An Integral Two Space-Variables Condition for Parabolic Equations
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1 Jornl of Mhemics nd Sisics 8 (): 85-9, ISSN Science Pblicions An Inegrl Two Spce-Vribles Condiion for Prbolic Eqions Mrhone, A.L. nd F. Lkhl Deprmen of Mhemics, Lborory Eqions Differenielles, Fcly of Ec Ciences, Universiy Menori Consnine, Algeri Deprmen of Mhemics, Universiy 8 Mi 45, P.O. Bo 4, Gelm 4, Algeri Absrc: In his sdy, n inegrl wo spce-vribles condiion for clss of prbolic eqions. The eisence nd niqeness of he solion in he fncionl weighed Sobolev spce were proved. The proof is bsed on wo-sided priori esimes nd on he densiy of he rnge of he operor genered by he considered problem. Key words: Inegrl bondry wo spce-vribles condiion, energy ineqliies, weighed sobolev spces INTRODUCTION In he domine = {(, ) (, ) (,T),T>}, we consider he eqion: L = ((, ) ) = f (, ) () where he fncion (, ) nd is derivive re bonded on he inervl [. T]: < < (, ) < (, ) 3 To Eq. nd we dd he iniil condiions: l = (,) = ϕ(), (,) () The bondry condiion Eq. 3: (, ) = (, ) (,T) (3) And inegrl condiion Eq. 4: ( ξ,)dξ + ( ξ,)dξ = >, >, < + = (, ) (4) Here, we ssmed h he known fncion ϕ sisfy he condiions given in (3) nd (4), i.e., ϕ () = ϕ(), ϕ ()d + ϕ ()d = When considering he clssicl solion of he problem ()-(4), long wih he condiion (4) shold be flfilled he condiions: f (,) f (,) = (7) nd Mrhone nd Lkhl (9)). Corresponding Ahor: Mrhone, A.L., Lborory Eqions Differenielles, Deprmen of Mhemics, Fcly of Ec Ciences, Universiy Menori Consnine, Algeri 85 α { (,) ϕ '() + (,) ϕ "()}d + { (,) ϕ '() + (,) ϕ "()}d = f (,)d + f (f (,)d Mhemicl modeling of differen phenomen leds o problems wih nonlocl or inegrl bondry condiions. Sch condiion occrs in he cse when one mesres n verged vle of some prmeer inside he domine. This problems rise in plsm physics, he condcion, biology nd demogrphy, modelling nd echnologicl process, see for emple (Smrskii, 98; Hieber nd Prss, 997; Ewing nd Lin, 3; Shi, 993; Mrhone, 99). Bondry-vle problems for prbolic eqions wih inegrl bondry condiion re invesiged by Ben (963); Bozini nd Benor (998); Cnnon (963); (984); Cnnon e l. (987); Ionkin (977); Kmynin (964); Field nd Komkov (99); Shi (993); Mrhone nd Bozi (5); Mrhone nd Hmeid (8); Denche e l. (994); Denche nd Mrhone (); Mrhone nd Lros (8); Yrchk (986) nd mny references herein. The problem wih inegrl one spce-vrible (respecively wo spce-vribles) condiion is sdied in Firweher nd Sylor (99) nd Denche nd Mrhone () (respecively in Mrhone
2 J. Mh. & S., 8 (): 85-9, The presen pper is devoed o he sdy of problem wih bondry inegrl wo-spce-vribles condiion for pril differenil eqion. We ssocie o problem ()-(4) he operor L = (L, l), defined from E ino F, where E is he Bnch spce of fncions L (), sisfying (3) nd (4), wih he finie norm Eq. 5: E = θ ()[ + ]dd + θ() d T + + d T T d (5) And F is he Hilber spce of vecor-vled fncions F = (f, ϕ) obined by compleion of he spce L () W (, ) wih respec o he norm Eq. 6: F = (f, ϕ ) = θ () f dd + θ() ϕ'd F F + ϕ d + ϕ d Where: < < θ () = ( ) ( ) (6) θ () ϕ ' d () θ d (9) T ϕ d d, ϕ d d, T T () Combining he in eqliies (8), (9) nd (), we obin (7) for E Lemm : Mrhone (7) for E we hve Eq. : ( ξ,)dξ d d 4 () Theorem 3: For ny fncion E, we hve he priori esime Eq. : E k L () F' Wih he consn: 4δ δ δ k = min,, ep(ct)( δ + 36) 6 where c nd δ is sch h Eq. 3: 4 δ = >,c < nd 8c (4 + δ ) (3) 3 3 Using he energy ineqliies mehod proposed in (Yrchk, 986), we esblish wo-sided priori esimes. Then, we prove h he operor L is liner homeomorphism beween he spces E nd F. Two-sided priori esimes: Theorem: For ny fncion E we hve he priori esime Eq. 7: L F k (7) E ' where he consn k is independen of. Proof: Using Eq. nd iniil condiions () we obin Eq. 8-: () L dd () θ 4 θ + dd + 4 α3 θ() d T (8) Proof: Define: ( ) M = ( ) + ( ) ( ξ,)dξ < α + ( ξ,)dξ < We consider for E he qdric forml Eq. 4: Re ep( c)lmdd (4) wih he consn c sisfying (4), obined by mliplying he Eq. by ep (-c) M, by inegring over, where = (, ) (, ),, wih T, nd by king he rel pr. Inegring by prs in (4) wih he se of bondry condiions (3) nd (4), we obin Eq. 5: 86
