The BV solution of the parabolic equation with degeneracy on the boundary

Huashui Zhan; Shuping Chen

doi:10.1515/math-2016-0025

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The BV solution of the parabolic equation with degeneracy on the boundary

Huashui Zhan und Shuping Chen

Veröffentlicht/Copyright: 10. Mai 2016

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 14 Heft 1

Abstract

Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

Keywords: Local BV Solution; Boundary degeneracy; Partial boundary condition; Stability

MSC 2010: 35L65; 35L85; 35R35

1 Introduction and the main results

Yin-Wang [1] first studied the equation

ut=div(ρα|∇u|p−2∇u), (x,t)∈QT=Ω×(0,T).(1)

where Ω is a bounded domain in R^N with appropriately smooth boundary, ρ(x) = dist(x, ∂Ω), p > 1, α > 0. An obvious character of the equation is that, the diffusion coefficient depends on the distance to the boundary. Since the diffusion coefficient vanishes on the boundary, it seems that there is no heat flux across the boundary. However, Yin-Wang [1] showed that the fact might not coincide with what we image. In fact, the exponent α, which characterizes the vanishing ratio of the diffusion coefficient near the boundary, does determine the behavior of the heat transfer near the boundary. One may refer to [1] for the details.

In our paper, we will consider the following equation

ut=div(ρα|∇u|p−2∇u)+∑i=1N∂bi(u)∂xi, (x,t)∈QT.(2)

The convection term ∑i=1N∂bi(u)∂xi not only brings the difference on operational skills, but more essentially, it makes the nature of boundary condition change. The equation (2) had been originally studied by the authors in [2, 3]. Instead of the whole boundary condition

u(x,t)=0, (x,t)∈∂Ω×(0,T),(3)

only a partial boundary condition

u(x,t)=0, (x,t)∈Σp×(0,T),(4)

matching equation (2) is considered. Here, denoting {n_i (x)} as the unit inner normal vector of ∂Ω, when b′i(0)ni(x)<0, ∀x∈∂Ω, then Σ_p = ∂Ω. But generally, it is just a portion of ∂Ω. However, we don’t need to pay too much attention to its explicit formula, we only need to remember it is just a subset of ∂Ω. Certainly, the initial value is always necessary,

u(x,0)=u0(x).(5)

In [2, 3], we said a bounded domain Ω has the integral non-singularity, if the constants α, p, satisfy

∫Ωρ−2αp−2dx⩽c.

We assumed that there are constants β, c such that

|bi(s)|⩽c|s|1+β, |b′i(s)|⩽c|s|β.(6)

If p > 2, we had obtained the existence of the solution of equation (2) with the initial boundary values (4)–(5), and if Σ_p = ∂Ω, we also had obtained the stability of the weak solutions. In our paper, we will promote the existence of the solution without the condition (6) but limiting that α ≥ 1. The most innovation of our paper is that the stability of the weak solutions can be obtained only based on the partial boundary condition (4). Comparing with the case of that Σ_p = ∂Ω in [2, 3] or Σ_p = ∅ in [1] (when α ≥ p – 1), how to obtain the stability of the weak solutions only based on the partial boundary condition (4) seems very difficult.

Let us give the basic definitions and the main results as following.

Definition 1.1

A function u(x, t) is said to be a local BV solution of equation(2)with the initial value(5), if

u∈BV(QTλ)∩L∞(QT),ρα|∇u|p∈L1(QT),

and for any functionφ∈C0∞(QT), the following integral equivalence holds

∬QT(−uφt+ρα|∇u|p−2∇u⋅∇φ+∑i=1Nbi(u)⋅φxi)dxdt=0.(7)

The initial value is satisfied in the sense of

limt→0∫Ω|u(x,t)−u0(x)|dx=0.(8)

Here, Q_Tλ = {(x, t) ∈ Q_T : ρ(x) = dist(x, ∂Ω) > λ} for small enough λ > 0.

Definition 1.2

A function u(x, t) is said to be a local BV solution of equation(2)with the initial boundary values(4)–(5), ifu satisfies Definition 1.1, and it satisfies the partial boundary condition(4)in the sense of the trace.

