INVERSE CAUCHY PROBLEMS FOR NONLINEAR FRACTIONAL PARABOLIC EQUATIONS IN HILBERT SPACE

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1 IJAS 6 (3 Febuay INVESE CAUCH POBLEMS FO NONLINEA FACTIONAL PAABOLIC EQUATIONS IN HILBET SPACE Mahmoud M. El-Boai Faculty of Sciece Aleadia Uivesit Aleadia Egypt yahoo.com ABSTACT This ote is devoted to study a ivese Cauchy poblem i a Hilbet space H fo factioal abstact diffeetial euatios of the fom: d = A f ( F( W ( with the iitial coditio u ( = u H ad the ovedetemiatio coditio: ( u ( = w( whee (.. is the ie poduct i H f is a eal ukow fuctio w is a give eal fuctio u v ae give elemets i H g is a give abstact fuctio with values i H < u is ukow ad A is a liea closed opeato defied o a dese subset of H W( = ( B (... B ( { B ( : i =... t J} is a family of liea closed opeatos defied o dese sets i S... S S espectively i H ito H F is a give abstact fuctio o J H ito H. It is supposed that A geeates a semigoup. A applicatio is give to study a ivese poblem i a suitable Sobolev space fo geeal oliea factioal paabolic patial diffeetial euatios with ukow souce fuctios. Keywods ad phases: Noliea factioal evolutio euatioivese Cauchy poblem. Mathematics Subect Classificatios: 35A G.. INTOUCTION Successful utilizatio of ay factioal diffeetial euatio as a modelig tool euies esults about eistece uiueess ad egulaity popeties of the solutio ude sufficietly geeal assumptios. The geeal fom of the euatio is kow ad the details must be detemied by ecocilig the model with the obsevatio of the pocess. I othe wods a ivese poblem must be solved to fid o the basis of the obsevatio the coefficiets fee tem the ight-had side ad sometimes iitial ad bouday coditios. Seveal authos [-5] studied the uiue solvability of ivese poblems fo vaious paabolic euatios with ukow souce fuctios ude a itegal ovedetemiatio coditio. Cao ad uchateu cosideed the idetificatio of a ukow state-depedet souce tem i the heat euatio[6]. I this ote the followig oliea model is cosideed: d = A f ( F( W (. u ( = u(. whee A is a liea closed opeato defied o a dese set S i a eal Hilbet space H ito H W( = ( B (... B ( { Bi ( : i =... t J} is a family of liea closed opeatos defied o dese sets S... S S espectively i H ito H F is a give abstact fuctio o J H ito H u is a give elemet i S g is a give abstact fuctio defied o J = [ T]( T > with values i H f is a ukow eal fuctio ad <. It is assumed that A geeates a aalytic semigoup Q (. This coditio implies Q( c fo all t whee. = (.. (.. is the ie poduct i H ad c is a positive costat. It is supposed also that 4

2 IJAS 6 (3 Febuay El-Boai Noliea Factioal Paabolic Euatios i Hilbet Space ( C Thee is a umbe ( such that K B i ( t Q( t h h t whee t ( T] t J hh ad K is a positive costa i =... ( C The fuctios B ( h... B ( h ae uifomly Hölde cotiuous i t J fo evey elemet h i S i i ( C 3 g is cotiuous i t o J with espect to the om i H ( C 4 F is cotiuous o J H with espect to the om i H. I sectio the ivese Cauchy poblem is studied ude the ovedetemiatio coditio: ( u ( = w( (.3 whee v is a give elemet i H ad w is a give eal fuctio defied o J. A of the closed opeato A eists ad that if We shall suppose that the adoit opeato d ( = ( the t ( = ( ( t s ( s ds ( whee ( is the gamma fuctio < ae abstact fuctios of t with values i H ad the itegal is take i Boche`s sese [7]. We shal also assume that : (i F is uifomly Hölde cotiuous i that is thee eist costats c > (] such that t t J fo all ad all W H (iithe Lipschitz coditio f ( t W f ( W c t t F( W F( W c ad all t J whee i= wi w is satisfied fo all W W H c > is a costat. We shall assume the followig coditios; A : u v S S fo all t J A : g ( g t J whee g ( = ( ad g is a positive costa 3 v A : The abstact fuctios g ad Ag ae cotiuous o J with espect to the om i H A 4 : dw C(J. I sectio 3 a applicatio is give to the ivese Cauchy poblem fo oliea factioal diffeetial euatios of paabolic type.. AN INVESE CAUCH POBLEM A pai of fuctios { u f } is said to be a stictly solutio of the ivese poblem (.-(.3 if d u S H fo each t ( T] f C( J ad the elatios (.-(.3 ae satisfied. I this case we say that the ivese poblem (.-(.3 is solvable. Let us coside the followig euatio: f = h P (. whee d w( h ( = g ( i 43

