AN APPLICATION OF THE THEORY OF SCALE OF BANACH SPACES. Łukasz Dawidowski. 1. Introduction

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1 Aales Mathematicae Silesiaae 29 (2015), Prace Naukowe Uiwersytetu Śląskiego r 3332, Katowice DOI: /amsil AN APPLICATION OF THE THEORY OF SCALE OF ANACH SPACES Łukasz Dawidowski Abstract. The abstract Cauchy roblem o scales of aach sace was cosidered by may authors. The goal of this aer is to show that the choice of the sace o scale is sigificat. We rove a theorem that the selectio of the saces i which the Cauchy roblem u t u = u u s with iitial boudary coditios is cosidered has a ifluece o the selectio of idex s. For the Cauchy roblem coected with the heat equatio we will study how the chage of the base sace ifluets the regularity of the solutios. 1. Itroductio Fractioal owers of a ositive sectorial oerator A defied o aach sace X ad scale of aach sace geerated by the owers of oerator A were cosidered i the aers [7], [2], [3]. We would like to rove that choice of the base sace affect the selectio of assumtios. Let us cosider the abstract Cauchy roblem (1.1) { ut + Au = F (u), t > 0, u(0) = u 0, where A is sectorial i a aach sace X. Without loss of geerality (addig if ecessary a term cu to both sides of (1.1)), we may assume that A is Received: Revised: (2010) Mathematics Subject Classificatio: Primary 4670; Secodary 35K10. Key words ad hrases: scale of aach saces, arabolic equatios.

2 52 Łukasz Dawidowski sectorial ad ositive; Reσ(A) > 0. We will search for local X z -solutio of the roblem (1.1) i the sese recised below. Defiitio 1.1. Let X be a aach sace, z [0, 1) ad u 0 be a elemet of X z. If, for some real τ > 0, a fuctio u C([0, τ), X z ) satisfies u(0) = u 0, u C 1 ((0, τ), X), u(t) D(A) for all t (0, τ), u solves the first equatio i (1.1) i X for all t (0, τ), the u is called a local X z -solutio of the roblem (1.1). The followig theorem gives coditios for the existece of local X z - solutio. Theorem 1.2. Let X be a aach sace, A: D(A) X a sectorial oerator i X with Re σ(a) > 0. Let F : X z X be Lischitz cotiuous o bouded subsets of X z for some z [0, 1). The, for each u 0 X z, there exists a uique X z -solutio u = u(t, u 0 ) of (1.1) defied o its maximal iterval of existece [0, τ u0 ), which meas that either τ u0 = + or (1.2) if τ u0 < + the lim su t τ u 0 u(t, u 0 ) X z = +. We ca fid the roof of this Theorem i [6, Sectio 3.3]. 2. Sectoriality of oerator Assume that (X, ) is a aach sace, ad let A: D(A) X X be a sectorial oerator i X. We cosider a scale of the aach saces (X α ) α R, geerated by oerator A i the sace X (for details see [7] or [3, Chater 1]). The, for α R, we ca cosider oerators A X α : X α+1 X α X α give by (i) if α = 0, the a oerator A X : X 1 X X coicides with A: D(A) X X, (ii) if α > 0, the X α = D(A α ) is a dese subset of X ad X α+1 is desely cotaied i X 1, so we ca cosider a oerator A X α : X α+1 X α X α as a restrictio of A: D(A) X X to the set X α+1, i.e. A X αx = Ax, x X α+1,

