Initial-boundary value problem for second order degenerate pseudohyperbolic equations

Transcription

1 PROCEEDINGS OF TE YEREVAN STATE UNIVERSITY Physcal ad Mathematcal Sceces,, p 3 Mathematcs Ital-boudary value problem for secod order degeerate pseudohyperbolc equatos G S akobya, Savash Ghorbaa Char of Theory of Effectve Maagemet ad Approxmate Methods YSU, Armea Islamc Azad Uversty Froozkooh, Ira The paper studes a tal-boudary value problem for a class of secod order degeerate pseudohyperbolc equatos We prove the exstece ad uqueess of the problem the approprately costructed fuctoal space Keywords: pseudohyperbolc degeerate equato, weak soluto Itroducto We cosder the followg tal-boudary value problem of Sobolev type u L ( ), + M u = t ( () u = u ( x), u ( ), () t t = u x = t= u =, t >, where =Ω, L ad M are dfferetal operators to be perused later We are terested the case, whe the ellptc operator L ca be degeerato the part of the tal hyperplae We treat ths problem wth the help of costructo of the correspodg fuctoal space ad by establshg ts equvalece to the Cauchy problem for some operator equato Ths problem was cosdered for the frst tme by Sobolev, coecto wth the study of small oscllatos of a rotatg deal flud, the partcular case whe L =Δ s the three-dmesoal Laplace operator Later smlar problems were cosdered by RA Aleksadrya [], SA Galper [] ad others (see, eg,[3 5]) Let Ω be a bouded doma -dmesoal vector space R located the half-space x > We suppose that the boudary of the doma has the form E-mal: gurgeh@ysuam E-mal: savash_ghorbaa@yahoocom

2 Proc of the Yereva State Uv Phys ad Mathem Sc,,, p 3 Ω = where =Ω { = } s a doma the hyperplae { x = },, x =Ω \, ad for the doma Ω the Sobolev embeddg theorems are vald We cosder the followg tal-boudary value problem the cylder Q=Ω R + for the degeerate pseudohyperbolc equato u L ( ), () + M u = t ( () u = u ( x), u ( ), () t t = u x = t= u =, t >, (3) u u where Lu = bj ( x, t) b ( x, t), x j x x u u Mu = aj ( x, t) a ( x, t) x j x x We assume that the coeffcets of the operators L ad M are symmetrc: a (,) xt = a (,), xt b(,) xt = b (,)(, xt j=,,, ) cotuous ad bouded j j j j Ω, cotuously dfferetable wth respect to the varables x, x,, x Q=Ω R +, there exst expoets α >, such that the products α x b( x, t), x a( x, t) are bouded from above ad below by postve costats ad for every x Ω ad every t the quadratc form Λ ( xt,, ξ ) = bj ( xt, ) ξξ j R + s postve-defte, where ξ = ( ξ,, ξ ) R s a part of the boudary, whch depedg o the order of degeeracy, represets ether the whole boudary Ω or cocdes wth the I the space L ( Ω ) we defe the operator wth the doma of defto C ( Ω ) by the formula ( ) u u L u = x = x x x It follows from the results of [5], that the operator s symmetrc ad postve-defte Defe the lbert space as the completo of the lear mafold C ( Ω ) the metrc geerated by the followg scalar product u v u v ( uv, ) = ( L ( u), v ) = x + dx (4) Ω = x x x x Let T >, Q= Ω (, T) be a cylder wth the base Ω, Σ = (, T ) be the lateral boudary of the cylder Q

