New Aspects for Non-linear Semigroups in L1 Applied to Heat Equation s Analysis

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1 Uiversal Joural of Itegral Equatios 3 5), - New Aspects for No-liear Seigroups i L Applied to Heat Equatio s Aalysis Iterpaper Research Orgaizatio 8, Diaki Str. Athes, GR - 6 7, Greece eladopoulos@iterpaper.org Abstract No-liear seigroups are studied ad ivestigated i order to prove the existece ad uiqueess of solutios for the o-liear partial differetial equatio defied i L. The above partial differetial equatio is derived fro the geeral heat equatio's aalysis at high teperature. The above differetial equatio has ay applicatios i potetial flow probles. I additio, the existece ad uiqueess of solutios for the o-liear heat equatio is proved, by presetig soe geeral boudary coditios. esides, soe properties of the solutios for the above oliear partial differetial equatio are fially proved. Matheatics Subject Classificatio : 35Q79. Key Word ad Phrases No-liear Partial Differetial Equatios, No-liear Seigroups, L Spaces, Heat Equatio, Dissipative Operator.. Itroductio Over the past years of sufficietly Icreasig iterest was the ivestigatio of o-liear seigroups i geeral aach spaces ad their applicatio to the existece ad uiqueess theory for differetial equatios associated with o-liear operators. Cosequetly, the fudaetal results o o-liear seigroups are applied to the solutio of several types of partial differetial equatios arisig i atheatical physics ad egieerig. esides, the study of o-liear seigroups was derived fro the exaiatio of o-liear parabolic equatios ad fro various o-liear boudary value probles. Geerally, the theory of o-liear seigroups is a geeralizatio of the Hille-Yosida theory for oe-paraeter seigroups of liear operators ad is further closely related to the theory of o-liear ootoe operators. The first work o seigroups was published by A.V.alakrisha [], whe studyig fractioal powers of closed operators. Soe years later T.Kato [] studied o-liear seigroups i coectio with evolutio equatios, while Y.Koura [3], [4] studied o-liear seigroups defied i Hilbert spaces. eyod the above, K.Sato [5] ivestigated o-egative cotractio seigroups i aach spaces, while a geeral theory of o-liear seigroups was ivestigated by M.G. Cradall et al. [6] - [8]. Also, J.Wataabe [9], [] studied seigroups of o-liear operators o closed covex sets ad H.rezis et al. [], [] itroduced a geeral seigroups forulatio. At the sae tie, M.Iaelli [3] proposed o-liear seigroups o coes of a o-reflexive aach space, while J.Meri [4] ad S.Oharu [5] ivestigated geeral theories of o-liear seigroups. Additioally, I.Miyadera [6] studied seigroups of o-liear operators ad.k.qui [7] ivestigated seigroups i L spaces. O the cotrary, Y.Koishi [8] studied o-liear seigroups associated with soe partial differetial equatios ad U.Westphal [9] ad S.Aizawa [] ivestigated soe forulatios for o-liear seigroups. Moreover, T.Kurtz [] studied seigroups of o-liear operators applied to gas kietics, while R.ruck [] ivestigated asyptotic covergece of o-liear cotractio seigroups i Hilbert spaces.

2 The theory of o-liear seigroups was geerated by Y.Kobayashi [3], [4] ad a oograph o the above subject was writte by V. arbu [5]. Moreover, J.M. all [6] studied strogly cotiuous seigroups, while.c.urch [7] ivestigated a seigroup treatet of the Hailto - Jacobi equatios i several space variables. At the sae tie, J.H.Lightboure ad R.H.Marti [8] ivestigated relatively cotiuous perturbatios of aalytic seigroups, whe A.T.Plat [9] studied o-liear seigroups of traslatios i aach spaces geerated by fuctioal differetial equatios. The theory of o-liear seigroups o geeral aach spaces was also ivestigated by J..aillo [3] ad A.Pazy [3], [3], while J.A.Goldstei [33] wrote a oograph o seigroups of liear operators with soe geeral applicatios. Also, a oograph o o-liear evolutio operators ad seigroups was writte by N.H.Pavel [34]. Thus, over the past years there has bee a icreasig use of seigroups techiques i solvig probles related to partial differetial equatios, defied i ifiite diesioal aach spaces. The ethod of seigroups has cosiderably siplified the proofs ad has uified the treatet of several differet classes of differetial equatios. These differetial equatios are solved successfully by usig seigroup techiques i dealig with discotiuous data ad regularity. y the curret research, the o-liear seigroups are used i order to prove the existece ad uiqueess of solutios for the o-liear partial differetial equatio defied i L spaces. This differetial equatio is derived fro the geeral theory of heat equatios aalysis. The above theory is a part of potetial flows aalysis as was ivestigated ad aalyzed by E.G.Ladopoulos [35] - [49] over the last two decades. So, a geeral theory is preseted i order to prove the existece ad uiqueess of solutios for the above o-liear partial differetial equatio defied i L spaces. This theory cosists to the use of o-liear seigroups, by applyig the to the existece ad uiqueess theores.. Heat Equatio's Ivestigatio at High Teperature Theore. Cosider by a bouded doai i R 3 with sooth boudary Γ ad by s a arbitrary subdoai of with sooth boudary s Figure ). Suppose that i there a gas with teperature u=u at the poit x x, x, x3 ) at tie t. The, i the case of high teperature, the heat equatio is equal to: u * u t,.) t i which deotes the Laplace operator:.) x x x3 ad is equal to: / c a ).3) with the theral coductio costat, c the specific heat costat ad a ] 4,5;5,5[, a. 3

