^{2\zeta_0}\int_0^m(-\varphi)^{-2} \Big|w\frac{\partial w}{\partial \psi}\Big|{\rm d}\psi{\rm d}\varphi \\ &\qquad+\nu_3k(-\zeta)^{-2} \int_0^m|w(\zeta,\psi)|{\rm d}\psi. \end{align*} |
Then the Hölder's inequality gives
\begin{align} &\int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +k^{-1/2}\int_\zeta^0\int_0^m \langle-\varphi\rangle^{1/2}\Big(\frac{\partial w}{\partial \psi}\Big)^2 {\rm d}\psi{\rm d}\varphi \\ \leq\,&\nu_4k^{1/2}\int_{2\zeta_0}^0\int_0^m (-\varphi)^{-1/2}w^2{\rm d}\psi{\rm d}\varphi +\nu_4k^{1/2}\int_\zeta^{2\zeta_0}\int_0^m (-\varphi)^{-4}w^2{\rm d}\psi{\rm d}\varphi \\ &\qquad+\nu_4k(-\zeta)^{-2} \int_0^m|w(\zeta,\psi)|{\rm d}\psi. \end{align} | (81) |
It follows from the Hölder's inequality and Cauchy inequality that
\begin{align} \int_{2\zeta_0}^0\int_0^m (-\varphi)^{-1/2}w^2{\rm d}\psi{\rm d}\varphi &\leq\int_{2\zeta_0}^0\int_0^m(-\varphi)^{-1/2} \bigg(\int_\varphi^0\frac{\partial w}{\partial \varphi}(s,\psi) {\rm d}s\bigg)^2{\rm d}\psi{\rm d}\varphi \\ &\leq\int_{2\zeta_0}^0(-\varphi)^{1/2} {\rm d}\varphi \int_{\zeta_0}^0\int_0^m \Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\varphi{\rm d}\psi \\ &\leq(-2\zeta_0)^{3/2}\int_\zeta^0\int_0^m \Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\varphi{\rm d}\psi, \end{align} | (82) |
\begin{align} \int_\zeta^{2\zeta_0}\int_0^m (-\varphi)^{-4}w^2{\rm d}\psi{\rm d}\varphi &\leq\int_\zeta^{2\zeta_0}\int_0^m(-\varphi)^{-4} \bigg(\int_\varphi^0\frac{\partial w}{\partial \varphi}(s,\psi) {\rm d}s\bigg)^2{\rm d}\psi{\rm d}\varphi \\ &\leq\int_\zeta^{2\zeta_0}(-\varphi)^{-3} {\rm d}\varphi \int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi \\ &\leq(-2\zeta_0)^{-2} \int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi, \end{align} | (83) |
and
\begin{align} \int_0^m|w(\zeta,\psi)|{\rm d}\psi &\leq\dfrac{m}{2}+\dfrac{1}{2} \int_0^mw^2(\zeta,\psi){\rm d}\psi \\ &\leq\dfrac{\delta_2}{2}+\int_0^m \bigg(\int_\zeta^0\Big|\frac{\partial w}{\partial \varphi}\Big| {\rm d}\varphi\bigg)^2{\rm d}\psi \\ &\leq\dfrac{\delta_2}{2}+(-\zeta) \int_\zeta^0\int_0^m \Big(\frac{\partial w}{\partial \varphi}\Big)^2{\rm d}\varphi{\rm d}\psi. \end{align} | (84) |
Substituting (82)–(84) into (81) to get
\begin{align} &\int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +k^{-1/2}\int_\zeta^0\int_0^m \langle-\varphi\rangle^{1/2}\Big(\frac{\partial w}{\partial \psi}\Big)^2 {\rm d}\psi{\rm d}\varphi \\ \leq\,&\nu_5k^{1/2}\int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +\nu_5k(-\zeta)^{-2} +\nu_5k(-\zeta)^{-1}\int_\zeta^0\int_0^m \Big(\frac{\partial w}{\partial \varphi}\Big)^2{\rm d}\varphi{\rm d}\psi \\ \leq\,&2\nu_5k^{1/2}\int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +\nu_5k(-\zeta)^{-2}. \end{align} | (85) |
