ENERGY DECAY OF A VARIABLE-COEFFICIENT WAVE EQUATION WITH NONLINEAR TIME-DEPENDENT LOCALIZED DAMPING

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1 Electroic Joural of Differetial Equatios, Vol (2015), No. 226, pp IN: URL: or ftp ejde.math.txstate.edu ENERGY DECAY OF A VARIABLE-COEFFICIENT WAVE EQUATION WITH NONLINEAR TIME-DEPENDENT LOCALIZED DAMPING JIEQIONG WU, FEI FENG, HUGEN CHAI Abstract. We study the eergy decay for the Cauchy problem of the wave equatio with oliear time-depedet ad space-depedet dampig. The dampig is localized i a bouded domai ad ear ifiity, ad the pricipal part of the wave equatio has a variable-coefficiet. We apply the multiplier method for variable-coefficiet equatios, ad obtai a eergy decay that depeds o the property of the coefficiet of the dampig term. 1. Itroductio Let 2 be a iteger. We cosider the eergy decay for the solutio to the Cauchy problem of the wave equatio with time-depedet ad space-depedet dampig, u tt div(a(x) u) + u + σ(t)ρ(x, u t ) = 0, x R, t > 0, (1.1) u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x R, (1.2) where A(x) = (a ij (x)) is a symmetric, positive matrix for each x R ; σ : R + R + is a o-icreasig fuctio of class C 1. Let Ω ad Ω 1 be two bouded domais such that Ω 1 Ω. We assume that { a(x)u t, x R \ Ω 1 ρ(x, u t ) = (1.3) a(x)h(u t ), x Ω 1 where h : R R is a oliear cotiuous odecreasig fuctio satisfyig h(s)s > 0 for all s 0 ad a( ) L (R ) is a oegative fuctio satisfyig Whe x Ω 1, a(x) ε 0 > 0, x Ω 1 Ω c. (1.) c 1 u t m h(u t ) c 2 u t 1/m, u t 1, (1.5) c 3 u t h(u t ) c u t, u t 1 (1.6) where m 1 ad c i > 0 (i = 1, 2, 3, ) are give umbers Mathematics ubject Classificatio. 35L05, 35L70, 93B27. Key words ad phrases. Eergy decay; time-depedet ad space-depedet dampig; localized dampig; Riemaia geometry method; variable coefficiet. c 2015 Texas tate Uiversity - a Marcos. ubmitted May 28, Published eptember 2,

2 2 J. WU, F. FENG,. CHAI EJDE-2015/226 ice (1.) implies that the dissipatio may vaish i Ω\Ω 1, we call the dampig σ(t)ρ(x, u t ) a time-depedet localized dampig. May studies cocerig eergy decay of wave equatios i the bouded domai are available i the literature. We refer the readers to [3, 8, 9, 16]. The author of [8] derived precise eergy decay estimates for the iitial-boudary value problem to the wave equatio with a localized oliear dissipatio which depeded o the time as well as the space variable. The result of [8] is a geeralizatio of the work [9], where the dissipatio was idepedet of the time. A wave equatio with timedepedet but space-idepedet dampig was ivestigated i [3] ad the decay rate was obtaied by solvig a oliear ODE. For the eergy decay of wave equatio i the exterior domai or o the whole space, may results have bee obtaied for the case of costat coefficiets, that is A(x) I (see [1, 2,, 10, 11, 12, 17]). The dampig i [, 11, 17] were timeidepedet ad localized i a sub-domai of R. Eergy decay of the Cauchy problem for the wave equatio with time-depedet dampig was studied i [2] where the coefficiet of the dampig did ot deped o the space variable. It is iterestig to study the case where the dampig is time-depedet ad exists o a local domai of R. [1] addressed the decay for a wave equatio o a exterior domai where the dampig is time-depedet ad effective oly i a ball of R. The dimesio of the space variable was restricted to be odd sice the Lax-Phillips scatterig theory was used i [1]. Nakao [10] cosidered the Cauchy problem of wave equatio with a dissipatio effective outside of a give ball of R. However, i the case of variable coefficiet, few results about the eergy decay of wave equatio i the exterior domai or o the whole space ca be see (see[5, 13]). Yao [13] studied the eergy decay for the Cauchy problem of the variable-coefficiet wave equatio with a liear dampig. The authors of [5] cosidered the eergy decay of variable-coefficiet wave equatio with oliear dampig i a exterior domai. The purpose of this paper is to derive the eergy decay for (1.1)-(1.2) with the assumptios (1.3)-(1.6). Our first mai goal is to dispese with the restrictio A(x) I. We will use the Riemaia geometry method which was itroduced by Yao [1] (see also [15]) to deal with cotrollability for the wave equatio with variable-coefficiet pricipal part ad has bee viewed as a importat method for variable-coefficiet models. Our secod goal is to geeralize the work [2], where the dampig exists o the whole space. I this paper, we assume that the dampig is localized ear the origi poit ad ear ifiity ad may disappear i a large area. We try to explai how the time-depedece of the dampig affect the decay whe the dissipatio is effective i a localized domai. Eergy decay is obtaied uder some coditios o A(x) ad the coefficiet of the dampig. We itroduce a Riemaia metric o R by g(x) = A 1 (x) for x R. (1.7) We deote by f ad g f the gradiets of f i the stadard metric of the Euclidea space R ad i the metric g, respectively. The eergy of the model (1.1)-(1.2) is defied by E(t) = 1 (u 2 t + g u 2 g + u 2 )dx. (1.8) 2 R

