Lie Group Classifications and Stability of Exact Solutions for Multidimensional Landau-Lifshitz Equations

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1 Applied Mahemaics Published Online April 06 in SciRes hp://wwwscirporg/journal/am hp://dxdoiorg/06/am ie Group Classificaions and Sabiliy of Exac Soluions for Mulidimensional andau-ifshi Equaions Jiali Yu Fuhi i Hui Yang Ganshan Yang * School of Mahemaics Yunnan Normal Universiy Kunming China Received 5 February 06; acceped 5 April 06; published 8 April 06 Copyrigh 06 by auhors and Scienific Research Publishing Inc This work is licensed under he Creaive Commons Aribuion Inernaional icense (CC BY hp://creaivecommonsorg/licenses/by/0/ Absrac In his paper based on classical ie group mehod we sudy muli-dimensional andau-ifshi equaion and ge is infiniesimal generaor symmery group and new soluions In paricular we build he connecion beween new exac soluions and old exac soluions A he same ime we also prove ha he iniial boundary value condiion of he hree-dimensional andau-ifshi equaion admis a unique soluion and discuss he sabiliy of he soluion Keywords ie Group Mulidimensional andau-ifshi Equaion Explici Soluions Sabiliy Inroducion In 95 he famous andau-ifshi equaions were proposed by andau and ifshi [] o describe he evoluion of spin fields in coninuum ferromagne [] In his paper we sudy wo imporan equaions as follows u = u u ( u = λu u+ λu u u ( n + where denoes he vecor cross-produc in u= ( uvw : is he spin densiy λ < 0 is a damping parameer Emphasiing is parabolic characer ( can also be considered as a quasilinear perurbaion of he hea flow for harmonic maps by he (conservaive precession erm u u The n-dimensional cylindrical symmerical form of ( is * Corresponding auhor How o cie his paper: Yu J i FZ Yang H and Yang GS (06 ie Group Classificaions and Sabiliy of Exac Soluions for Mulidimensional andau-ifshi Equaions Applied Mahemaics hp://dxdoiorg/06/am067706

2 J Yu e al where r = x + x + + x n n u = u urr + u u r ( r For he mulidimensional case Zhou and Guo proved he global exisence of weak soluion for he generalied andau-ifshi equaions a absence of Gilber erm [] Chang e al considered he iniial value problem for he -dimensional cylindrical symmeric andau-ifshi equaion wihou exernal magneic field [] The solion soluions o he andau-ifshi equaions wih and wihou exernal magneic field have been sudied by many physiciss and mahemaicians [5]-[7] For he Equaion ( when n = Guo and Yang have consruced an exac soluion in uni sphere [8] In [9] and [0] Guo and Han as well as Yang have also obained an exac blow up soluion for he n-dimensions form In [] and [] Yang considered he relaions beween ( and ( I is of grea imporance o find exac soluions of andau-ifshi equaions Bu i is difficul o solve andau-ifshi equaions As is known he symmery group echnique is one of he powerful ools for solving a nonlinear differenial equaion (see []-[]: he classical ie group mehod [5] [6] he non-classical ie group mehod [7] [8] Xu and iu have sudied n-dimensional radial symmeric andau-ifshi equaion wih exernal magneic field in [9] In his paper he symmery group of he n-dimensional andau-ifshi equaion is obained by using he classical mehod in Secion The ransformaions leave he soluions invarian In Secion we give he new soluions of andau-ifshi equaion from he known soluions [0] Finally he uniqueness and sabiliy of he andau-ifshi equaion and he andau-ifshi-gilber equaion are given [0] respecively in Secion and 5 ie Symmery Group of he andau-ifshi Equaion Here are four independen variables x ( xy dependen variables he velociy field u ( uvw u = v w w v v = w u u w w = u v v u = being spaial coordinaes and he ime ogeher wih four = In vecor noaion he sysem has he form According o he mehod of deermining he infiniesimal generaor of nonlinear parial differenial equaion [6] we ake he infiniesimal generaor of equaion as follows: v = ξ + η + ς + τ + ϕς + ψ + χ x y u v w where ξ χ are funcions of xu ( is of second order n = Applying he firs prolongaion pr v we find ( x y x y pr v = v+ ϕ + ϕ + ϕ + ϕ + ψ + ψ ux uy u u vx vy x y + ψ + ψ + χ + χ + χ + χ v v w w w w Applying x y xx yy xx yy pr v = pr v + φ + φ + φ + φ + ψ + ψ + ψ u u u u v v v xx yy x y + ψ + χ + χ + χ + χ + φ + φ v wxx wyy w w ux uy x y x y + φ + ψ + ψ + ψ + χ + χ + χ u v v v w w w xx yy xx yy x y x y ( pr v o ( we find he following sysem of symmery equaions ( 666