3 3 Fig. A bouded doai i R with sooth boudary Γ iside which there is a gas with teperature u=u at the poit x x, x, x ) D Proof Let the aout of heat i s at tie t: 3. Q c t, u) dx I D s.4) ad the aout of heat outflowig fro s per uit of tie: Q II D s R t, u, u), ds.5) where deotes the outward oral to s ad ds the eleet of area of s. esides, cosider the aout of heat fro sources F i per uit of tie: s Q F t, u) dx III D s.6) Cosequetly, the balace of heat is equal to: 4

4 d dt Q Q Q.7) I additio, by usig the forula of Gauss the.5) ca be writte as: I II III Q II DS R, ds Rdx DS.8) The, by substitutig eqs.4),.6) ad.8) ito.7) oe obtais: c t, u) R t, u, u) F t, u).9) t Furtherore, we assue the followig forula to be valid: c t, u) cu.) i which c deotes the specific heat costat. I the case of high teperature, the the radiatio of heat is equal to: R u a u.) where deotes the theral coductio costat ad a ]4,5;5,5 [. So, fro.) we obtai: R u a u.) which is further equal to: a a R u.3) ad fially to: a a R u.4) i which deotes the Laplace operator. Hece, fro eqs.9) ad.4) by eglectig the source ter F, we obtai the required heat equatio.). 3. The use of No-liear Seigroups for the Existece ad Uiqueess Theores of Noliear Partial Differetial Equatios i L 5

5 Theore 3. Let u=u the teperature fuctio at the poit the followig operator E: x x, x, x3) at tie t. Fig.). The, D E) { u L ) ; u W ), u L )} 3.) Eu u, for u DE) 3.a) is - dissipative i L ), with D E ) L ). Proof I order to prove the dissipativity of E i L ), we have to prove the followig iequality: u v u v t Eu Ev), t, u, v D E) 3.) L Cosequetly, we choose the followig sequece h C R) : with the properties: h L s s), s R 3.3) s h ) h s) h s) li h s) sig s, s R 3.4) I additio, by choosig u, v D E), oe has: h [ v ] h[ v ] [ u v ] 3.5) So, by usig Gree's forula we obtai: dx Eu Ev) h [ ] u v dx h[ u v ] [ u v ] 3.6) Thus, oe obtais for every t>: v) h[ u v ] dx [ u v t Eu Ev)] h [ u u v ] dx u v t Eu Ev) dx u v t Eu Ev) L 3.7) 6

6 eyod the above, oe has: sig[ u v ] sig[ u v] 3.8) ad thus, by lettig i 3.7), we obtai the required forula 3.). Also, we have to prove that for each h L ) there exists a uique u DE ), such that: u u h 3.9) So, with v u, the eq 3.9) is equivalet to the followig equatio: v v h 3.) Cosequetly, the Laplace operator:, D ) { v W ), v L )} 3.) is dissipative i L ). Furtherore, eq 3.) has a uique solutio v D ) ad thus, u v is the solutio of 3.9). Thus, D E) { u L ) ; u D )}, which is dese i L ) ad thus D E) L ), which copletes the proof. Theore 3. Cosider by u=u the teperature fuctio at the poit x x, x, x3) at tie t Fig. ). Also, for every h L, T ; L )), T>, ad u L ) let the heat equatio: with the boudary coditios: u * u t, h i], T[, x 3.) t u t,, o ], T [, x 3.3) u, u, i 3.4) i which Γ deotes the boudary of ad is give by.3) ad >. The, the heat equatio 3.) has a uique itegral solutio u C {[, T ] ; L )}. Proof Let E be the ifiitesial geerator of a give by 3.). C cotractio seigroup S, while the operator E is The, the followig fuctio: 7