3 EJDE-2015/226 ENERGY DECAY 3 where g u = A(x) u ad g u 2 g = g u, g u g = A(x) u, u = ij=1 a ij (x) u x j u x i. We refer the readers to [1] ad [15] for more iformatio about the metric g(x). We use the followig assumptio i this article (H1) There is a vector field H o (R, g) such that D g H(X, X) σ X 2 g, X R x, x Ω, (1.9) where σ > 0 is a costat ad D g metric g. Our mai result read as follows. is the Levi-Civita coectio of the Theorem 1.1. Let (u 0, u 1 ) H 1 (R ) L 2 (R ). I additio to (H1) assume that σ : R + R + is a o-icreasig C 1 fuctio ad σ(t) σ 0 > 0 for all t 0. The (1.1)-(1.2) admits a uique solutio u satisfyig ad ( 1 ) 2 E(t) CE(0) t 0 σ(s)ds E(t) CE(0) exp( ω for some ω > 0 ad C > 0. t 0 m 1, t > 0 if m > 1, σ(s)ds), t > 0 if m = 1, 2. Proof of mai result To prove our mai result we use the followig lemma. Lemma 2.1 ([6]). Let E : R + R + be a o-icreasig fuctio ad φ : R + R + a strictly icreasig C 1 fuctio such that φ(0) = 0, φ(t) + as t +. Assume that there exist r 0 ad ω > 0 such that The + E 1+r (t)φ (t)dt 1 ω Er (0)E(), 0 < +. ( 1 + r ) 1/r E(t) E(0) for all t 0, if r > 0, 1 + ωrφ(t) E(t) CE(0)e 1 ωφ(t) for all t 0, if r = 0. To apply Lemma 2.1, we eed some estimates o the eergy terms, which are based o the idetities below. Multiply (1.1) by u t ad itegrate by parts over R with respect to the variable x to obtai d dt E(t) = R σ(t)ρ(x, u t )u t dx. (2.1)

4 J. WU, F. FENG,. CHAI EJDE-2015/226 Let H be a vector field o R. Multiply (1.1) by H(u) ad itegrate by parts over R with respect to the variable x to obtai d u t H(u)dx + D g H( g u, g u)dx dt R R + H(u)u dx + 1 (u 2 t g u 2 R 2 g)divhdx (2.2) R = σ(t)ρ(x, u t )H(u)dx. R Let q C 1 (R ) be a fuctio. Multiply (1.1) by qu ad itegrate by parts over R with respect to the variable x to obtai d quu t dx + q ( g u 2 g u 2 ) t dx u 2 div A q dx + qu 2 dx dt R R R R (2.3) = quσ(t)ρ(x, u t )dx. R Proof of Theorem 1.1. Let the vector field H be such that the assumptio (H1) holds. Let R ad R be two bouded domais such that Ω. Let ϕ ad ψ be two C0 fuctios ad 0 ϕ 1, 0 ψ 1, ad { { 1, x Ω, 1, x, ϕ = 0, x c = R \, ψ = (2.) 0, x R \. Let q 0 = div(ϕh) 2 σϕ. Replacig H by ϕh i (2.2) ad replacig q by q 0 i (2.3), respectively, we ca obtai two idetities. The we add them up ad obtai d ( ) ut ϕh(u) + q 0 uu t dx + D g (ϕh)( g u, g u)dx dt R R + ϕh(u)u dx + σϕ(u 2 t g u 2 g)dx R R (2.5) u 2 div( g q 0 )dx + q 0 u 2 dx R R = q 0 uσ(t)ρ(x, u t )dx σ(t)ρ(x, u t )ϕh(u)dx. R R Let k be a large costat determied later. Combiig (2.1) with (2.5), we have d ( ) ut ϕh(u) + q 0 uu t dx + ke (t) + D g (ϕh)( g u, g u)dx dt R R + ϕh(u)u dx + σϕ(u 2 t g u 2 g)dx u 2 div( g q 0 )dx R R R + q 0 u 2 dx (2.6) R = q 0 uσ(t)ρ(x, u t )dx σ(t)ρ(x, u t )ϕh(u)dx R R k σ(t)ρ(x, u t )u t dx. R