3 ( xx yy ( xx yy ( xx yy ( xx yy ( xx yy ( xx yy xx yy xx yy φ = v χ + χ + χ w ψ + ψ + ψ + w + w + w ψ v + v + v χ xx yy xx yy ψ = w φ + φ + φ u χ + χ + χ + u + u + u χ w + w + w φ xx yy xx yy χ = u ψ + ψ + ψ v φ + φ + φ + v + v + v φ u + u + u ψ J Yu e al x which mus be saisfied whenever u saisfy ( Here φ ψ ec are he coefficiens of he firs order derivaives ec appearing in pr v ux vx J ( n i α i α According o he formula φα ( xu = DJ( φα Σ ξ ui +Σ ξ uji we have xx φ = D φ u D ξ u D η u D ς u D τ u D ξ u Dη u D ς u Dτ x x x y x x x xx x xy x x x x x yy xx yy xx yy Similarly we can ge φ ψ χ φ φ ψ ψ ψ χ χ χ we find he deermining equaions for he symmery group of he ( Equaion (5 o be he following: w v χxx + χyy + χ ψ xx + ψ yy + ψ φ = 0 w( τ ξx + χ = 0 v( τ ξx + ψ = 0 u( τ ξx + φ = 0 v( χwx ξxx ξyy ξ = 0 v( χwy ηxx ηyy η = 0 v( χw ςxx ς yy ς = 0 ηx = ξy ςx = ξ ς y = η Since we have now saisfied all he deermining equaions we conclude ha mos general infiniesimal symmery of ( has coefficien funcions of he form: ξ = cx + c y + c + c 5 η = cx + cy + c + c 6 ς = cx c y + c + c 7 τ = c+ c 8 φ = cu ψ = cv χ = cw where c c8 are arbirary consans Thus he ie-algebra of infiniesimal of he andau-ifshi equaion is spanned by eigh vecor fields: so we have The one-parameer groups v = x + y u + v + w x y u v w v = y x v = x x x v = y 5 x v = 6 x v = 7 y v = 8 v = v= cv + c v + + c v 8 8 G i generaed by he v i The enries give he ransformed poin y (5 667

4 J Yu e al exp v i xyuvw = x y uv w : G : e x e y e e e u e v e w ( + G : x yy xuvw ( + G : x y xuvw ( + G : xy yuvw ( + G : x yuvw 5 ( + G : 6 xy uvw ( + G : 7 xy uvw ( + G : 8 xy uvw where G is a Galiean ransformaion G G G G5 G6 G 7 are space ranslaions G 8 is a ime ranslaion is an arbirary consan Theorem If u= f( xy v= g( xy w= h( xy are known soluions of ( hen by using G i = 8 so are he funcions he symmery groups i ( ( ( u = e f e xe ye e v = e g e xe ye e w = e h e xe ye e x y x+ y x y x+ y x y x+ y u = f v = g w = h x x+ x x+ x x+ u = f y v = g y w = h y y y+ y y+ y y+ u = f x v = g x w = h x ( ( ( u = f x y v = g x y w = h x y ( ( ( u = f xy v = f xy w = f xy ( ( ( u = f xy v = f xy w = f xy ( ( ( u = f xy v = f xy w = f xy where is any real number For he known soluions u= f( xy v= g( xy w= h( xy by using one-parameer symmery groups Gi ( i = 8 coninuously we can obain a new soluion which can be expressed as he following form: u = f + x y e e e + y+ x e + + x+ y 7 e