3 J. Mh. & S., 8 (): 85-9, θ() Re ep( c)lmdd = ep( c) dd + ep( c) ( ξ, ) dξ dd + ep( c) ( ξ, )dξ dd ep( c) (, )d ξ ξ dd + ( ) ep( c) ( ξ, )dξ dd + Re ep( c) θ() dd Re ep( c) () dd Re ep θ ( c) θ () dd Re ep( c) ( ξ, )dξ dd Re ( ) ep( c) (, ξ )dξ dd (5) On he oher hnd, by sing he elemenry ineqliies, we ge Eq. 6: θ() Re ep( c)lm dd ep ( c) dd + Re ep( c) θ() dd Re ep( c) θ() dd ep( c) (, )d dd ξ ξ ep( c) dd + ( ) ep( c) dd + Re ep( c) dd + Re ep( c) dd (6) Inegring by prs he second, hird nd forh erms of he righ-hnd side of he ineqliy (7) nd king ino ccon he iniil condiion () nd he condiion (3) give Eq. 8: Re ep( c)lmdd + ϕ d + ϕ d. + θ() ϕ' d δ ep( c) θ() dd + () ep( c) (, ) d ep θ + ( c ) (, ) d + ep( c ) (, ) d (8) By sing he elemenry ineqliies on he firs inegrl in he lef-hnd side of (8) we obin: 3δ ep( c) () 34 θ dd + θ()ep( c ) (, ) d + ep( c ) (, ) d + ep( c ) (, ) d θ() ep( c) θ () L dd + δ θ() ϕ ' d + ϕ d + ϕ d Now, from Eq. we hve: δ θ() ep( c) 6 ep( c) θ() dd δ ep( c) () L dd 4 θ δ + ep( c) θ () dd + 4 δ 6 dd (9) () Agin, sing he elemenry ineqliies nd lemm we obin: Re ep( c)lmdd δ ep( c) θ () dd + Re ep( c) () θ dd + Re ep( c) dd Re ep( c) dd 5 3 ep( c) dd (7) 87 δ + Combining ineqliies (9), () we ge Eq. : ep( ct) θ () L dd + θ() ϕ' 4δ d + ϕ d + ϕ d δ θ () dd + () (, ) θ d + (, ) d + (, ) δ d + θ() dd ()
4 J. Mh. & S., 8 (): 85-9, As he lef-hnd side of () is independen of, by remplcing he righ-hnd side by is pper bond wih respec o in he inervl [, T], we obin he desired ineqliy. Solvbiliy of he problem: The proof of eisence of solion is bsed on he following lemm. Lemm 4: Le: D (L) = { E: ϕ=} If for D nd some ω L (), we hve Eq. : φ ()L ϖ dd = () where Then, ω =. < v φ () = ( ) ( ) < Proof: From () we hve: φ() ϖ dd = φ()((, ) ) ϖdd (3) Now, for given ω, we inrodce he fncion: ω( ξ,) ω dξ < ξ v(, ) = ω ω( ξ,) ω d ξ < ξ Inegring by prs wih respec o ξ, we obin: v + v( ξ,)dξ < Nv = φ() ω = ( )v Which implies h Eq. 4: + ξ ξ < ( )v v(,)d Where: A() = ( φ ()(, ) ) If we inrodce he smoohing operors wih respec o (Yrchk, 986; Mrhone nd Lkhl, 9), J = (I + ) nd (J ), hen hese operors provide he solions of he respecive problems Eq. 6: (g ) () + g () = g() g () = = And Eq. 7: (g ) () + g () = g() g () = = T (6) (7) And lso hve he following properies : for ny g L (, T), he fncions g = (J )g nd g = (J )g re in W (,T) sch h g = = nd g = T =. Moreover, J commes wih, g g d nd so T T g g d for. Ping ep(c ) (, )d in (5), where he consn c 3 sisfies c 3 nd sing (7), we obin Eq. 8: ep(c) Ndd = + A() ep( c) dd A()( ) dd (8) Inegring by prs ech erm in he lef-hnd side of (8) nd king he rel prs yield Eq. 9 nd 3: Re A() ep( c)dd = (,)ep( c) φ() (,T) d + ep( c) φ()(c(, ) (, )) dd (9) v( ξ,)dξ = v( ξ,)dξ = (4) Re A() ep( c)(v ) dd Then, from eqliy (3) we obin Eq. 5: = Re (,) () (v ) dd φ Ndd A()vdd (5) = 88 (, ) ep( c) () (v ) dd + φ (3)