7 u t, S u ) S t s) h s) ds, t t T 3.5) is said to be the solutio of the o-liear partial differetial equatio 3.). I additio, cosider the sequece h C, T, L )) be such that: li h h 3.6) y choosig u D ), with u u ) as, the the followig E x proble: u t, [ u t, ] h i ], T [, x 3.7) t with the coditios: u t,, o ], T [, x 3.8) u, u, i 3.9) has a uique strog solutio u t, equal to: t u t, S u ) S t s) h s) ds, t T 3.) where the last ter of eq 3.) is cotiuously differetiable. Furtherore, by settig: w t S t s) h s) ds 3.) the it is easily proved that: S ) w w w t ) w t t S t s) h s) ds 3.) for all sufficiatly sall ad t T. Hece, sice: u t, S u ) w 3.3) 8

8 follows that 3.) satisfies 3.7). Fially, the solutio u t, is the uique solutio to eq 3.7). Hece, passig to the liit, follows that the solutio 3.5) is the uique solutio to the o-liear partial differetial equatio 3.). 4. Coclusios A geeral heat equatio's aalysis was preseted, by applyig a bouded doai i R 3, iside which there is a gas with teperature u u at soe poit x. This proble is reduced to the solutio of a o-liear partial differetial equatio. Cosequetly, a o-liear seigroup techique was itroduced i order to prove the existece ad uiqueess of solutios for the o-liear partial differetial heat equatio, whe soe geeral boudary coditios are preseted. Hece, the fudaetal results ad properties of the o-liear seigroups were preseted i order to uify the treatet of the oliear partial differetial heat equatio, defied i L, by usig soe geeral boudary coditios. Fially, the ethod of o-liear seigroups has clearly siplified the proofs of the existece ad uiqueess of solutios for the o-liear partial differetial equatio, which are of ai iterest for the solutio of potetial flow probles. Refereces. alakrisha A.V., "Fractioal powers of closed operators ad the seigroups geerated by the", Pacific J. Math., 96), Kato Τ., "Noliear seigroups ad evolutio equatios", J. Math. Soc. Japa, 9 967), Koura Υ., "Noliear seigroups i Hilbert space", J. Math. Soc. Japa, 9 967), Koura Υ., "Differetiability of oliear seigroups", J. Math. Soc. Japa, 969), Satο Κ., "O the geerators of o-egative cotractio seigroups i aach lattices", J. Math. Soc. Japa, 968), Cradall M.G. ad Pazy Α., "Sei-groups of oliear operators ad dissipative sets", J. Fuct. Aal., 3969), Cradall M.G. ad Liggett Τ. Μ., "Geeratio of seigroups of oliear trasforatios o geeral aach spaces", Aer. J. Math., 93 97), Cradall M.G. "A geeralized doai for seigroups geerators", Proc. Aer. Math, Soc., ), Wataabe J., "Seigroups of oliear operators o closed covex sets", Proc. Japa. Acad.. Sci., ), Wataabe J., "Autooous oliear fuctioal differetial equatios ad oliear seigroups", J. Math. Aal, Appl., ), -.. rezis H..ad Pazy A., "Seigroups of oliear cotractios o covex sets", J. Fuct. Aal., 6 97), rezis Η., "Operateurs Maxiaux Mootoes et Seigroups de Cotractios das les Espaces de Hilbert", Math, Studies, 5, North Hollad, Iaelli M., "No-liear seigroups o coes of a o reflexive aach space", oll. U.. Mat. Ital., 3 97), Meri J., "O expoetial liit forula ad oliear seigroups", Tras. Aer. Math. Soc., 5 97) Oharu S., "O the geeratio of seigroups of oliear cotractios", J. Math. Soc. Japa, 97), Miyadera J., "Soe rearks o sei-groups of oliear operators", Tohaku Math. J., 3 97), Qui. Κ., "Solutios with shocks: A exaple of L cotractive seigroups", Co. Pure Appl. Math., 4 97),

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