3 A alicatio of the theory of scale of aach saces 53 (iii) if α < 0, the the sace X α is a comletio of the sace X i the orm X α, so we ca cosider a oerator A X α : X α+1 X α X α as a realizatio (extesio) of A: D(A) X X, i.e. A X αx = Ax, x X α+1 X. We reset lemma which says that whe a oerator A: D(A) X X is sectorial, the all the oerators A X α : X α+1 X α X α for α R are also sectorial. Lemma 2.1. Whe A X : X 1 X X is a sectorial oerator with Reσ(A X ) > 0, the all oerators A X α : X α+1 X α X α for α R are sectorial. The roof of this Lemma ca be foud i [3, ]. 3. Alicatios The oerator defied i the sace L () with D( ) = W 2, () W 1, 0 (), where R is a bouded domai ad [1, ), is sectorial (see [8, Theorems 2.12 ad 2.13,. 77,79]). The usig Theorem 1.2 to get a solutio of roblem (3.1) u t u = f(u), u = 0, u(0, x) = u 0 X z, we have to show that the fuctio f : X z X satisfies Lischitz coditio o bouded subsets of X z (z [0, 1)). It is ofte easier to verify this coditio i a bigger sace o the scale. Let us take X = L () where R is a bouded domai with C 2 ad [1, ). The the aach scale geerated by the sace X ad oerator is i the form X γ = W 2γ, 2γ, () for γ R (see [2, 3]), where W () deotes the fractioal Sobolev sace with Dirichlet boudary coditios (for more details see [8, Sectio ]). If we choose the base sace as Y = X γ = W 2γ, () for some γ [0, 1 2 ], the we will have to rove that the fuctio f : Y z Y, so f : X γ+z X γ is Lischitz cotiuous o bouded subsets of X γ+z (z [0, 1)).

4 54 Łukasz Dawidowski Further, we will cosider exemlary oliearity f give by: f(u) = u u s, where s R +. We would like to fid the largest ossible value of s such that f is Lischitz cotiuous o bouded subsets of Y z = X γ+z with certai z [0, 1). Lemma 3.1. Let R be a bouded domai with C 2 ad [1, ). Let z [0, 1). The fuctio f : W 2γ+2z, () W 2γ, (), f(u) = u u s with s R + is Lischitz cotiuous o bouded subsets of W 2γ+2z, () 2z if s s 0 = +2γ 2z (or for each s R+ if + 2γ 2z 0). Proof. We have (usig mea value theorem) the followig estimate: f(u) f(v) = u u s v v s u u s v s + u v v s u s u v ( u s 1 + v s 1) + u v v s = u v ( s u s + s u v s 1 + v s) c s u v ( u s + v s ). Our aim is to estimate the differece f(u) f(v) W 2γ, (). For that urose, goig through the dual saces, we first observe that L +2γ () W 2γ, (). To rove this fact, we will show the corresodig embeddig for dual saces W 2γ, () L r () where is Hölder cojugate exoet to ( = 1) ad r to +2γ. From the Sobolev s theorem (see Theorem 6 (Geeral Sobolev iequalities) i [4], age 270) we kow that this embeddig holds whe 1 r 1 2γ ad 1 < 2γ. If we take r = 2γ +, we will have 1 2γ + = = 2γ r + = 1 2γ = 1 2γ, so that the embeddig W 2γ, () L r () holds. Let us fix a bouded set W 2γ+2z, () ad let u, v. Let us choose R = 2γ+ +2γ 2z, Q = 2γ+ 2z so that 1 R + 1 Q = 1. The usig Hölder

5 A alicatio of the theory of scale of aach saces 55 iequality, with exoets R ad Q, we obtai: f(u) f(v) W 2γ, () c f(u) f(v) L +2γ () (3.2) = c u u s v v s L +2γ () = c c c c s u u s v v s +2γ u v c s ( u s + v s ) u v +2γ R dx ) +2γ dx +2γ ) 1 +2γ R (2 max ( u, v )) s +2γ Q dx ) +2γ dx ) 1 +2γ Q. To assure that the secod itegral is fiite we have to fid the largest s such that W 2γ+2z, () L s 2γ+ Q (). For that urose, if we take 2γ+2z, we verify that W 2γ+2z, () L () L s 2γ+ Q () for each s 0, sice 0 2γ + 2z ad is a bouded set. O the secod had if we take 2γ + 2z <, s 2z 0 = +2γ 2z ad it is eough to check for s s 0 the coditio s 2γ+ Q 2z +2γ 2z 2γ+ + 2γ 2z = = 2γ 2z. 2γ+ 2z Sice u, v are elemets of the set which is bouded i W 2γ+2z, (), hece bouded i L s 2γ+ Q () for s s 0, we observe that the secod itegral (2 max ( u, v ))s +2γ Q dx is fiite ad bouded by some costat. Therefore our estimatio has the form f(u) f(v) W 2γ, () c s, u v +2γ R dx = c s, u v L +2γ R (). ) 1 +2γ R