Choose
\begin{align} \int_\zeta^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +k^{-1/2}\int_\zeta^0\int_0^m \langle-\varphi\rangle^{1/2}\Big(\frac{\partial w}{\partial \psi}\Big)^2 {\rm d}\psi{\rm d}\varphi \leq2\nu_5k^{1/2}(-\zeta)^{-2}. \end{align} | (86) |
Taking
\int_{-\infty}^0\int_0^m\Big(\frac{\partial w}{\partial \varphi}\Big)^2 {\rm d}\psi{\rm d}\varphi +k^{-1/2}\int_{-\infty}^0\int_0^m \langle-\varphi\rangle^{1/2}\Big(\frac{\partial w}{\partial \psi}\Big)^2 {\rm d}\psi{\rm d}\varphi \leq0, |
which implies
\begin{align} \frac{\partial w}{\partial \varphi}(\varphi,\psi) = \frac{\partial w}{\partial \psi}(\varphi,\psi) = 0, \quad(\varphi,\psi)\in(-\infty,0)\times(0,m). \end{align} | (87) |
It follows (77) and (87) that
w(\varphi,\psi) = 0,\quad (\varphi,\psi)\in(-\infty,0]\times[0,m]. |
Therefore,
First we prove the existence of the solution to the problem (16)–(20) by a fixed point argument.
Theorem 4.1. Assume that
\begin{gather} q\in C^\infty((-\infty,0)\times(0,m)) \cap C^1((-\infty,0)\times[0,m]) \cap C^{1/2}((-\infty,0]\times[0,m]) \\ \Big|\frac{\partial q}{\partial \psi}(\varphi,\psi)\Big| \leq k\sigma_3(\min\{-\varphi,-\zeta_0\})^{1/2}, \quad(\varphi,\psi)\in(-\infty,0)\times(0,m), \end{gather} | (88) |
\begin{gather} |A(q(\varphi_1,\psi_1))-A(q(\varphi_2,\psi_2))| \leq k\sigma_4(|\varphi_1-\varphi_2|^{1/2}+|\psi_1-\psi_2|), \\ (\varphi_1,\psi_1),\, (\varphi_2,\psi_2)\in(-\infty,0]\times[0,m], \end{gather} | (89) |
\begin{gather} c_*-\sigma_6k^{1/2}(\min\{-\varphi,-\zeta_0\})^{1/2} \leq q(\varphi,\psi)\leq c_*-\sigma_5k^{1/2}(\min\{-\varphi,-\zeta_0\})^{1/2}, \\ (\varphi,\psi)\in(-\infty,0]\times[0,m], \end{gather} | (90) |
where
\begin{align} m = q_\infty\rho(q_\infty^2)(f_k(-l_0)+l_1),\quad c_*-\sigma_6k^{1/2}(-\zeta_0)^{1/2}\leq q_\infty \leq c_*-\sigma_5k^{1/2}(-\zeta_0)^{1/2}, \end{align} | (91) |
and
\begin{align} \Big|\frac{\partial q}{\partial \varphi}(\varphi,\psi)\Big| \leq\sigma_7k^{1/4}(-\varphi)^{-1/2},\quad (\varphi,\psi)\in[2\zeta_0,0)\times(0,m), \end{align} | (92) |
and for any positive integer
\begin{align} \begin{split} \Big|\frac{\partial q}{\partial \varphi}(\varphi,\psi)\Big| \leq\sigma_8'k^{1-\lambda/4}(-\varphi)^{-\lambda},\quad &\Big|\frac{\partial q}{\partial \psi}(\varphi,\psi)\Big| \leq\sigma_8'k(-\varphi)^{-\lambda}, \\ &(\varphi,\psi)\in(-\infty,2\zeta_0)\times(0,m), \end{split} \end{align} | (93) |
and
\begin{align} \|q(\varphi,\psi)-q_\infty\|_ {L^\infty((-\infty,\zeta)\times(0,m))} \leq\sigma_9'k(-\zeta)^{-\lambda}, \quad\zeta\in(-\infty,2\zeta_0), \end{align} | (94) |