5 EJDE-2015/226 ENERGY DECAY 5 et Y (t) = D g (ϕh)( g u, g u)dx+ R σϕ(u 2 t g u 2 g)dx+k R σ(t)ρ(x, u t )u t dx. R et σ 1 = sup D g (ϕh)(x, X). Noticig (1.3),(1.),(1.9) ad (2.), we have the followig estimates o Y (t): Y (t) D g (ϕ)h( g u, g u)dx + \Ω + k R \Ω ( σ 1 σ) + σu 2 t dx + 3k Ω ɛ 0 σ(t)u 2 t dx + Ω g u 2 gdx + \Ω R \Ω σu 2 t dx + 3k R \Ω R σ(t)ρ(x, u t )u t dx. σϕu 2 t dx \Ω σϕ g u 2 g dx R σ(t)ρ(x, u t )u t dx (k ε 0σ(t) + σϕ ) u 2 t dx (2.7) Noticig σ(t) σ 0 > 0, we choose k large eough so that kσ0ε0 > σ. The we have Y (t) σ u t 2 dx 1 + σ) g u R \Ω(σ 2 g + 3k σ(t)ρ(x, u t )u t dx. (2.8) R Now we estimate the last term of the right side of (2.8). σ(t)ρ(x, u t )u t dx R ψσ(t)ρ(x, u t )u t dx = ψσ(t)ρ(x, u t )u t dx R = ψσ(t)a(x)u 2 t dx + ψσ(t)ρ(x, u t )u t dx \Ω 1 Ω 1 = ψσ(t)a(x)u 2 t dx σ(t)a(x)u 2 t dx + ψσ(t)ρ(x, u t )u t dx. Ω 1 Ω 1 (2.9) For the first term o the right of (2.9), oticig σ(t) σ 0 ad usig (1.), (2.) ad (2.3) for q = σ 0 ψa(x), we have ψσ(t)a(x)u 2 t dx σ 0 ψa(x)u 2 t dx = σ 0 ψa(x)u 2 t dx R d σ 0 a(x)ψuu t dx + σ 0 ε 0 g u 2 g dx dt R \Ω σ 0 u R 2 div A ( a(x)ψ ) dx + σ 0 a(x)ψu 2 dx R + σ 0 R a(x)ψσ(t)ρ(x, u t )u dx. (2.10) sice supp ψ ad a(x) ε0 > 0 for x Ω c.