3 Proc of the Yereva State Uv Phys ad Mathem Sc,,, p 3 Defto Twce dfferetable trajectory ut ( ) s called a weak or ( geeralzed soluto of the problem () (3), f u = t= u ( x), u () t = u ( x) ad t= for every v C ( Ω ) ad every t > Ω du v v b (, ) (, ) j x t + b x t dx+ dt x j x x j j v v + u aj ( x, t) + a ( x, t) dx= Ω x j x x Theorem For ay tal values ( () u L ad u L there exsts a uque geeralzed soluto of the problem () (3) Where for the case < we have =Ω ad = =Ω\ for Proof Let t [, ) be a fxed umber I the space L ( Ω ) we defe the operator Lt ( wth the doma of defto C ( Ω ) by the formula u v v Lt ( v= bj ( xt, b ( xt, x j x x Sce the quadratc form Λ ( xt,, ξ ) = b( xt, ) ξξ + b ( xt, ) ξ s postvedefte for every x Ω\ ad Λ ( xt,, ξ ) =Λ ( xt,, ξ) + b ( xt, ) ξ Λ ( xt,, ξ) c ξ, we coclude that the operator Lt ( s symmetrc It s easy to verfy that the operator Lt ( s postve-defte L ( Ω ) Мoreover, sce the quadratc form ( xt,, Λ ξ ) s postve-defte, the product x b ( x, t ) ad the fuctos bj ( x, t,, j =,,,, are bouded, we deduce that there exst costats c ad C, such that for every v C ( Ω ) we have v v c( L( v), v) = c + x dx Ω = x x v v v bj ( x, t + b ( x, t dx C ( L( v), v) Ω = x xj x O the lear mafold C ( Ω ) we defe the operator At ( by the formula At ( : v L ( t M( t v, where Lt ( s the Fredrch s exteso of Lt ( It s easy to check that At ( s bouded the space Ideed, let v ( x), v ( x) C ( Ω ) are arbtrary fuctos L (5)

4 Proc of the Yereva State Uv Phys ad Mathem Sc,,, p 3 3 The we have ( At ( ) v, v ) ( v, At ( ) v ) Lt ( ) =, e At ( s symmetrc the space Lt ( Lt ( Moreover, there s a costat c > such that for every v C ( Ω ) we have v v v ( At ( vv, ) = (, ( (, Lt aj xt + a xt dx Ω j, = x xj x (6) v v v v α v aj ( x, t + a ( x, t dx c + x dx Ω j, = x x x j Ω = x x α Sce α, we obta x cx cb ( xt, ), where c ad c are some postve costats Cosequetly, from equalty (6) we coclude that there exsts a costat c > such that v α v At vv = x Lt ( ) + dx Ω = x x ( ( ), ) v v v c bj ( x, t + b ( x, t dx= c( L( t) v, v) = c v L( t ), Ω x xj x e, the operator At ( s bouded the space A( t ) From the equalty (5) we deduce that At ( s bouded the space We exted the operator At ( by cotuty from the lear mafold C ( Ω ) to the whole space The exteso wll be deoted by At ( I the lbert space we cosder the followg auxlary Cauchy problem du = At ( u, dt (7) ( () ut= = u, ut t= = u It s easy to see, that every soluto of the problem (7) s a weak soluto for the problem () (3) ad vce-versa The boudedess of the operator At () the space mples that the problem (7) has a uque soluto Thus, the Theorem s proved REFERENCES Receved Aleksadrya RA // Trudy MMO, 96, v 9, p ( Russa) Galper SA // Trudy MMO, 96, v 9, p 4 43 ( Russa) 3 Aleksadrya RA, Berezask JuM, Il VA, Kostjucheko AG // Math Soc Trasl, 976, v 5(), p 53 4 Gaevsk Kh, Greger K, Zakharas K Nolear Operator Equatos ad Operator Dfferetal Equatos M: Mr, 978 ( Russa) 5 akobya GS // Uhcee Zapsk EGU, 986,, p 8 86 ( Russa)

5 4 Proc of the Yereva State Uv Phys ad Mathem Sc,,, p 3 ԳՍ Հակոբյան, Սիավաշ Գորբանիան Սկզբնկանան-եզրային խնդիր երկրորդ կարգի վերասերումով պսևդոհիպերբոլական հավասարումների համար Աշխատանքում ուսումնասիրվում է երկրորդ կարգի վերասերումով պսևդոհիպերբոլական հավասարումների մի դասի համար սկզբնական-եզրային խնդիր: Ապացուցվում է այդ խնդրի լուծման գոյությունը և միակությունը համապատասխան ֆունկցիոնալ տարածությունում: ГС Акопян, Сиаваш Горбаниан Начально-краевая задача для вырождающегося псевдогиперболического уравнения второго порядка В работе исследуется начально-краевая задача для одного класса псевдогиперболических уравнений второго порядка с вырождением Доказывается существование и единственность решения этой задачи в построенном соответствующим образом функциональном пространстве