3 IJAS 6 (3 Febuay El-Boai Noliea Factioal Paabolic Euatios i Hilbet Space ad P is defied o C (J by: P( = ( A ( F( W (.(. g ( g ( Theoem.. Suppose that the coditios ( A A4 ae satisfied. The the followig assetios ae valid : (I If the ivese poblem (. is solvable the so euatio (. has a solutio f C(J (II If euatio (. has a solutio f C(J ad the compatibility coditio ( u o = w((.3 holds the the ivese poblem (. - (.3 is solvable. Poof. Assume that the ivese poblem (. - (.3 is solvable. Multiplyig both sides of (. by v scalaly i H we obtai the elatio d ( = ( A f ( ( F( W (.(.4 To pove assetio (II we otice that by the assumptio euatio (. has a solutio f C(J. Whe isetig this fuctio i (. the esultig poblem (. (. ca be teated as a diect poblem. Usig pevious esults [8] this solutio is give by u ( = ( Q( t u d t ( t s ( Q(( t s [ f ( s s F( s W( s] d ds (.5 Usig pevious esults [89] we ca see that the solutio u ( of euatio (.5eists ad uiue i S. Let us pove ow that u satisfies the ovedetemiatio coditio (.3. I this case u ad f ae kow coseuetly (.4 will epeset a idetit d w( f ( = ( Au ( F( W.(.6 Subtactig euatio (.4 fom (.6 oe gets d w( d = (. applyig the factioal itegal of ode ad takig ito accout the compatibility coditio (.3 we fid out that u satisfies the ovedetemiatio coditio (.3 ad that the pai { u f } is a stictly solutio of the ivese poblem (. - (.3. This completes the poof of the theoem [-6]. Theoem.. Let the coditios ( A4 stictly solutio of the ivese poblem (. - (.3. Poof. Substitutig fom (. ito (. oe gets: d = A F( W ( h( g ( A ad the compatibility coditio (.3 hold the thee eists a uiue ( u A ( F( W (.(.7 g ( Usig simila techiues as i [8] we deduce that the diect Cauchy poblem (. (.7 has a uiue stog solutio u. To pove the uiueess of u ad f we assume to the cotay that thee wee two diffeet solutios { u f} ad { u f} of the ivese poblem (. - (.3. We claim that i this case f f fo all poits of J. I fact if f = f o J the applyig the uiueess theoem to the coespodig diect poblem (. (.7 we would have u = u. Sice both pais satisfy idetity (.4 the fuctios f ad f give two diffeet solutios of euatio (.7. But this cotadicts the uiueess of solutios to euatio (.. This completes the poof of the theoem. 3. APPLICATIONS Coside the oliea itego-patial diffeetial euatio of factioal ode; a( = F( W f ( t = m < t T(3. 44

4 IJAS 6 (3 Febuay El-Boai Noliea Factioal Paabolic Euatios i Hilbet Space with the iitial coditio whee dimesioal multi - ide u o ( = u ( (3. is a - is the -dimesioal Euclidem space g =... = = (... i i =... W = ( w... w ad w ( = b ( c ( dy m whee is a ope subset of. Let L be the set of all suae itegable fuctios o ( m. We deote by m y C the set of all cotiuous eal valued fuctios defied o which have cotiuous patial deivatives of ode less tha o eual to m. By C m we deote the set of all fuctios f C m with compact suppots. Let H m be the completio of C m with espect to the om It is supposed that the opeato f = [ f ( d]. m m A = = m a ( is uifomly elliptic. I othe wods it is supposed that all the coefficiets o ad that thee is a positive umbe c such that m m ( a ( c = m is satisfied fo all that all the coefficiets a = m ae cotiuous ad bouded ad all = (... ( =... =... a = m satisfy a Hölde coditio o.. It is supposed also Ude these coditios the opeato A with the domai of defiitio S = H m geeates a aalytic semigoup Q ( [7-]. It is well kow that m If H m g the ( g = H is dese i L. ( Q ca be witte i the fom ( = G( y dy whee G is the fudametal solutio of the Cauchy poblem u = Au =. t K It ca be poved that Q( g g t whee < < K is a positive costa m t >. ad g = g ( d. The popeties of the fudametal solutio G ca be foud i []. To solve the Cauchy poblem (3. (3. we suppose the followig coditios; (a All the coefficiets b m =... ae bouded cotiuous o J ad satisfy the Hölde coditio sup b( t b( t K( t t whee K is a positive costat ad < <. 45