6 6 J. WU, F. FENG,. CHAI EJDE-2015/226 Combiig (2.8) with (2.9) ad (2.10), we have ( kσ0 ε ) \Ω 0 Y (t) (σ 1 + σ) g u 2 g dx + σu 2 t dx R + k σ(t)ρ(x, u t )u t dx + σ [ 0k d a(x)ψuu t dx 2 R dt R u 2 div A ( a(x)ψ ) dx + a(x)ψu 2 dx R R ] + a(x)ψσ(t)ρ(x, u t )udx k σ(t)a(x)u 2 t dx R Ω 1 + k σ(t)ρ(x, u t )u t dx Ω 1 (2.11) Choose k large eough so that kσ0ε0 > σ + σ 1. The it follows from (2.11) ad (2.6) that d ( ( ) σ 0 k ) ut ϕh(u) + q 0 uu t dx + ke(t) + a(x)ψuu t dx dt R R + σu 2 t dx + k σ(t)ρ(x, u t )u t dx k σ(t)a(x)ψu 2 t dx R 2 Ω 1 Ω 1 + k σ(t)ρ(x, u t )u t dx + σ [ 0k ( ( ) aψ div A a(x)ψ) u 2 dx Ω 1 R ] (2.12) + a(x)ψσ(t)ρ(x, u t )udx + ϕh(u)u dx R R ( ) + q0 div g q 0 u 2 dx + q 0 uσ(t)ρ(x, u t )dx R R + σ(t)ρ(x, u t )ϕh(u)dx 0. R We may have equality by settig q = σ 2 i (2.3). The add that equality to (2.12) to yield d ( ( ) σ 0 k ut ϕh(u) + q 0 uu t dx + ke(t) + a(x)ψuu t dx dt R R σ ) + R 2 uu σ ( t dx + g u 2 g + u 2 t + u 2 + σ(t)ρ(x, u t )u ) dx R 2 + k σ(t)ρ(x, u t )u t dx k σ(t)a(x)ψu 2 t dx 2 Ω 1 Ω 1 + k σ(t)ρ(x, u t )u t dx + σ [ 0k ( aψ div A ( a(x)ψ )) u 2 dx (2.13) Ω 1 R ] + a(x)ψσ(t)ρ(x, u t )udx + ϕh(u)u dx R R ( ) + q0 div g q 0 u 2 dx + q 0 σ(t)ρ(x, u t )u dx R R + σ(t)ρ(x, u t )ϕh(u)dx 0. R

7 EJDE-2015/226 ENERGY DECAY 7 et ( ) σ 0 k X(t) = ut ϕh(u) + q 0 uu t dx + ke(t) + a(x)ψuu t dx + R R From (2.13) ad the defiitio of E(t), we have where σe(t) d dt X(t) σ R 2 σ(t)ρ(x, u t)u dx σ(t)ρ(x, u t )ϕh(u) dx R q 0 σ(t)ρ(x, u t )udx k σ(t)ρ(x, u t )u t dx R 2 R k σ(t)ρ(x, u t )u t dx σ [ 0k ( aψ div A ( a(x)ψ )) u 2 dx Ω 1 R ] + a(x)ψσ(t)ρ(x, u t )u dx + k σ(t)a(x)ψu 2 t dx R Ω 1 ( ) ϕh(u)u dx q0 div g q 0 u 2 dx R R d dt X(t) + σ σ(t)ρ(x, u t )u dx k R 2 E (t) + ( σ 0kσ 2 + σ 3 ) u 2 dx + k σ(t)a(x)u 2 t dx + σ 5 g u g u dx Ω 1 + σ 5 R σ(t) ρ(x, u t ) g u g dx σ 2 = sup aψ div g (aψ), x σ = max { σ 0 k a, σ 2, sup σ 3 = sup q 0 div g q 0, x q 0 }, σ 5 = sup ϕh. R σ 2 uu t dx. (2.1) For the secod term, the sixth term ad the last term o the right of (2.1), we use Youg s iequality to obtai σ σ(t)ρ(x, u t )udx R (Cε σ(t)ρ(x, σ ut) 2 + εu2)dx R (2.15) σ C ε σ(t)ρ(x, u t ) 2 + 2εC E(t), R σ 5 g u g udx σ 5 ε g u R 2 g dx + C ε σ 5 u 2 dx, (2.16) 2εC 5 E(t) + C ε σ 5 u 2 dx, ad σ 5 σ(t) ρ(x, u t ) g u g dx R εσ 5 g u R 2 g dx + C ε σ 5 σ(t)ρ(x, u t ) 2 dx R 2εC 5 E(t) + C ε σ 5 R σ(t)ρ(x, u t ) 2 dx (2.17)

