TIME-PERIODIC SOLUTIONS OF A PROBLEM OF PHASE TRANSITIONS

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1 Far East Joural o Mathematical Scieces (FJMS) 6 Pushpa Publishig House, Allahabad, Idia Published Olie: Jue 6 Volume 99, umber, 6, Pages ISS: Proceedigs o 4th Abu Dhabi Uiversity Aual Iteratioal Coerece: Mathematical Sciece ad its Applicatios, December 3-6, 5, Abu Dhabi, UAE IME-PERIODIC SOLUIOS OF A PROBLEM OF PHASE RASIIOS S. K. Jaabekova *, Valéry Covachev ad Haydar Akça Abai Kazakh atioal Pedagogical Uiversity Kazakhsta shaa_@mail.ru Istitute o Mathematics ad Iormatics Bulgaria Academy o Scieces Soia, Bulgaria vcovachev@hotmail.com Departmet o Applied Scieces ad Mathematics College o Arts ad Sciece Abu Dhabi Uiversity P. O. Box 599 Abu Dhabi, U. A. E. haydar.akca@adu.ac.ae Abstract We ivestigate the time-periodic solutio o a mathematical model o phase trasitios with relaxatio. he mathematical model describes Received: April 4, 6; Accepted: April 8, 6 Mathematics Subject Classiicatio: 34K3, 35B, 35B5. Keywords ad phrases: humidity, ice, bouded domai, time-periodic uctios, Schauder s theorem, time-periodic solutio. Correspodig author Commuicated by Haydar Akça, Guest Editor

2 948 S. K. Jaabekova, Valéry Covachev ad Haydar Akça the temperature chage processes i the soil, takig ito accout humidity. he correctess o the mathematical model has bee previously studied ad it has bee proved that the mathematical model is a regularized Stea-type problem.. Statemet o the Problem Let Ω be a bouded domai i R with a suicietly smooth boudary S, Q = Ω (, S = S (, ). I [], it is required to id uctios θ ( x,, w( x,, l( x, (temperature, humidity ad rost o soil) deied i Q satisyig the ollowig system o equatios: θ l c = k Δθ + χ, () w l l = λ Δw, = α( w H ( θ) () (3) with iitial ad boudary coditios θ( x, ) = θ( x), x Ω, (4) w ( x, ) = w ( x), x Ω, (5) l( x ) = l ( x), x Ω, θ( x, = θ ( x,, ( x, S, (6), s w( x = w ( x,, ( x, S. (7), s Istead o the boudary coditios (6) ad (7 we ca cosider the ollowig coditios: θ w = θs ( x, ( x, S,, =, ( x, S. (8) (9)

3 ime-periodic Solutios o a Problem o Phase rasitios 949 Usig (3 we exclude l rom () ad ( the we obtai a system o two equatios or θ ( x, ad w ( x, : θt = k Δθ + χα( w H ( θ) (a) w t = λ Δw α( w H ( θ) ). (a) I what ollows, Problem I reers to the problems (a (a (4)-(7 ad Problem II to the problems (a (a (4 (5 (8) ad (9). I this paper, we propose a method o determiig the time-periodic solutio o the ollowig problem: i the edless bad Q = Ω (, + the temperature θ ( x, ad the humidity w ( x, are kow time-periodic uctios o the boudaries o the domai Q with period >. As it ollows rom the geeral theory, it is possible to solve the problem i which the iitial coditios are replaced by the coditios o periodicity θ( x, = θ( x, t + x Ω, () w ( x, = w( x, t + x Ω. (). Mai Result Deiitio. A solutio o the Problem I is a pair o uctios { θ ( x,, w ( x, } such that, () θ( x,, w( x, W ( Q q >. q () Equatios (a) ad (a) are satisied almost everywhere i Q. (3) he iitial ad boudary coditios or θ ad w are take i the sese o the uctios trace i the speciied class. W, q ( Q ) deotes the set o time-periodic uctios rom Wq, ( Q ) s that have period >. he spaces Lp ( Q Wp ( Q s, p, q > are deied i the same way.

4 95 S. K. Jaabekova, Valéry Covachev ad Haydar Akça Here ad below the otatios o the orms ad spaces coicide with []. heorem. Let θ s ( x, ad w s ( x, be time-periodic boudary data with period >. he { θ w} L (, ; W ( Ω)) L (, ; L ( ) (), Ω ( v) + λ( w, v) + α ( w, v) = α ( H ( θ), v), w t,, Ω, Ω, Ω, Ω, v W ( Q (3) w ( x, ) = w( x, (4) ( θ u) + k ( θ, u) χα( w H ( θ), u), t,, Ω, Ω, Ω = u W, ( Q (5) θ ( x, ) = θ( x, ). (6) he proo o the theorem is carried out by the irst boudary value problem. he uctio H ( θ) is approximated by cotiuous mootoe uctios H ( θ) which coicide with the uctio H ( θ) i θ >. Here, istead o the uctio H ( θ the uctio H ( θ) is cosidered. For each, the origial problem is solved usig Schauder s theorem or a ixed poit o,, the operator Λ : W ( Q ) W ( Q ). By deiitio g ( x, = Λ( g( x, ) i g ( x, is the solutio o the ollowig problem:, Here W ( ) ad it satisies Q g kδg = χα( H ( g )). (7) t t λδ + α = αh ( g). (8) Without loss o geerality ad or coveiece o urther calculatios, the