5 IJAS 6 (3 Febuay El-Boai Noliea Factioal Paabolic Euatios i Hilbet Space (b The itegals c ( dy d eit fo all t J m =.... c m... (c All the coefficiets = satisfy the Hölde coditio [ c( t c( t] dy d K( t t whee K is a positive costat ad < <. The Cauchy poblem (3. (3. ca be witte i the abstact fom (. (. whee A is the opeato a ( S = H m.the opeatos B ( ae defied by = m with domai of defiitio B ( u = u =... whee u ( = b ( m m The domai of defiitio of the opeatos B (... B ( ca be take S =... = S = H m [4][5]. t c ( y dy We suppose that F satisfies the coditios (.3 ad (.4 with espect to the om i ( L. Now it is clea that we ca apply theoems (. (. ad (.3 to the Cauchy poblem (3. (3.. I othe wods the Cauchy poblem (3. (3. has a uiue solutio i the space H m. Also the cosideed poblem is coectly fomulated. We suppose that u satisfies the itegal ovedetemiatio coditio: The fuctios o v( d = w( (3.3 u v w ad g ae kow ad satisfy the coditios of theoem i the space L fuctio f is ukow. Applyig theoems (. ad (. we ca see that the ivese poblem (3.- (3.3 is uiuely solvable i the cosideed Sobolev space. 4. EFEENCES []. I.A. Vasi V.L. Kamysi. O the asymptotic behavio of solutios of ivese poblems fo paabolic euatios. Sibeia Math. J. 38 ( []. V.L. Kamysi I.A. Vasi Asymptotic behavio of solutios of ivese poblems fo paabolic euatios with iegula coefficiets Soboik Math. 8 ( [3]. A.I. Pileplo..G. Olovsk L.A. Vasi Methods fo solvig ivese poblems i mathematical physics Macel ekke Ic [4]. A.F. Guveili ad V.K. Kalatov The asymptotic behavio of solutios to a ivese poblem fo diffeetial opeato euatios Mathematical ad compute Modelig 37( [5]. A.I. Pilepko ad.s. Tkacheko Popeties of solutios of paabolic euatio ad the uiueess of the solutio of the ivese souce poblem with itegal ovedetemiatio compt. Math.Math. phys. 43(4 ( [6]. J.. cao P.uchateav Stuctual idetificatio of a tem i a heat euatio Ivese pobl. 4 ( [7]. Mahmoud M. El-Boai Some pobability desities ad fudametal solutio of factioal evolutio euatios Chaos Solito ad Factals 4 ( [8]. Mahmoud M. El-Boai Semigoup ad some oliea factioal diffeetial euatios Appl. Math. ad Computatios 49( [9]. Mahmoud M. El-Boai O some factioal evolutio euatios with olocal coditios It. J. Pue Appl. Math. 4(3( []. Mahmoud M. El-Boai The fudametal solutios fo factioal evolutio euatios of paabolic type J. of Appl.Math. Stochastic Aalysis (JAMSA ( []. Mahmoud M. El-Boai O some factioal evolutio euatios With o local coditios Iteatioal J. of Pue ad Applied Mathematics Vol. 4 No. 3 ( []. Mahmoud M. El-Boai O some factioal diffeetial euatios i the Hilbet-space Joual of iscete ad Cotiuous yamical Systems Seies A ( [3]. Mahmoud M. El-Boai O some stochastic itego-diffeetial euatios Advaces i yamical Systems ad Applicatios Volume Numbe ( [4]. Mahmoud M. El-Boai Khaiia El-Said El-Nadi Hoda A. Fouad O some factioal stochastic delay diffeetial euatios Computes ad Mathematics With Applicatios 59( [5]. Mahmoud M. El-Boai Khaiia El-Said El-Nadi Ema G. El-Akabaw O some factioal evolutio euatios Computes ad Mathematics With Applicatios 59( [6]. Mahmoud M. El-Boai O the solvability of a ivese factioal abstact Cauchy poblem IJAS 4(4 Septembe ( [7]. Khaiia El-Said El-Nadi O some stochastic paabolic diffeetial euatios i a Hilbet space J. of Applied Mathematics ad Stochastic Aalysis Vol. ( [8]. Khaiia El-Said El-Nadi O some stochastic diffeetial euatios ad factioal Bowia motio. It. J. of Pue Ad Applied Math. Vol.4 No.3. ( [9]. Khaiia El-Said El-Nadi Asymptotic methods ad diffeece factioal diffeetial euatios It. J. Cotemp. Math. Sci. Vol No. ( []. S.. Eidelma Bouds fo solutios of paabolic systems ad some applicatios Math.Sb []. S.. Eidelma O the fudametal solutio of paabolic systems Math. Sb.( (. The 46

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special