8 8 J. WU, F. FENG,. CHAI EJDE-2015/226 where ε > 0 is small ad C ε is a costat depedet o ε. Let φ be a fuctio satisfyig the coditios i Lemma 2.1. Let r 0 be a costat determied later. Multiplyig (2.1) by E r φ ad itegratig o [, T ] with respect to t, from (2.15) -(2.17) we have T E r+1 (t)φ σdt T T E r φ X (t)dt + (σ C ε + σ 5 C ε ) T + (2εσ + εσ 5 ) E r+1 φ dt k 2 + ( kσ T 0σ 2 + σ 3 + C ε σ 5 ) E r φ + k T E r φ Ω 1 σ(t)a(x)u 2 t dx dt. T E r φ R σ(t)ρ(x, u t ) 2 dx dt E r φ E dt u 2 dx dt (2.18) Usig a similar argumet as i [13] ad [7], we ca prove that if T is large eough, the T T E r φ u 2 dx dt C η E r φ ( σ(t)ρ(x, u t ) 2 dx R ) T + σ(t)u 2 t dx dt + η E r+1 (t)φ dt Ω 1 (2.19) holds for ay η > 0. For the rest of this article, let φ(t) = t σ(s)ds. It is clear that φ(t) satisfies the 0 coditios of Lemma 2.1. Noticig that X(t) CE(t) ad E (t) 0, we have by itegratig by parts with respect to t, T E r φ X (t)dt = E r (t)φ (t)x(t) T + T (E r φ ) X(t)dt T Cσ(0)E r+1 () + re r 1 E φ X(t) dt + Cσ(0)E r+1 () + Cσ(0) CE r+1 (). T re r ( E )dt + T T E r φ X(t) dt E r+1 ( φ )dt (2.20) We also have k 2 T E r φ E dt CE r+1 (). (2.21) Now we estimate the secod term o the right of (2.18). et Ω + 1 = {x Ω 1; u t 1} ad = {x Ω 1; u t < 1}. The Ω 1 = Ω + 1 Ω 1. Noticig (1.5), usig Hölder

9 EJDE-2015/226 ENERGY DECAY 9 iequality ad Youg s iequality, we have T E r φ σ(t)ρ(x, u t ) 2 dx dt T = E r φ T c 2 E r φ σ 2 (t) T = c 2 E r φ C a 2 2 T σ 2 (t) a(x)h(u t ) 2 dx dt a 2 (x) ( h(u t )u t ) 2 dx dt ( σ(t)a(x)h(ut )u t ) 2 ( a(x)σ(t) ) 2 2 dx dt E r φ σ(t) 2 2 T C a 2 2 σ 2 (0) [ε 1 T + C ε1 ( ( E r φ σ(t) ) 2 m 1 dt ] σ(t)a(x)h(u t )u t dx dt. σ(t)a(x)h(u t )u t ) dx ) 2 dt Now, we set r = m 1 2. Thus, we have from (2.22) ad (2.1) T E r φ σ(t)ρ(x, u t ) 2 dx dt T C a 2 2 σ 2 (0) [ε 1 E 2 φ dt + C ε1 E() ]. Noticig (1.6) ad (2.1), it is easy to obtai T T E r φ σ(t)ρ(x, u t ) 2 dx dt c σ(0) a E r φ ( E (t))dt Ω + 1 O the other had, it is easy to obtai T sice ρ(x, u t ) = a(x)u t whe x R \ Ω 1. From (2.23)-(2.25), we have T CE r+1 (). (2.22) (2.23) (2.2) E r φ R \Ω 1 σ(t)ρ(x, u t ) 2 dx dt CE r+1 () (2.25) E r φ R σ(t)ρ(x, u t ) 2 dx dt Cε 1 a 2 2 σ 2 (0) T E 2 φ dt + C ε1 E() + CE r+1 (). imilarly, for the last term o the right side of (2.18), we have T E r φ Ω 1 σ(t)a(x)u 2 t dx dt Cε 1 a m 1 T σ(0) E 2 φ dt + C ε1 E() + CE r+1 (). (2.26) (2.27)