where
Proof. Choose
\begin{align*} k_0 = \min\bigg\{k_3,\,k_4,\, \frac{c_*^2}{4\sigma_6^2\delta_4},\, \frac{1}{\sigma_6^4\delta_4^2},\, \frac{N_1}{2\sigma_4\delta_5^{1/2}}\bigg\}. \end{align*} |
For
\mathscr{Q} = \left\{(m,Q_{\rm up})\in [\delta_1,\delta_2]\times C^{1/4}((-\infty,0]):\, \hbox{$Q_{\rm up}$ satisfies $(22)$}\right\} |
with the norm
\|(m,Q_{\rm up})\|_{\mathscr{Q}} = \max\left\{m,\, \|Q_{\rm up}\|_{L^\infty(-\infty,0)}\right\}. |
For a given
\hat{m} = q_\infty\rho(q_\infty^2)(f_k(-l_0)+l_1),\quad \widehat{Q}_{\rm up}(x) = q(\Phi_{\rm up}(x),m), \quad x\in(-\infty,0]. |
From (56)–(58), (66) and the choice of
\mathcal{K}:\,\mathscr{Q}\to\mathscr{Q},\quad (m,Q_{\rm up}) \mapsto(\hat{m},\widehat{Q}_{\rm up}). |
is a self-mapping. Furthermore, one can prove the compactness of
From Theorem 4.1, for
\begin{align} \begin{split} &\max\left\{\dfrac{c_*}{2},\, c_*-M_1k^{1/2}(\min\{-\varphi,-\zeta_0\})^{1/2}\right\} \\ \le\,&q(\varphi,\psi)\le c_*-M_2k^{1/2}(\min\{-\varphi,-\zeta_0\})^{1/2}, \\ & \quad (\varphi,\psi)\in(-\infty,0)\times(0,m) \end{split} \end{align} | (95) |
and
\|q(\varphi,\psi)-q_\infty\|_{L^\infty((-\infty,\zeta)\times(0,m)} \leq M_3k(-\zeta)^{-2}, \quad\zeta < 2\zeta_0, |
where
\begin{gather*} m = q_\infty\rho(q_\infty^2)(f_k(-l_0)+l_1), \\ \max\left\{\dfrac{c_*}{2},\,c_*-M_1k^{1/2}(-\zeta_0)^{1/2}\right\} \le q_\infty\le c_*-M_2k^{1/2}(-\zeta_0)^{1/2}, \end{gather*} |
and
Theorem 4.2. Assume that
Proof. In the proof, we use
\left\{\begin{array}{ll} x = X_{{\rm up},i}(\varphi),&\varphi\in(-\infty,0], \\ y = \dfrac{\psi}{m^{(i)}},&\psi\in[0,m^{(i)}], \end{array}\right.\qquad \left\{\begin{array}{ll} \varphi = \Phi_{{\rm up},i}(x),&x\in(-\infty,0], \\ \psi = m^{(i)}y,&y\in[0,1]. \end{array}\right. |
Define
W_i(x,y) = A(q^{(i)}(\Phi_{{\rm up},i}(x),m^{(i)}y)),\quad (x,y)\in(-\infty,0]\times[0,1],\quad i = 1,\,2. |
Then
\begin{align} &\frac{\partial}{\partial x}\Big(m^{(i)}X_i(x)\frac{\partial W_{i}}{\partial x}\Big) +\frac{\partial}{\partial y}\Big(\frac{1}{m^{(i)}X_i(x)}\frac{\partial B\left(A^{-1}\left(W_{i}\right)\right)}{\partial y}\Big) = 0, && \\ &&&(x,y)\in(-\infty,0)\times(0,1), \end{align} | (96) |
\begin{align} &\frac{\partial W_{i}}{\partial y}(x,0) = 0, &&x\in(-\infty,0), \end{align} | (97) |
\begin{align} &\frac{1}{m^{(i)}X_i(x)}\frac{\partial B\left(A^{-1}\left(W_{i}\right)\right)}{\partial y}(x,1) = \frac{f''_k(x)}{1+(f'_k(x))^2}, &&x\in(-\infty,0), \end{align} | (98) |
\begin{align} &W_i(0,y) = 0, &&y\in(0,1), \end{align} | (99) |
where
X_i(x) = \frac{1}{(1+(f'_k(x))^2)^{1/2} A^{-1}(W_i(x,{f_k(-L_0)}))},\quad x\in(-\infty,0]. |
Set
W(x,y) = W_1(x,y)-W_2(x,y),\quad (x,y)\in(-\infty,0]\times[0,1]. |