Definition 1.3

If u is a local BV solution of equation(2), satisfies that

|u(x,t)|≤cρ(x), |∇u|≤cρ−α(x),(9)

when x is near ∂Ω, then we say u is a regular solution.

Remark 1.4

If b_i(s) ≡ 0, we had proved that the solution of equation(1)is regular in [4].

Theorem 1.5

Let 1 < p, 1 ≤ α < p – 1, b_i(s) ∈ C²(R¹). If

u0(x)∈C0∞(Ω),(10)

then equation(2)with initial boundary values(4)–(5)has a local BV solution u, and u_t ∈ L^∞(Q_T).

Theorem 1.6

Let α < p – 1, u and v be two local BV solutions of equation(1)with the same partial homogeneous boundary value

u|Σp×(0,T)=0=υ|Σp×(0,T),(11)

and with the different initial values u(x, 0) = v(x, 0) respectively. If b_i(s) is a Lipschitz function, and moreover

|∇u|≤cρ−α(x), |∇υ|≤cρ−α(x)(12)

then

∫Ω|u(x,t)−v(x,t)|dx⩽∫Ω|u0−v0|dx+c∫Σ′p|u−v|dΣ+lim supn→∞∫Σ′pgn(u−v)|u−v|dΣ, ∀t∈[0,T).(13)

Here, n > 0 is a nature number, the details of the definition and the properties of the function g_n(s) is in Section 3, in particular, |g_n(s)s| ≤ c.

Theorem 1.7

Let α ≥ p – 1, and u, v be two local BV solutions of(1)with the initial values u₀(x), v₀(x) respectively. If u and v are regular, and

|bi(u)−bi(v)|≤c|u−v|α+2,(14)

then

∫Ω|u(x,t)−v(x,t)|2dx≤c∫Ω|u0−v0|2dx,a.e.t∈(0,T).(15)

The most important character of Theorem 1.7 is in that we obtain the stability (15) without any boundary value condition. However, since the solutions considered in the theorem are regular, we can easily obtain the conclusion (15) in a similar way as in [1]. So we omit the details of the proof of the theorem in our paper.

Recently, the author has been interested in the initial-boundary value problem of the following strongly degenerate parabolic equation

∂u∂t=∂∂xi(aij(u,x,t)∂u∂xj)+∂bi(u,x,t)∂xi,(x,t)∈QT.(16)

The stability of the solutions based on a partial boundary condition (4) has been established in [5–7] et. al. Actually, many mathematicians have been interested in the problem, and have obtained many important results of the the stability of the solutions based on a partial boundary condition, one may see the Refs. [8–11]. Unlike the equation (16), to the best knowledge of the authors, considering the parabolic equation related to the p-Laplacian, our paper is the first one to study the stability of the solutions based on a partial boundary condition (4). Of course, whether the condition (12) in Theorem 1.6 and the assumption that u, v are regular in Theorem 1.7 are necessary or not? This is a very interesting problem to be studied in the future. Some other related references, one can refer to Refs. [12–16]. The paper is arranged as following. In Section 1, we have introduced the problem and given the main results of the paper. In Section 2, we prove the existence of the local BV solution. In Section 3, only based on a partial boundary condition, we prove the Theorem 1.6.

2 The localBV solution

To study equation (2), we consider the following regularized problem

uεt−div(ρεα(|∇uε|2+ε)p−22∇uε)−∑i=1N∂bi(uε)∂xi=0,(x,t)∈QT,(17)

uε(x,t)=0, (x,t)∈∂Ω×(0,T),(18)

uε(x,0)=u0(x), x∈Ω.(19)

where ρ_∊ = ρ * δ_∊ + ∊, ∊ > 0, δ_∊ is the mollifier as usual. It is well-known that the above problem has a unique classical solution [17, 18]. Hence, for any φ∈C0∞(QT), u_∊ satisfies

∬QT(−uεφt+ρεα|∇uε|p−2∇uε⋅∇φ+∑i=1Nbi(uε)⋅φxi)dxdt=0.(20)

Lemma 2.1

Ifu0∈C0∞(Ω), α≥1, then the solution u_∊ of the initial boundary value problem(17)–(19)converges locally in BV(Q_T), and its limit function u is the local BV solution of equation(2)with the initial value(5).