5 ime-periodic Solutios o a Problem o Phase rasitios 95 boudary data is zero, i.e., θ ( x =, ( x, S, (9) s s, w ( x =, ( x, S. (), he g ( x, satisies coditios (6) ad (9 ad ( x, t ) satisies coditios (4) ad (). Approximate solutios o the problems (7 (8) ad (9) are obtaied by Galerki s method. Based o [3, 4], we will use a special basis o the uctios ω i ( x) that belog to W ( Ω) ad the ollowig coditios: ( ωi, v) ( ) = ℵi ( ωi, v), v W ( Ω). () W Ω We cosider the Cauchy problem, Ω ( F i, ω i ), Ω + λ(, ωi ), Ω + α(, ωi ), Ω = α( H ( g) ω ) Ω, i, (), i, ( x, ) = w ( x (3) w ( x) is ay elemet rom [ ω, ω,..., ω ]. As it is kow [5], the solutio ( x, exists o the iterval [, ]. We show that there exists R idepedet o such that ( x, ) R as soo as w ( x ), Ω R. (4), Ω Multiply () by li () t ad sum over i rom to. he, rom ( it ollows that d dt, Ω + λ, Ω + α, Ω α α = α( H ( g), ), Ω ( ),, Ω + H g, Ω

6 95 S. K. Jaabekova, Valéry Covachev ad Haydar Akça ad, Ω,, Ω ( mes Ω) thus, d dt, Ω + C, Ω α H, Ω. (5) he we obtai C e,, Ω, Ω ( x ) w ( x) + α e H ( g) C, Ωdt, C = w ( x) Ω +. (6) I we choose R so that R C exp( C ), (7) the rom the last iequality there ollows (4). hus, the mappig w I ( w ) = ( x, ) maps B R (the ball with ceter at the origi ad radius R i the space [ ω, ω,..., ω ]) ito itsel. Sice this mappig is cotiuous, there exists a poit w BR such that I ( w ) = w. A similar mappig is created or the uctios g ( x,. Sice g ( x, ) are bouded i W ( Ω) ad cosiderig (4 we obtai exactly the same estimate as i the case o equatios with the iitial coditios. { g, } are bouded i ( L, ; W ( Ω )) L (, ; L ( Ω ) { i, gi } are bouded i L (, ; L ( )). From this, it ollows that oe ca choose a Ω p p subsequece {, g } such that p i ( L, ; W ( Ω )) weakly, ad i L (, ; L ( Ω) )- weakly,

7 ime-periodic Solutios o a Problem o Phase rasitios 953 g p g i ( L, ; W ( Ω)) weakly, ad i L (, ; L ( Ω) )-weakly, p i ( ) t t L Q weakly, g p i ( ) t gt L Q weakly, p where, i particular, ( x, ) ( x, ( x, ) ( x, ) weakly i L ( Q g ( x, ) g( x, ), g ( x, ) g( x ) weakly i L ( Q ). p p p, 3. Coclusio We ca pass to the limit with respect to i the correspodig itegral idetities. hus, we get θ ( x, ) = θ( x, w( x, ) = w( x, ). Reereces [] I. A. Kaliev, S.. Mukhambetzhaov ad E.. Razikov, Correctess o oe mathematical model o oequilibrium phase trasitio o water i porous medium, Diaamika Sploshoi Sredy, Sb. auch. tr./akad. auk SSSR, Sibirsk. Otd., Istitut Gidrodiamiki, 989, vyp. 93, 94, pp (i Russia). [] S.. Mukhambetzhaov ad S. K. Jaabekova, Mathematical ad umerical modelig o luid low i porous media with phase trasitios, Proceedigs o the II Iteratioal Scietiic-Practical Coerece Fudametal Sciece ad Educatio, Biysk, 4, pp [3] M. Sh. ilepiev, E. W. Urazmagabetova ad S. K. Jaabekova, O the properties o solutios o oe model o seepage theory with ree boudaries, Proceedigs o the II Iteratioal Scietiic-Practical Coerece Fudametal Sciece ad Educatio, Biysk, 4, pp [4] J.-L. Lios, Some Methods o Solvig oliear Boudary Value Problems, Duod-Gauthier-Villars, Paris, 969. [5] O. A. Ladyzheskaya, he Mathematical heory o Viscous Icompressible Flow, Mathematics ad its Applicatios, Vol., Gordo ad Breach, ew York, 969.