One can verify from that
\begin{align} &\frac{\partial}{\partial x}\Big(m^{(1)}X_1(x)\frac{\partial W}{\partial x}\Big) +\frac{\partial}{\partial y}\Big(\dfrac{1}{m^{(1)}X_1(x)}H(x,y)\frac{\partial W}{\partial y}\Big) \\[1.5 mm] &\qquad+\frac{\partial}{\partial x}\Big(m^{(1)}X(x)\frac{\partial W_{2}}{\partial x}\Big) +\frac{\partial}{\partial x}\Big(mX_2(x)\frac{\partial W_{2}}{\partial x}\Big) \\[1.5 mm] &\qquad+\frac{\partial}{\partial y}\Big(\dfrac{1}{m^{(1)}X_1(x)}\frac{\partial Z}{\partial y}(x,y)W\Big) -\frac{\partial}{\partial y}\Big(\dfrac{m}{m^{(1)}m^{(2)}X_1(x)}\frac{\partial B\left(A^{-1}\left(W_{2}\right)\right)}{\partial y}\Big) \\[1.5 mm] &\qquad-\frac{\partial}{\partial y}\Big(\dfrac{X(x)}{m^{(2)}X_1(x)X_2(x)} \frac{\partial B\left(A^{-1}\left(W_{2}\right)\right)}{\partial y}\Big) = 0,\quad(x,y)\in(-\infty,0)\times(0,1), \end{align} | (100) |
where
\begin{gather*} m = m^{(1)}-m^{(2)}, \\ X(x) = X_1(x)-X_2(x),\quad x\in(-\infty,0], \\ H(x,y) = \int_0^1\dfrac{B'(A^{-1} (\eta W_1(x,y)+(1-\eta)W_2(x,y)))} {A'(A^{-1}(\eta W_1(x,y)+(1-\eta)W_2(x,y)))}{\rm d}\eta, \quad(x,y)\in(-\infty,0)\times(0,1). \end{gather*} |
It follows from (13), (59), (88) and (90)–(93) that
\begin{gather} C_1k^{-1/2}\langle-x\rangle^{-1/2} \leq H(x,y)\leq C_2k^{-1/2}\langle-x\rangle^{-1/2}, \quad(x,y)\in(-\infty,0)\times(0,1), \end{gather} | (101) |
\begin{gather} \Big|\frac{\partial H}{\partial y}(x,y)\Big| \leq\left\{\begin{array}{ll} C_2(-x)^{-1/2},&(x,y)\in[-L_0,0)\times(0,1), \\ C_2(-x)^{-2},&(x,y)\in(-\infty,-L_0)\times(0,1), \end{array}\right. \end{gather} | (102) |
\begin{gather} \Big|\frac{\partial W_{i}}{\partial x}(x,y)\Big| \leq\left\{\begin{array}{ll} C_2k^{3/4},&(x,y)\in[-L_0,0)\times(0,1), \\ C_2k(-x)^{-2},&(x,y)\in(-\infty,-L_0)\times(0,1), \end{array}\right.\quad i = 1,\,2, \end{gather} | (103) |
\begin{gather} \Big|\frac{\partial B\left(A^{-1}\left(W_{2}\right)\right)}{\partial y}(x,y)\Big| \leq\left\{\begin{array}{ll} C_2k(-x)^{1/2},&(x,y)\in[-L_0,0)\times(0,1), \\ C_2k(-x)^{-2},&(x,y)\in(-\infty,-L_0)\times(0,1), \end{array}\right. \end{gather} | (104) |
\begin{gather} |X(x)|\leq\left\{\begin{array}{ll} C_2k^{-1/2}(-x)^{-1/2}|W(x,1)|, &(x,y)\in[-L_0,0)\times(0,1), \\ C_2k^{-1/2}|W(x,1)|, &(x,y)\in(-\infty,-L_0)\times(0,1), \end{array}\right. \end{gather} | (105) |
\begin{gather} |m|\leq C_2\bigg(\int_{-L_0}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x\bigg)^{1/2}, \end{gather} | (106) |
where
\langle-x\rangle = \min\{-x,\,L_0\},\quad L_0 = 3l_0\left(1+\|f'\|_{L^\infty((-l_0,0))}^2\right)^{1/2}. |
Fix