Proof. By the maximum principle, there is a constant c only dependent on ‖u₀‖_L^∞(Ω) but independent on ε, such that

‖uε‖L∞(QT)⩽c.(21)

Multiplying (17) by u_ε and integrating it over Q_T, we have

12∫Ωuε2dx+∬QTρεα(|∇uε|2+ε)p−22|∇uε|2dxdt+∬QTuε∑i=1N∂bi(uε)∂xidxdt=12∫Ωu02dx.

By the fact

∬QTuε∑i=1N∂bi(uε)∂xidxdt=−∑i=1N∂uε∂xibi(uε)dxdt=−∑i=1N∫Ω∂∂xi∫0uεbi(s)dsdt=0,

then

12∫Ωuε2dx+∬QTρεα(|∇uε|2+ε)p−22|∇uε|2dxdt⩽c.(22)

For small enough λ > 0, let Ω_λ = {x ∈ Ω : dist(x, ∂Ω) > λ}. Since p > 1, by (22),

∫0T∫Ωλ|∇uε|dxdt≤c(∫0T∫Ωλ|∇uε|pdxdt)1p≤c(λ).(23)

Differentiating (17) with t, and denoting w = u_∊t, then

∂w∂t=(p−2)ρεα(|∇uε|2+ε)p−42uxkuxiwxkxi+ρεα(|∇uε|2+ε)p−22wxixi+(p−2)∇ρεα⋅∇uε(|∇uε|2+ε)p−42uxkwxk+∇ρεα(|∇uε|2+ε)p−22⋅∇w +(p−2)(p−4)ρεα(|∇uε|2+ε)p−62uxjuxixjuxiwxkuxk+(p−2)ρεα(|∇uε|2+ε)p−42(uxiuxixkwxk+uxkuxkxiwxi+uxkuxixiwxk) +bi″(u)uxiw+bi′(u)wxi,

rewriting it as

∂w∂t=aij∂2w∂xi∂xj+∑i=1Nfi(x,t,w)wxi+bi″(u)uxiw,

where

aij=ρεα(|∇uε|2+ε)p−22(δij+(p−2)(|∇uε|2+ε)−1uxiuxj).fi(x,t,w)=(p−2)∇ρεα⋅∇uε(|∇uε|2+ε)p−42uxi+(|∇uε|2+ε)p−22(ρεα)xi +(p−2)(p−4)ρεα(|∇uε|2+ε)p−62uxjuxkxjuxkuxi+(p−2)ρεα(|∇uε|2+ε)p−42(uxkuxkxi+uxiuxixk+uxiuxkxk)+b′i(u),

Clearly, w satisfies that

w(x,t)=0,(x,t)∈∂Ω×[0,T),w(x,0)=div(ρεα(|∇u0|2+ε)p−22∇u0)+∂bi(u0)∂xi,x∈Ω.

Denoting that

a=(|∇uε|2+ε)p−22,

then

min{p−1,1}a|ξ|2≤aijξiξj≤max{p−1,1}a|ξ|2.

By the maximum principle, due to α ≥ 1, we have

supΩ×(0,T)|uεt|≤supΩ|div(ρεα(|∇u0|2+ε)p−22+∂bi(u0)∂xi|≤c.(24)

By (23)–(24), we know that u_ε ∈ BV(Q_T_λ), and

∫0T∫Ωλ|∇uε|dxdt≤c, ∫0T∫Ωλ|uεt|dxdt≤c.(25)

Then by Kolmogoroff’s theorem, there exists a subsequence (still denoted as u_ε) of u_ε, which is strongly convergent to u ∈ BV(Q_T_λ). In particular, by the arbitrary of λ, u_ε → u a.e. in Q_T.

Hence, by (22), (25), there exists a function u and n dimensional vector function ζ→=(ζ1,⋯,ζn) satisfying that

u∈BV(QTλ)∩L∞(QT), |ζ→|∈Lpp−1(QT),

and

uε⇀*u, in L∞(QT),∇uε⇀∇u in Llocp(QT),ρεα|∇uε|p−2∇uε⇀ζ→ in Lpp−1(QT).