5 J Yu e al where ( i 8 v = g + x y e e e + y+ x e + + x+ y 7 e w= h + x y i = are arbirary consans In vecor noaion he sysem has he form when λ λ λ( λ 0 = = > in ( In view of he vecor ideniies Equaion (6 can equivalenly be wrien as We ransformed equaions as follows: Applying e e y+ x e e + + x+ y 7 e u = u u λu u u (6 = u u u u u u u u = λ u u u u u u u u ( ( ( u = w v v w λ uv v + uw w v u w u v = u w w u λ uv u+ vw w u v w v w = v u u v λ uw u+ vw v u w v w ( pr v o (8 we find he following sysem of symmery equaions w v v w xx yy xx yy λ φv v + ψu v + uv ( ψ + ψ + ψ + φw w + χu w + uw( χ + χ + χ xx yy xx yy ψv u v ( φ + φ + φ χw u w ( φ + φ + φ u w w u xx yy λ xx yy φv u+ ψu u+ uv( φ + φ + φ + χv w+ ψw w + wv ( χ + χ + χ xx yy xx yy φu v u ( ψ + ψ + ψ χw v w ( ψ + ψ + ψ v u u v xx yy xx yy λ φw u+ χu u+ uw( φ + φ + φ + ψw v+ χv + vw( ψ + ψ + ψ xx yy xx yy φu w u ( χ + χ + χ ψv w v ( χ + χ + χ = xx yy xx yy φ ψ ψ ψ χ χ χ χ ψ = xx yy xx yy ψ χ χ χ φ φ φ φ χ = xx yy xx yy χ φ φ φ ψ ψ ψ ψ φ (7 (8 (9 669

6 J Yu e al Then use he same mehod we can find mos generaed infiniesimal symmery of (6 has coefficien funcions of he form: ξ = c η = c ς = c τ = c φ = c 5 ψ = c 6 χ = c 7 where c c7 are arbirary consans Thus he ie-algebra of infiniesimal of he andau-ifshi-gilber equaion equaion is spanned by seven vecor fields: So we have v = x v = y v = v = v = 5 6 u v = 7 v v = v = cv + c v + c v + c v + c v + c v + c v where G G G are space ransformaions G is a space ranslaion G5 G6 G 7 are Galiean ranslaions is an arbirary consan Theorem If u= f( xy v= g( xy w= h( xy are known soluions of (6 hen by using G i = 7 so are he funcions he symmery groups i u = f( x y v = g( x y w= h( x y w ( ( ( u = f xy v = g xy w = h xy ( ( ( u = f xy v = g xy w= hxy ( ( ( u = f xy v = g xy w = h xy u = f xy + v = g xy w = h xy u = f xy v = g xy + w = h xy u = f xy v = g xy w = h xy where is any real number For he known soluions u= f( xy v= g( xy w= h( xy by using one-parameer symmery groups Gi ( i = 7 coninuously we can obain a new soluion which can be expressed as he following form: u = f x y + 5 v = g x y

7 J Yu e al where i ( i 7 Remark If u ( uvw w= h x y + 7 = are arbirary consans = is a known soluion we can ge v = Au is a new soluion hrough ie group mehod where A is arbirary consan orhogonal marices Exac Soluions of he andau-ifshi Equaion In his secion we choose he known blow-up soluions and explici dynamic spherical cone symmeric soluions from [0] and [] o ge he relevan group invarian soluions According o [0] has a blow-up soluion: where u C [ 0 T + u = u uu x+ y+ x+ y+ u ( x 0 = k( x+ y+ k cos k sin C ( kt kt = ( + + u xy k x y x+ y+ v( xy = k cos k ( T x+ y+ w( xy = k sin k ( T Case Using u in Theorem we ge he new blow-up soluions of ( as follows: where u C [ 0 T ( e ( e ( x+ y+ = ke cos k ( T ( e ( x+ y+ = k k ( T u xy = k x+ y+ v xy w xy e sin saisfying he following iniial value: ( 0 e = k e ( x+ y+ u xy = k x+ y+ v xy 0 e cos kt e ( x+ y+ wxy ( 0 k = e sin kt Case Using u in Theorem we ge he new blow-up soluions of ( as follows: (0 ( ( ( where u C [ 0 T + + u( xy = k x+ y x+ y+ v( xy k cos + + = k ( T + x+ y+ w( xy = k sin + + k ( T saisfying he following iniial value: ( 67