\begin{align*} &\int_{-L}^0\int_0^1m^{(1)}X_1(x)\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +\int_{-L}^0\int_0^1\frac{1}{m^{(1)}X_1(x)}H(x,y) \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \\ = \,&-\int_{-L}^0\int_0^1m^{(1)}X(x)\frac{\partial W}{\partial x}\frac{\partial W_{2}}{\partial x} {\rm d}y{\rm d}x -\int_{-L}^0\int_0^1mX_2(x)\frac{\partial W}{\partial x}\frac{\partial W_{2}}{\partial x} {\rm d}y{\rm d}x \\ &\qquad-\int_{-L}^0\int_0^1\dfrac{1}{m^{(1)}X_1(x)} \frac{\partial H}{\partial y}(x,y)W\frac{\partial W}{\partial y}{\rm d}y{\rm d}x \\ &\qquad+\int_{-L}^0\int_0^1\dfrac{m}{m^{(1)}m^{(2)}X_1(x)} \frac{\partial B\left(A^{-1}\left(W_{2}\right)\right)}{\partial y}\frac{\partial W}{\partial y}{\rm d}y{\rm d}x \\ &\qquad+\int_{-L}^0\int_0^1 \dfrac{X(x)}{m^{(2)}X_1(x)X_2(x)}\frac{\partial B\left(A^{-1}\left(W_{2}\right)\right)}{\partial y} \frac{\partial W}{\partial y}{\rm d}y{\rm d}x \\ &\qquad+\int_0^1W(-L,y)\Big(m^{(1)}X_1(-L)\frac{\partial W_{1}}{\partial x}(-L,y) -m^{(2)}X_2(-L)\frac{\partial W_{2}}{\partial x}(-L,y)\Big){\rm d}y, \end{align*} |
which, together with (23), (90), (101) and (103), yields
\begin{align} \begin{split} &\int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \\ \leq\,&C_3\underbrace{\int_{-L}^0\int_0^1 \Big|X(x)\frac{\partial W}{\partial x}\frac{\partial W_{2}}{\partial x}\Big| {\rm d}y{\rm d}x}_{J_1} +C_3\underbrace{\int_{-L}^0\int_0^1 \Big|m\frac{\partial W}{\partial x}\frac{\partial W_{2}}{\partial x}\Big| {\rm d}y{\rm d}x}_{J_2} \\ &\qquad+C_3\underbrace{\int_{-L}^0\int_0^1 \Big|\frac{\partial H}{\partial y}(x,y)W\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x}_{J_3} \\ &\qquad+C_3\underbrace{\int_{-L}^0\int_0^1 \Big|m\frac{\partial B\left(A^{-1}\left(W_{2}\right)\right)}{\partial y}\frac{\partial W}{\partial y}\Big| {\rm d}y{\rm d}x}_{J_4} \\ &\qquad+C_3\underbrace{\int_{-L}^0\int_0^1 \Big|X(x)\frac{\partial B\left(A^{-1}\left(W_{2}\right)\right)}{\partial y}\frac{\partial W}{\partial y}\Big| {\rm d}y{\rm d}x}_{J_5} \\ &\qquad+C_3k(-L)^{-2} \underbrace{\int_0^1|W(-L,y)|{\rm d}y}_{I_L}. \end{split} \end{align} | (107) |
Below, let us make estimates on
\begin{align} &\int_{-L_0}^0\int_0^1(-x)^{-\vartheta_1}W^2 {\rm d}y{\rm d}x \\ \leq\,&\int_{-L_0}^0\int_0^1(-x)^{-\vartheta_1} \bigg(\int_x^0\left|\frac{\partial W}{\partial x}(s,y)\right| {\rm d}s\bigg)^2{\rm d}y{\rm d}x \\ \leq\,&\int_{-L_0}^0(-x)^{1-\vartheta_1}{\rm d}x \int_{-L_0}^0\int_0^1\left(\frac{\partial w}{\partial x}\right)^2 {\rm d}y{\rm d}x \\ \leq\,&\dfrac{L_0^{2-\vartheta_1}}{2-\vartheta_1} \int_{-L}^0\int_0^1\left(\frac{\partial W}{\partial x}\right)^2 {\rm d}y{\rm d}x,\quad\vartheta_1\in[0,2), \end{align} | (108) |
and