In order to prove u satisfies equation (2), we notice that for any function φ∈C0∞(QT),

∬QT(−uεφt+ρεα(|∇uε|2+ε)p−22∇uε⋅∇φ+∑i=1Nbi(uε)⋅φxi)dxdt=0.(26)

and u_ε → u is almost everywhere convergent, so b_i(u_ε) → b_i(u) is true. Then

∬QT(∂u∂tφ+ς→⋅∇φ+∑i=1Nbi(u)⋅φxi)dxdt=0.(27)

Now, if we can prove that

∬QTρα|∇u|p−2∇u⋅∇φdxdt=∬QTζ→⋅∇φdxdt.(28)

for any function φ∈C0∞(QT), then u satisfies equation (2).

Let 0⩽ψ∈C0∞(QT) and ψ = 1in suppφ. Let v ∈ BV(Q_T_λ) ∩ L^∞(Q_T), ρ^α|∇v|^p ∈ L¹(Q_T). It is well-known that

∬QTψρεα(|∇uε|p−2∇uε−|∇v|p−2∇v)⋅(∇uε−∇v)dxdt⩾0.(29)

By choosing φ = ψu_∊ in (26), we can obtain

∬QTψρεα(|∇uε|2+ε)p−22|∇uε|2dxdt=12∬QTψtuε2dxdt−∬QTρεαuε(|∇uε|2+ε)p−22∇uε⋅vψdxdt −∑i=1N∬QTbi(uε)(uεxiψ+uεψxi)dxdt.(30)

Noticing that when p ≥ 2,

(|∇uε|2+ε)p−22|∇uε|2≥|∇uε|p,(|∇uε|2+ε)p−22|∇uε|≤(|∇uε|p−1+1),

and when 1 < p < 2,

(|∇uε|2+ε)p−22|∇uε|2≥(|∇uε|2+ε)p2−εp2,(|∇uε|2+ε)p−22|∇uε|≤(|∇uε|2+ε)p−12,(31)

then in both cases, by (29), we have

12∬QTψtuε2dxdt−∬QTρεαuε(|∇uε|2+ε)p−22∇uε⋅∇ψdxdt−∑i=1N∬QTbi(uε)(uεxiψ+uεψxi)dxdt +εp2c(Ω)−∬QTρεαψ|∇uε|p−2∇uε∇vdxdt−∬QTρεαψ|∇v|p−2∇(uε−v)dxdt≥0.(32)

Thus

12∬QTψtuε2dxdt−∬QTρεαuε(|∇uε|2+ε)p−22∇uε⋅∇ψdxdt−∑i=1N∬QTbi(uε)(uεxiψ+uεψxi)dxdt +εp2c(Ω)−∬QTρεαψ|∇uε|p−2∇uε∇vdxdt−∬QTψρα|∇v|p−2∇v⋅(∇uε−∇v)dxdt +∬QTψ(ρα−ρεα)|∇v|p−2∇v⋅(∇uε−∇v)dxdt⩾0.(33)

Noticing

|∬QTψ(ρα−ρεα)|∇v|p−2∇v⋅(∇uε−∇v)dxdt|⩽sup(x,t)∈QT|ψ(ρα−ρεα)|ρα∬QTρα|∇v|p−1|∇uε−∇v|dxdt⩽sup(x,t)∈QT|ψ(ρα−ρεα)|ρα(∬QTρα|∇v|pdxdt+∬QTρα|∇v|p−1|∇uε|dxdt)(34)

and

(|∇uε|2+ε)p−22∇uε=|∇uε|p−2∇uε+p−22ε∫01(|∇uε|2+εs)p−42ds∇uε

then

limε→0∬QTp−22ε∫01(|∇uε|2+εs)p−42ds∇uε∇ψuεdxdt=0.(35)

By Hölder inequality, there holds

∬QTρα|∇v|p−1|∇uε|dxdt⩽(∬QT(ρm|∇v|p−1)sdxdt)1/s⋅(∬QT(ρn|∇uε|)pdxdt)1/p,

where m=α(p−1)p, n=αp, s=pp−1. Due to ρ^α|∇|^p, ρ^α|∇|^p ∈ L¹(Q_T), then

∬QTρα|∇v|pdxdt+∬QTρα|∇v|p−1|∇uε|dxdt⩽c.

Let ε → 0 in (33). It converges to 0.