5 J. Mh. & S., 8 (): 85-9, Using -ineqliiesweobin Eq. 3: CONCLUSION Re A() ep( c)(v ) dd ep( c) φ() (, ) dd Re Combining (9) nd (3) we ge Eq. 3: ( ep(c)vnvdd ) ep( c) φ() c 3 dd Now, sing (3), we hve: Re ep(c)vnvdd Then, for we obin: Re ep(c)vnvdd = ep(c) φ() v dd (3) (3) We conclde h v = ; hence, ω =, which ends he proof of he lemm. Theorem 5: The rnge R (L) of L coincides wih F. Proof: Since F is Hilber spce, we hve R (L) = F if nd only if he relion Eq. 33: θ ()Lfdd + dl dϕ θ () d + lϕ d + lϕ d = d d (33) For rbirry E nd (fϕ) F, implies h f =, nd ϕ =. Ping D (L) in (33), we conclde from he lemm 3 h Ψf =, hen f = where: f < ψ f = ( )f ( )f < Tking E in (33) yield: dl dϕ θ d + lϕ d + lϕ d = d d ( ) The rnge of he rce operor l is everywhere dense in Hilber spce wih he norm: dϕ θ () + ϕ d + ϕ d ; hence, ϕ = d 89 From esimes (7) nd () i follows h he operor L: E F is coninos nd is rnge is closed in F. Therefore, he inverse operor L - eiss nd is coninos from he closed sbspce R (L) ono E, which mens h L is homeomorphism from E ono R (L). The heorem 5 chow h R (L) = F. So he eisence nd niqeness of he solion of he problem is proved. REFERENCES Ben, G.W. Jr., 963. Second-order correc bondry condiions for he nmericl solion of he mied bondry problem for prbolic eqions. Mh. Comp., 7: Bozini, A. nd N.E. Benor, 998. Mied problem wih inegrl condiions for hird order prbolic eqion. Kobe J. Mh., 5: Cnnon, J.R., 963. The solion of he he eqion sbjec o he specificion of energy. Qr. Applied Mh., : Cnnon, J.R., 984. The one-dimensionl he eqion. s Edn., Cmbridge Universiy Press, Cmbridge, ISBN: , pp: 483. Cnnon, J.R., S.P. Esev nd J.V.D. Hoek, 987. A Glerkin procedre for he diffsion eqion sbjec o he specificion of mss. Sim J. Nmer. Anl., 4: Denche, M. nd A. Kor, Kmynin, N.I., 964. Bondry vle problem for second-order differenil operors wih nonreglr inegrl bondry condiions. U.S.S.R. Comp. Mh Phys., 4: Denche, M. nd A.L. Mrhone,. A hree-poin bondry vle problem wih n inegrl condiion for prbolic eqions wih he Bessel operor. Applied Mh. Le., 3: DOI:.6/S ()6-4 Denche, M. nd A.L. Mrhone,. Mied problem wih nonlocl bondry condiions for hird-order pril differenil eqion of mied ype. IJMMS, 6: Ewing, R.E. nd T. Lin, 3. A clss of prmeer esimion echniqes for flid flow in poros medi. Adv. Wer Ressorces, 4: DOI:.6/39-78(9)955-S Firweher, G. nd R.D. Sylor, 99. The reformlion nd nmericl solion of cerin nonclssicl iniil-bondry vle problems. SIAM J. Sci. S. Comp., : DOI:.37/97
6 J. Mh. & S., 8 (): 85-9, Field, D.A. nd V. Komkov, 99. Theoreicl Aspecs of Indsril Design. s Edn., SIAM, Phildelphi, ISBN: 89879, pp: 33. Hieber, M. nd J. Prss, 997. He Kernels nd miml Lp-Lq Esimes for prbolic evolion eqions. Comm. Pril Differenil Eq., : Ionkin, N.I., 977. Solion of bondry-vle problem in he condiion wih nonclsicl bondry condiion. Differensil'nye Urvneniy, 3: Mrhone, A.L., 99. A hree-poin bondry vle problem wih n inegrl wo-spce-vribles condiion for prbolic eqions. Comp. Mh. Appli., 53: DOI:.6/j.cmw Mrhone, L. nd M. Bozi, 5. High order differenil eqions wih inegrl bondry condiion. Fr Es J. Mh. Sci., 8: Mrhone, A.L. nd A. Hmeid, 8. Mied problem wih n inegrl spce vrible condiion for hird order prbolic eqion of mied ype. Fr Es J. Mh. Sci., 9: Mrhone, A.L., 7. A hree-poin bondry vle problem wih n inegrl wo-spce-vribles condiion for prbolic eqions. Comp. Mh. Appli., 53: DOI:.6/j.cmw Mrhone, A.L. nd C. Lros, 8. A srong solion of high order mied ype pril differenil eqion wih inegrl condiions. Applicble Anl., 87: Mrhone, A.L. nd F. Lkhl, 9. A bondry vle problem wih mlivribles inegrl ype condiion for prbolic eqions. J. Applied Mh. Sochsic Anl. DOI:.55/9/9756 Smrskii, A.A., 98. Some problems in differenil eqions heory. Differ. Urvn., 6: -8. Shi, P., 993. Wek solion o evolion problem wih nonlocl consrin. Sim J. Anl., 4: DOI:.37/544 Yrchk, N.I., 986. Mied problem wih n inegrl condiion for cerin prbolic eqions. differenil eqions, :
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