8 J Yu e al for According o [0] + + u( xy 0 = k x+ y x+ y+ v( xy 0 k cos + + = kt + x+ y+ wxy ( 0 = k sin + + kt n u = u urr + u u r r n+ n+ n r r u ( x 0 = r k k cos k sin k( n T k( n T + + n u u k ku u ( k k r= = + r= = + r = x + y + has a blow-up soluion: n ( r r ( k k θ k θ u = cos sin n+ r θ = k ( n+ ( T Case Using u in Theorem we ge he new blow-up soluions for he cylindrical symmeric andau- ifshi Equaion ( ypically n = as follows: for ( r ( kr kr θ kr θ u = e cos sin 5 5 e r θ = 5k ( T e r = x + y + saisfying he following iniial value: According o [] e r e r = r k k k 5kT 5kT u x 0 e cos sin 6 u u e ( k k u u e ( k k r= = + r= = + ( T u = u u 0 c ( x y + ( x + ( y u( x0 ( cos θ0sin θ00 θ = 0 = + c has he explici dynamic spherical cone symmeric soluions: u = ( cos θsin θ c + c + ( c + c c ( x y + ( x + ( y θ = + ( c + c Case Using u in Theorem we ge he new explici dynamic spherical cone symmeric soluions of ( (5 (6 (7 (8 (9 (0 ( 67

9 J Yu e al as follows: saisfying he following iniial value: u = ( cos θ sin θ ce + c + ( ce + c ce ( x y + ( x + ( y θ = + ( ce + c ( θ θ u x0 = cos sin ce ( x y + ( x + ( y θ 0 = + c Case 5 Using u in Theorem we ge he new explici dynamic spherical cone symmeric soluions of ( as follows: u = ( cos θsin θ c + c + ( c + c c ( x y ( x y ( x ( y x ( y ( xy θ = ( + + ( c + c saisfying he following iniial value: ( θ θ u x0 = e cos sin ce ( x y + ( x + ( y θ 0 = + c Remark Similarly we can uilie he differen seed soluions of [0] [] repeaedly using u ( 8 o obain differen group invarian soluions so exend he known exac soluions in [9] [] Uniqueness and Sabiliy of he andau-ifshi Equaion i i = In his secion we sudy he uniqueness and sabiliy of he iniial boundary value problem for ( and we have he following resuls and i should be resuls ha maer insead Uniqueness Theorem There exiss following iniial value problem: is a bounded domain and j ( e u( x ( on [ 0 T has a unique smooh blow-up soluion u C [ 0 T ( ( ( (5 k j = are nonero consans Then we he + u = u u u e ( x+ y+ e ( x+ y+ u( x 0 = ke ( x+ y+ ke cos ke sin kt kt x+ y+ e ( x+ y+ = ke x+ y+ ke cos ke sin k T k( T Proof To prove he uniqueness we consider wo smooh soluions uu C ( 0 < T (6 e heir 67

10 J Yu e al difference be ρ = each oher in ( we have u u where ρ = ( ρ ρ ρ = ( uvw = ( uvw Muliplying he firs equaion of (7 by obain u u Then subracing he equaions ( T ρ = ρ u+ u ρin 0 ρ ( x0 = ( 00 in ρ = ( 00 (7 ρ inegraing over and using he Gauss formula [] we d ρ ρ = ρ (8 d ( ρ ρ = ( ρ + ρ ρ = ρ ( ρ d for u = ( u v w u = ( u v w ρ = ( ρ ρ ρ By using u( xy ke ( x y e ( x+ y+ e ( x+ y+ v( xy = ke cos w( xy = ke sin in case we obain k ( T k ( T u u u x (9 = + + where e θ = ( x+ y+ k ( T ke ke ke k k k = sinθ sinθ sinθ k( T k( T k( T k k k cosθ cosθ cos θ k( T k( T k( T u (0 Insering ( ino (9 i follows ha k k u = max ke sin θ cosθ k( T k( T e k = sup c ( k ( T d d ( u u ρ = ρ + ρ ρ ( ρ c ρ u Thanks o he Gronwall inequaliy [] we have he following: ( c ( x 0 ρ exp τ d τ ρ 0 = 0 herefore we can prove he uniqueness of he soluion in he sense of C [ 0 T In a similar way by using in case we obain ( r ( kr kr θ kr θ u = e cos sin 5 5 e r θ = 5k ( T e ( ( 67