Thus, we have

12∬QTψtu2dxdt−∬QTuζ→⋅∇ψdxdt−∑i=1N∬QTbi(u)(uxiψ+uψxi)dxdt−∬QTψζ→⋅∇vdxdt−∬QTψρα|∇v|p−2∇v⋅(∇uε−∇v)dxdt⩾0.

Let v = ψu in (2), we get

∬QTψζ→⋅∇udxdt−12∬QTu2ψtdxdt+∬QTuζ→⋅∇ψdxdt+∑i=1N∬QTbi(u)(uxiψ+uψxi)dxdt=0.

Thus

∬QTψ(ζ→−ρα|∇v|p−2∇v)⋅(∇u−∇v)dxdt⩾0.(36)

Let v = u – λφ, λ > 0, φ∈C0∞(QT), then

∬QTψ(ζ→−ρα|∇(u−λφ)|p−2∇(u−λφ))⋅∇φdxdt⩾0,

If λ → 0, then

∬QTψ(ζ→−ρα|∇u|p−2∇u)⋅∇φdxdt⩾0.

Moreover, if λ < 0, similarly we can get

∬QTψ(ζ→−ρα|∇u|p−2∇u)⋅∇φdxdt⩽0.

Thus

∬QTψ(ζ→−ρα|∇u|p−2∇u)⋅∇φdxdt=0,

Noticing that ψ = 1 on suppφ, (28) holds. At the same time, we can prove (5) as in [19], we omit the details here. The lemma is proved. □

Corollary 2.2

Ifu0∈C0∞(Ω), α ≥ 1 there exists a local BV solution u of equation(2)with the initial value(5), such that u_t ∈ L^∞(Q_T).

Lemma 2.3

If α < p – 1, u is a weak solution of equation(1)(also equation(2)) with the initial value(5). Then u has trace on the boundary ∂Ω.

Lemma 2.3 had been proved in [1, 2]. Clearly, Theorem 1.5 is the directly corollary of Lemma 2.1–2.3.

Remark 2.4

The condition α ≥ 1 is only used to prove that |u_∊t| ≤ c, which implies that∫QTλ|uεt|dxdt≤c. Maybe one can prove the later conclusion directly. u0∈C0∞(Ω)is the simplest condition, but it is not the mostgeneral condition. However, we mainly concern with how the degeneracy of diffusion coefficient ρ^α affects the boundary value condition.

3 The stability when α < p – 1

Proof of Theorem 1.6. For a small positive constant λ > 0, let

Ωλ={x∈Ω:ρ(x)=dist(x,∂Ω)>λ},

and let

ϕ(x)={ 1, if x∈Ω2λ,1λ(ρ(x)−λ), x∈Ωλ\Ω2λ 0,if x∈Ω\Ωλ.(37)

For any given positive integer n, let g_n(s) be an odd function, and

gn(s)={1,s>1n,n2s2e1−n2s2,0≤s⩽1n.(38)

Clearly,

limn→0gn(s)=sgn(s),s∈(−∞,+∞),(39)

and

0≤g′n(s)≤cs, 0<s<1n,(40)

where c is independent of n.

By a process of limit, we can choose g_n(ϕ(u – v) as the test function, then

∫Ωgn(ϕ(u−v))∂(u−v)∂tdx+∫Ωρα(|∇u|p−2∇u−|∇v|p−2∇v⋅ϕ∇(u−v)g′ndx+∫Ωρα(∇u)|p−2∇u−|∇v|p−2∇v)⋅∇ϕ(u−v)g′ndx+∑i=1N∫Ω(bi(u)−bi(v))⋅(u−v)xig′nϕdx+∑i=1N∫Ω(bi(u)−bi(v))⋅(u−v)g′ndx=0.(41)

Thus

limn→∞limλ→0∫Ωgn(ϕ(u−v))∂(u−v)∂tdx=ddt‖u−v‖1,(42)

∫Ωρα(|∇u|p−2∇u−|∇v|p−2∇v)⋅∇(u−v)g′nϕ(x)dx≥0.(43)

By L’Hospital rule,

limλ→0∫Ωλ\Ω2λg′n(ϕ(u−v))(u−v)dxλ=limλ→0∫λ2λ∫ρ=ζg′n(ϕ(u−v))(u−v)dΣdζλ=limλ→0∫ρ=2λg′n(u−v)(u−v)dΣ=∫∂Ωg′n(2(u−v))(u−v)dΣ=∫Σ′pg′n(u−v)(u−v)dΣ.(44)