\begin{align} &\int_{-L}^{-L_0}\int_0^1 (-x)^{-\vartheta_2}W^2{\rm d}y{\rm d}x \\ \leq\,&\int_{-L}^{-L_0}\int_0^1(-x)^{-\vartheta_2} \bigg(\int_x^0\Big|\frac{\partial W}{\partial x}(s,y)\Big|{\rm d}s \bigg)^2{\rm d}y{\rm d}x \\ \leq\,&\int_{-L}^{-L_0}(-x)^{1-\vartheta_2}{\rm d}x \int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x \\ \leq\,&\dfrac{L_0^{2-\vartheta_2}}{\vartheta_2-2} \int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x, \quad\vartheta_2\in(2,+\infty). \end{align} | (109) |
Then from the Cauchy's inequality, (108) and (109), we have
\begin{align} &\int_{-L_0}^0W^2(x,1){\rm d}x \\ \leq\,&\int_{-L_0}^0\int_0^1W^2{\rm d}y{\rm d}x +2\int_{-L_0}^0\int_0^1\Big|W\frac{\partial W}{\partial y}\Big| {\rm d}y{\rm d}x \\ \leq\,& L_0^2\int_{-L}^0\int_0^1\left(\frac{\partial W}{\partial x}\right)^2 {\rm d}y{\rm d}x +k^{1/2}L_0^{1/2}\int_{-L_0}^0\int_0^1W^2 {\rm d}y{\rm d}x \\ &\qquad+k^{-1/2}\int_{-L_0}^0\int_0^1(-x)^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \\ \leq\,&(L_0^2+L_0^{5/2})\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x \\ &\qquad+k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x, \end{align} | (110) |
\begin{align} &\int_{-L_0}^0(-x)^{-1}W^2(x,1){\rm d}x \\ \leq\,&\int_{-L_0}^0\int_0^1(-x)^{-1}W^2 {\rm d}y{\rm d}x +2\int_{-L_0}^0\int_0^1(-x)^{-1}\Big|W\frac{\partial W}{\partial y}\Big| {\rm d}y{\rm d}x \\ \leq\,&L_0\int_{-L_0}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +k^{1/2}\int_{-L_0}^0\int_0^1(-x)^{-3/2}W^2 {\rm d}y{\rm d}x \\ &\qquad+k^{-1/2}\int_{-L_0}^0\int_0^1(-x)^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \\ \leq\,&(L_0+2L_0^{1/2})\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x \\ &\qquad+k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x, \end{align} | (111) |
and
\begin{align} &\int_{-L}^{-L_0}(-x)^{-4}W^2(x,1){\rm d}x \\ \leq\,&\int_{-L}^{-L_0}\int_0^1 (-x)^{-4}W^2{\rm d}y{\rm d}x +2\int_{-L}^{-L_0}\int_0^1(-x)^{-4} \Big|W\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x \\ \leq\,& L_0^{-2}\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x +k^{1/2}L_0^{1/2}\int_{-L}^{-L_0}\int_0^1(-x)^{-8}W^2 {\rm d}y{\rm d}x \\ &\qquad+k^{-1/2}L_0^{-1/2}\int_{-L}^{-L_0}\int_0^1 \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \\ \leq\,&(L_0^{-2}+L_0^{-5/2}) \int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x \\ &\qquad+k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x. \end{align} | (112) |
It follows from Cauchy's inequality with