10 10 J. WU, F. FENG,. CHAI EJDE-2015/226 At this poit, we choose ε ad η small eough so that 2εσ + εσ 5 + η( kσ 0σ 2 + σ 3 + C ε σ 5 ) < σ 2. The usig (2.19)-(2.21), (2.18) becomes σ 2 T E r+1 (t)φ dt CE r+1 () + ( σ C ε + σ 5 C ε + C η ( kσ 0σ 2 + σ 3 + C ε σ 5 ) ) T E r φ σ(t)ρ(x, u t ) 2 dx dt k T (2.28) E r φ E dt R 2 + ( k + C η( kσ 0σ 2 + σ 3 + C ε σ 5 ) ) T E r φ σ(t)a(x)u Ω 2 t dx dt. 1 Oce ε ad η are fixed, we pick ε 1 small eough so that Cε 1 a m 1 ( σ(0) + σ 2 (0) ) (σ C ε + σ 5 C ε + k + C η( kσ 0σ 2 + σ 3 + C ε σ 5 ) ) σ. The usig (2.26)-(2.27) i (2.28), we have T E 2 φ dt CE(). (2.29) Let T i (2.29). The Theorem 1.1 follows from Lemma 2.1 ad (2.29) with r = m 1 2 ad φ(t) = t 0 σ(s)ds. Ackowlegmets. This work was supported by the Natioal ciece Foudatio of Chia for the Youth ( , ), by the Youth ciece Foudatio of haxi Provice ( ), by the Natioal ciece Foudatio of Chia ( , ad ). Refereces [1] A. Bchatia, M. Daoulatli; Local eergy decay for the wave equatio with a oliear timedepedet dampig, Applicable Aalysis, 92(11) (2013), [2] A. Beaissa,. Beazzouz; Eergy decay of solutios to the Cauchy problem for a odissipative wave equatio, Joural of Mathematical Physics 51, (2010) [3] M. Daoulatli; Rates of decay for the wave systems with time depedet dampig, Discrete ad Cotiuous Dyamical systems, 31(2) (2011), [] M. Daoulatli, B. Dehma, M. Kheissi; Local eergy decay for the elastic system with oliear dampig i a exterior domai, IAM J Cotrol. Optim., 8(8) (2010), [5] Jig Li, huge Chai, Jieqiog Wu; Eergy decay of the wave equatio with variable coefficiets ad a localized half-liear dissipatio i a exterior domai, Z. Agew. Math. Phys. 66 (2015) [6] P. Martiez; A ew method to obtai decay rate estimates for dissipative systems, EAIM Cotrol Optim. Calc. Var., (1999), 19-. [7] M. Nakao, I. H. Jug; Eergy decay for the wave equatio i exterior domais with some half-liear dissipatio, Differetial ad Itegral Equatios, 18(8) (2003), [8] M. Nakao; O the decay of solutios of the wave equatio with a local time-depedet oliear dissipatio, Advace i Mathematical ad Applicatios Gakkotosho, Tokyo, 7(1) (1997), [9] M. Nakao; Decay of solutios of the wave equatio with a local oliear dissipatio, Math. A. 305 (3) (1996), [10] M. Nakao; Decay of solutios to the Cauchy problem for the Klei-Gordo equatio with a localized oliear dissipatio, Hokkaido Mathematical Joural, 27 (1998)

11 EJDE-2015/226 ENERGY DECAY 11 [11] M. Nakao; tabilizatio of local eergy i a exterior domai for the wave equatio with a localized dissipatio, Joural of Differetial Equatios, 18 (1998), [12] M. Nakao; Eergy decay for the liear ad semiliear wave equatios i exterior domais with some localized dissipatios, Mathematische Zeitschrift, 238 (2001), [13] P. F. Yao; Eergy decay for the Cauchy problem of the liear wave equatio of variable coefficiets with dissipatio, Chi. A. of Math., ser. B, 31(1) (2010), [1] P. F. Yao; Observatility iequality for exact cotrollability of wave equatios with variable coefficiets, IAM J. Cotrol Optim. 37 (1999), [15] P. F. Yao; Modelig ad cotrol i vibratioal ad structrual dyamics, A differetial geometric approach, Chapma ad Hall/CRC Applied Mathematics ad Noliear ciece eries, CRC Press, [16] E. Zuazua; Expoetial decay for the semiliear wave equatio with locally distributed dampig, Comm. PDE. 15 (1990), [17] E. Zuazua; Expoetial decay for the semiliear wave equatio with localized dampig i ubouded domais, J. Math. Pure et Appl. 70 (1992), Jieqiog Wu (correspodig author) chool of Mathematical cieces, haxi Uiversity, Taiyua, haxi , Chia address: jieqiog@sxu.edu.c Fei Feg chool of Mathematical cieces, haxi Uiversity, Taiyua, haxi , Chia address: fegfei599@126.com huge Chai chool of Mathematical cieces, haxi Uiversity, Taiyua, haxi , Chia address: sgchai@sxu.edu.c