11 J Yu e al k e xr k e yr k e r = M M M u ( N N N which e r 5 e kr x k θ = M = e kr xcosθ + sin θ N = cosθ 5k T e ( 5k ( T k ( T u = sup e 5kr + 0kr x+ y+ + 05kr x+ y+ ( k ( 8 6r x y r x y r x y k T c Insering ( ino (9 i follows ha d ( ρ ρ c ρ d Thanks o he Gronwall inequaliy we have he following: ( u (5 ( c 0 x ρ exp τ d τ ρ 0 = 0 herefore we can ge he uniqueness of he soluion from his Theorem There exiss is a bounded domain Then we he following iniial boundary value problem: ( T u = u u 0 ce ( x y + ( x + ( y u( x0 = ( cos θ0sin θ00 θ0 = + c u( x = ( cos θsin θ ce + c on ( 0 T (6 + ( ce + c ce ( x y + ( x + ( y θ = + ( ce + c has a unique explici dynamic spherical cone symmeric soluion ([ 0 ; ( [ 0 ; ( u C T C C T H Proof To prove he uniqueness we consider wo soluions [ ( e heir difference be ρ = equaions each oher in ( we have Muliplying he firs equaion of (7 by ([ ( uu C 0 T ; C C 0 T ; H u u where ρ = ( ρ ρ ρ = ( uvw = ( uvw u u Then subracing he ( T ρ = ρ u+ u ρin 0 ρ ( x0 = ( 00 in ρ = ( 00 ρ inegraing over and using he Gauss formula we obain (7 675

12 J Yu e al d ρ ρ = ρ (8 d ( ρ ρ = ( ρ + ρ ρ = ρ ( ρ d for u = ( u v w u = ( u v w ρ = ( ρ ρ ρ ce ( x y + ( x + ( y By using we obain where M u = ( cos θsin θ ce + c θ = + e + ( c c u u u x (9 ( c c + e + ( ( ( in case M x y M y x M x y u = N x y N y x N x y ( e c sinθ e c cosθ = = 6 e 6 e N ( + ( c + c + ( c + c 5 e c u = sup 6 ( x y 9( xy x y x yx y 6xy e c ( ( + ( c + c Insering ( ino (9 i follows ha d d ( u u ρ = ρ + ρ ρ c ( ( u ρ ( ρ Thanks o he Gronwall inequaliy we have he following: ( c ( x 0 ρ exp τ d τ ρ 0 = 0 herefore we prove uniqueness of he soluion in he sense of [ Sabiliy In his secion we discuss he sabiliy of he soluion in C ( and ively ( 0 ; ([ 0 ; C T C C T H for he problem ( respec- e u = ( uvw is a soluion of ( where u C H [ 0 T from case u ( uvw he soluion of a lile disurbance where u C H ( [ 0 T e u u u and u wih iniial value ρ0 ( x 0 in he sense of ( and boundary value u u ( ( ( 0 T] Then subracing one equaion from he oher we can ge ρ = ρ u + u ρin ( 0 T ρ( x0 = ρ0 ( x in ρ = ϕ( x for is a bounded domain Muliplying his by = denoe ρ = be he difference of in he sense of ( ρ inegraing over and using he Gauss formula 676

13 J Yu e al we obain d = ( d ( ρ ρ ρ ( ρ ρ = ( ρ + ρ ρ = ρ ρ = ρ ( ρ n + ρ ( ρ u u u dx u ds u dx (5 By using u( xy = ke ( x+ y+ v( xy e w( xy = ke sin ns ( ( ( ρ ρ u d u ρ ρ 0 (6 ( x+ y+ k ( T u ( ( ρ ρ u dx ρ (7 e = ke cos in case we obain ( x+ y+ k ( T k k u = max ke sin θ cosθ k( T k( T e k = sup c ( k T ( (8 Hence d ρ c ρ ρ ρ d + u (9 ( ( ( ( since u u in he sense of ( ( we can make for every given > 0 u ρ ρ (50 ( ( ( Using he Gronwall inequaliy in (-(7 for every ( 0 T : ρ exp ( d ( x 0 0 exp ( d c τ τ ρ s c τ τ d s 0 as u u + ( ( ( 0 ] and T In a similar way by using in case and by using in case we can ge he same conclusions in he sense of ρ0 x0 0 So we reach he sabiliy of he soluion in finie ime ( r ( kr kr θ kr θ u = e cos sin 5 5 e r θ = 5k ( T e u = ( cos θ sin θ ce + c + ( ce + c ce ( x y + ( x + ( y θ = + ( ce + c 677