\begin{align} J_1&\leq\varepsilon\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x +\dfrac{1}{\varepsilon}\int_{-L}^0\int_0^1 |X(x)|^2\Big|\frac{\partial W_{2}}{\partial x}\Big|^2{\rm d}y{\rm d}x \\ &\leq\varepsilon\int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +\dfrac{C_2^2k^{1/2}}{\varepsilon}\int_{-L_0}^0 (-x)^{-1}W^2(x,1){\rm d}y{\rm d}x \\ &\qquad\quad+\dfrac{C_2^2k}{\varepsilon}\int_{-L}^{-L_0} (-x)^{-4}W^2(x,1){\rm d}y{\rm d}x \\ &\leq C_4\Big(\varepsilon+\dfrac{k^{1/2}}{\varepsilon}\Big) \int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x \\ &\qquad\quad+\dfrac{C_4k^{1/2}}{\varepsilon} \cdot k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x, \end{align} | (113) |
\begin{align} J_2&\leq\varepsilon\int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +\dfrac{1}{\varepsilon}\int_{-L}^0\int_0^1m^2 \Big(\frac{\partial W_{2}}{\partial x}\big)^2{\rm d}y{\rm d}x \\ &\leq\varepsilon\int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +\dfrac{C_2^2}{\varepsilon}m^2\bigg(k^{3/2} +L_0k^2\int_{-L}^{-L_0}(-x)^{-4}{\rm d}x\bigg) \\ &\leq C_4\Big(\varepsilon+\dfrac{k^{1/2}}{\varepsilon}\Big) \int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x, \end{align} | (114) |
\begin{align} J_3&\leq C_2\int_{-L_0}^0\int_0^1 (-x)^{-1/2}\Big|W\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x +C_2\int_{-L}^{-L_0}\int_0^1 (-x)^{-2}\Big|W\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x \\ &\leq\dfrac{C_2k^{1/2}}{\varepsilon}\int_{-L_0}^0\int_0^1 (-x)^{-1/2}W^2{\rm d}y{\rm d}x +\dfrac{C_2L_0^{1/2}k^{1/2}}{\varepsilon} \int_{-L}^{-L_0}\int_0^1 (-x)^{-4}W^2{\rm d}y{\rm d}x \\ &\qquad\quad+C_2\varepsilon k^{-1/2}\int_{-L}^0\int_0^1 \langle-x\rangle^{-1/2}\Big(\frac{\partial W}{\partial y}\Big)^2 {\rm d}y{\rm d}x \\ &\leq\dfrac{C_4k^{1/2}}{\varepsilon}\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x +C_4\varepsilon k^{-1/2}\int_{-L}^0\int_0^1 \langle-x\rangle^{-1/2}\Big(\frac{\partial W}{\partial y}\Big)^2 {\rm d}y{\rm d}x, \end{align} | (115) |
\begin{align} J_4&\leq C_2k\int_{-L_0}^0\int_0^1 (-x)^{1/2}\Big|m\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x +C_2k\int_{-L}^{-L_0}\int_0^1 (-x)^{-2}\Big|m\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x \\ &\leq\dfrac{C_2k^{5/2}}{\varepsilon}m^2 \bigg(\int_{-L_0}^0(-x)^{3/2}{\rm d}x +\int_{-L}^{-L_0}(-x)^{-4}{\rm d}x\bigg) \\ &\qquad\quad+C_2\varepsilon k^{-1/2}\int_{-L}^0\int_0^1 \langle-x\rangle^{-1/2}\Big(\frac{\partial W}{\partial y}\Big)^2 {\rm d}y{\rm d}x \\ &\leq\dfrac{C_4k^{1/2}}{\varepsilon}\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x +C_4\varepsilon k^{-1/2}\int_{-L}^0\int_0^1 \langle-x\rangle^{-1/2}\Big(\frac{\partial W}{\partial y}\Big)^2 {\rm d}y{\rm d}x, \end{align} | (116) |
and