Since we assume that ρ|∇u|^p < ∞, ρ|∇v|^p < ∞, we have

limλ→0|∫Ωρα(|∇u|p−2∇u−|∇v|p−2∇v)⋅∇ϕ(u−v)g′n(ϕ(u−v))dx|=limλ→0|∫Ωλ\Ω2λρα(|∇u|p−2∇u−|∇v|p−2∇v)⋅∇ϕ(u−v)g′n(ϕ(u−v))dx|≤climλ→0|∫Ωλ\Ω2λg′n(ϕ(u−v))(u−v)dxλ=∫∑′pg′n(u−v)(u−v)dΣ.(45)

While,

|∫Ω(bi(u)−bi(v))g′n(ϕ(u−v))(u−v)ϕxi(x)dx|≤∫Ωλ∖Ω2λ|bi(u)−bi(v)||u−v|gn′(ϕ(u−v))cλdx.(46)

By |b_i(u) – b_i(v)| ≤ c|u – v|, and by (40), according to the definition of the trace, we have

limλ→0⁡|∫Ω(bi(u)−bi(v))gn′(ϕ(u−v))(u−v)ϕxi(x)dx|≤limλ→0⁡∫Ωλ∖Ω2λ|u−v|2g′n(ϕ(u−v))cλdx=∫∂Ω|u−v|2gn′((u−v))d∑≤c∫∂Ω|u−v|dΣ=c∫∑′p|u−v|dΣ.(47)

Moreover,

limλ→0⁡|∫Ω(bi(u)−bi(v))gn′(ϕ(u−v))(u−v)xiϕ(x)dx|=|∫{x∈Ω:|u−v|<1n}[bi(u)−bi(v)]gn′(u−v)(u−v)xidx⩽c∫{x∈Ω:|u−v|<1n}|bi(u)−bi(v)u−v||(u−v)xi|dx=c∫{x∈Ω:|u−v|<1n}|ρ−αpbi(u)−bi(v)u−v||ραp(u−v)xi|dx⩽c[∫{x∈Ω:|u−v|<1n}(|ρ−αpbi(u)−bi(v)u−v|)pp−1dx]pp−1⋅[∫{x∈Ω:|u−v|<1n}|ρα∇(u−v)|pdx]1p.(48)

Since α < p – 1,

∫{x∈Ω:|u−v|<1n}(|ρ−αpbi(u)−bi(v)u−v|)pp−1dx⩽c∫Ωρ−αp−1dx⩽c.(49)

In (47), let n → ∞. If {x ∈ Ω : |u – v| = 0} is a set with 0measure, then

limn→∞∫{x∈Ω:|u−v|<1n}ρ−αp−1dx=∫{x∈Ω:|u−v|=0}ρ−αp−1dx=0.(50)

If the set {x ∈ Ω : |u – v| = 0} has a positive measure, then,

limn→∞∫{x∈Ω:|u−v|<1n}ρα|∇(u−v)|pdx=∫{x∈Ω:|u−v|=0}ρα|∇(u−v)|pdx=0.(51)

Therefore, in both cases, we have

limn→∞limλ→0⁡∫Ω(bi(u)−bi(v))gn′(ϕ(u−v))(u−v)xiϕ(x)dx=0.(52)

Now, after letting λ → 0, let n → ∞ in (41). Then

ddt‖u−v‖1⩽c∫Σ′p|u−v|dΣ+lim supn→∞∫Σ′pgn(u−v)|u−v|dΣ.

It implies that

∫Ω|u(x,t)−v(x,t)|dx⩽∫Ω|u0−v0|dx+c∫Σ′p|u−v|dΣ+lim supn→∞∫Σ′pgn(u−v)|u−v|dΣ, ∀t∈[0,T).

Theorem 1.6 is proved.

Acknowledgement

The paper is supported by NSF of China (no.11371297, 11302184), Natural Science Foundation of Fujian province (no: 2015J01592), supported by Science Foundation of Xiamen University of Technology, China (no:XYK201448).

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Received: 2016-1-15

Accepted: 2016-4-9

Published Online: 2016-5-10

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.