14 J Yu e al 5 Uniqueness and Sabiliy of he andau-ifshi-gilber Equaion = in direcor fields where ypically n = we find he Gilber damping erm λu ( u u = λ( u+ u u i would be easier and he sabiliy of i has been done We can see i in [0] In his secion we sudy he uniqueness and sabiliy of he iniial boundary value problem for he andau-ifshi-gilber equaion below: Because of he andau-ifshi-gilber equaion u u u λu ( u u n u = ( 0 wih values in he uni sphere observe ha ( u + u = u u λu u u u (5 u u u = u u u u and we have he following resuls and i should be resuls ha maer insead 5 Uniqueness Theorem 5 There exiss is a bounded domain Then we he following iniial value problem: + v = v v λv v v v v0 = Av0 = Au( xy 0 H( (5 v = Au( xy on [ 0 T has a unique smooh soluion v = Au H [ 0 T Proof e heir difference be ρ = v v where ρ = ( ρ ρ ρ = A = Auvw ( H [ 0 T v = Au = Auvw ( H( [ 0 T Then subracing he equaions each oher in (5 we have ρ = v v+ v v λ v ( v v v ( v v ( 0 T ρ ( x0 = ( 00 in ρ = ( 00 n we obain ha if vv 0 T ; (5 by ρ inegraing over and using he Gauss formula we obain As d d vv are known soluions we can ge ρ ρ λ v ρ v v ρ ( ( ( ( ( ( ( ( λ c( c( ρ ( v u ( v ( (5 Muliplying he firs equaion of = v v+ v ρ v ρ + λ v v v + v v ρ v v v ρdx = ρ v ρ + λ v ρ v + v v ρ + ρ v v ρdx ρ v ρ + λ v ρ v + ρ v v ρdx v ρ v v ρ = ρ v ρ v v ρ = ρ v v ρ 0 for = A = A( u v w = A = A( u v w ρ = ( ρ ρ ρ v u v u Thanks o he Gronwall inequaliy we have he following: ( c ( c ( x 0 ρ exp τ + λ + τ d τ ρ 0 = 0 herefore we can prove he uniqueness of he soluion in he sense of 5 Sabiliy H 0 T In his secion we discuss he sabiliy of he soluion in C ( and (5 for he problem (5 respec- 678

15 J Yu e al ively Assume v = Au = Auvw ( where v H ( 0 T e v Av Auvw ( a lile disurbance where v H ( ( 0 T e v v ρ0 ( x 0 in he sense of ( and boundary value v v in he sense of ( ( ( 0 T] Then subracing one equaion from he oher we can ge ρ = + λ ( ( ( T ρ ( x0 = ( 00 in ρ = ϕ( x for is a bounded domain Muliplying his by we obain d d ρ = = denoe he soluion of ρ = be differen of v v wih iniial value v v v v v v v v v v 0 (55 ρ inegraing over and using he Gauss formula ( ( v ( = v v+ v ρ v v ρ + λ v v + v v ρ v v v ρdx = ρ v ρ + λ v ρ v + v v ρ + ρ v v ρdx ρ v ρ + λ v ρ v + ρ v v ρdx v ρ λ v ρ v v ρ ( ( ( ( ( ( ( ( v v v v v ( ( ( ( ρ ρ ( ( ( c( c( ( + λ + ρ + since v v in he sense of ( ( we can make for every given > 0 : ( v v v v v ( ( ( ( ( ( ( + + ρ ρ = = = = = Thanks o he Gronwall inequaliy we have he following: where v A u A( u v w v A u A( u v w ρ ( ρ ρ ρ ( c ( c 0 s ( c ( c 0 0 s exp d x 0 ρ τ + λ + τ τ ρ + exp τ + λ + τ dτ d 0 herefore we prove he sabiliy of he soluion in he sense of H ( 0 T 6 Conclusion In his paper we sudy he symmery reducions and explici soluions by means of classical ie group mehod Firs we ge he infiniesimal generaor and group invarian soluions o mulidimensional andau-ifshi equaion Then we build he relaions beween new soluions and olds have been found Finally via hese explici soluionswe sudy he uniqueness and sabiliy of iniial-boundary problem on mulidimensional andau- ifshi equaion Acknowledgemens This work was suppored by he Naural Foundaion of China (No No 056 References [] andau D and ifshi EM (95 On he Theory of he Dispersion of Magneic Permeabiliy Inferroagneic Bodies Phys Z Sowj (Reproduced in Colleced Papers of andau D Pergamon Press New York 965 pp 0- (56 679

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