\begin{align} J_5&\leq C_2^2k^{1/2}\int_{-L_0}^0\int_0^1 \Big|W(x,1)\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x \\ &\qquad\quad+C_2^2k^{1/2}\int_{-L}^{-L_0}\int_0^1(-x)^{-2} \Big|W(x,1)\frac{\partial W}{\partial y}\Big|{\rm d}y{\rm d}x \\ &\leq\dfrac{C_2^2L_0^{1/2}k^{3/2}}{\varepsilon} \int_{-L_0}^0W^2(x,1){\rm d}x +\dfrac{C_2^2L_0^{1/2}k^{3/2}}{\varepsilon} \int_{-L}^{-L_0}(-x)^{-4}W^2(x,1){\rm d}x \\ &\qquad\quad+C_2^2\varepsilon k^{-1/2}\int_{-L}^0\int_0^1 \langle-x\rangle^{-1/2}\Big(\frac{\partial W}{\partial y}\Big)^2 {\rm d}y{\rm d}x \\ &\leq\dfrac{C_4k^{1/2}}{\varepsilon}\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x \\ &\qquad\quad+C_4\Big(\varepsilon+\dfrac{k^{1/2}}{\varepsilon}\Big) k^{-1/2}\int_{-L}^0\int_0^1 \langle-x\rangle^{-1/2}\Big(\frac{\partial W}{\partial y}\Big)^2 {\rm d}y{\rm d}x, \end{align} | (117) |
where
\begin{align} I_L&\leq1+\int_0^1W^2(-L,y){\rm d}y \\ &\leq1+\int_0^1 \bigg(\int_{-L}^0\Big|\frac{\partial W}{\partial x}\Big|{\rm d}x\bigg)^2{\rm d}y \\ &\leq1+(-L)\int_{-L}^0\int_0^1 \Big(\frac{\partial W}{\partial x}\Big)^2{\rm d}y{\rm d}x. \end{align} | (118) |
Substituting (113)–(118) into (107) to get
\begin{align} &\int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \\ \leq\,&C_5\Big(\varepsilon+\dfrac{k^{1/2}}{\varepsilon}\Big) \bigg(\int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x\bigg) \\ &\qquad\quad+C_5(-L)^{-1} +C_5k\int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x. \end{align} | (119) |
Choose
\begin{align} \int_{-L}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +k^{-1/2}\int_{-L}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \leq 2C_5(-L)^{-1}. \end{align} | (120) |
Taking
\begin{align*} \int_{-\infty}^0\int_0^1\Big(\frac{\partial W}{\partial x}\Big)^2 {\rm d}y{\rm d}x +k^{-1/2}\int_{-\infty}^0\int_0^1\langle-x\rangle^{-1/2} \Big(\frac{\partial W}{\partial y}\Big)^2{\rm d}y{\rm d}x \leq0, \end{align*} |
which shows that
\frac{\partial W}{\partial x}(x,y) = \frac{\partial W}{\partial y}(x,y) = 0,\quad (x,y)\in(-\infty,0)\times(0,1). |
Then
In terms of the physical variables, Theorems 4.1 and 4.2 can be transformed as
Theorem 4.3. Assume that
\begin{align*} \max\left\{\frac{c_*}{2},\,c_*-\widetilde{M}_2(k\,{\rm dist}_S(\langle x\rangle,y))^{1/2}\right\} \leq|\nabla\varphi(x,y)|\leq c_*-\widetilde{M}_1(k\,{\rm dist}_S(\langle x\rangle,y))^{1/2}, \\ (x,y)\in\varOmega_k, \end{align*} |
where
\|\varphi(x,y)-q_\infty x\|_ {C^1(\varOmega_k\cap\{x < -R\})} \leq\widetilde{M}_3kR^{-\lambda},\quad R > l_0, |
where
\max\left\{\frac{c_*}{2},\,c_*-\widetilde{M}_2(kl_0)^{1/2}\right\} \leq q_\infty\leq c_*-\widetilde{M}_1(kl_0)^{1/2}. |
Therefore, the flow is uniformly subsonic at the far fields.
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