6 56 Łukasz Dawidowski I the ed we otice that W 2γ+2z, () L +2γ R (), sice +2γ R = 2γ+ +2γ +2γ 2z + 2γ 2z = = 2γ + 2z. This fiishes the roof of Lischitz cotiuity of f o bouded subsets of () because W 2γ+2z, f(u) f(v) W 2γ, () c s, u v L +2γ R () c s, u v W 2γ+2z, (). Remark 3.2. Let us cosider a critical exoet z = 1. The we ca rove (i a very similar way) that oliearity f : W 2γ+2, () W 2γ, (), f(u) = u u s is Lischitz cotiuous o bouded subsets of W 2γ+2, () if 2 s s 0 = +2γ 2 (or for each s R+ if + 2γ 2 0). As a cosequece of revious lemma, followig theorem holds: Theorem 3.3. Let R be a bouded domai with C 2, [1, ) ad X γ = W 2γ, () be the aach scale. Let z [0, 1). Let us cosider the roblem (3.1) i the sace Y = X γ = W 2γ, () with fuctio f : W 2γ+2z, () W 2γ, (), f(u) = u u s with s R +. The the roblem (3.1) has a local Y z 2z -solutio if s s 0 = +2γ 2z (or for each s R+ if + 2γ 2z 0). Remark 3.4. These results are geeralizatio of examles give i the moograh [3] (Examle ad Examle 8.1.7). If we take γ = 0, we see that the fuctio f : W 2, () L (), f(u) = u u s is Lischitz cotiuous o bouded subsets whe s 2 2. ut whe we take γ = 1 2, we will obtai that f : W 1, () W 1, (), f(u) = u u s will be Lischitz cotiuous o bouded subsets if s 2. We ca remark, that whe γ [0, 1 2 ] is bigger, so the sace W 2γ, () is bigger, the we will be able to take smaller idex s i oliearity f.

7 A alicatio of the theory of scale of aach saces More regular solutios I this sectio we show that selectio of base sace ca affect the regularity of solutio of cosidered roblem. I the followig examles exected solutio has to be more regular. Examle 4.1. Let R be a bouded domai with C 2 ad let us cosider the Cauchy roblem (4.1) { ut u = 0, u = 0. A abstract oerator A: D(A) X X i the base sace X will be defied by the differetial oerator subjected to suitable boudary coditios. Usually we take set X = W 1, 0 () ad the domai D(A) = W 3, (). However we ca select the base sace i aother way, takig ad defiig the base sace as X 1 = D(A) = W 3, () X α = [X, X 1 ] α, α (0, 1), where [, ] α deote the comlex iterolatio fuctor. The we ca cosider a oerator Ã i the base sace X = X α with the domai D(Ã) = X 1 = X α+1 as a restrictio of the oerator A, which meas that Ãu = Au for all u Xα+1. Usig Sobolev s embeddig theorem (see [1, Theorem 5.4, ]) we < 1. It kow that W k, () C m+µ () if ad oly if 0 < µ k m imlies that k = m + 1. We will fid the coditios which allow us to select [1, ) such that the solutios of the roblem (4.1) will belog to the sace C m+µ (). Takig 0 = µ+m k we have k m k m µ + m k = µ, what shows, that 0 is the smallest exoet which assures the embeddig of the Sobolev sace W k, 0 () i the sace of cotiuous fuctios. I this examle oerator A is the Lalace oerator with Dirichlet boudary coditios, so to fid cotiuous solutios of (4.1) we have to kow that the base sace is cotaied i C ε () for some ε (0, 1), which is satisfied if the domai D(A) is cotaied i C 2+ε (). Takig the base sace tyically as

8 58 Łukasz Dawidowski W 1, 0 () ad D(A) = W 3, (), we have to show that W 3, () C 2+ε (). This is true for 0 = 1 ε. It meas that whe we cosider the roblem (4.1) i the base sace W 1, 0 () with 0, the every solutio u has the roerty that u is cotiuous o R ad so is u t. The theorem reseted below gathers what was show i the examle. Theorem 4.2. Let R be a bouded domai with C 2, [1, ). Let us cosider the Cauchy roblem (4.1) i the base sace W 1, 0 (). The, if 0 = 3, 1 ε, every solutio u W () of (4.1) has the roerty that u t ad u are cotiuous o R. (4.2) Examle 4.3. Let us cosider followig roblem u t u = 0 i D T = [0, T ], u = 0, u(0, x) = u 0 (x), x, where T > 0 ad R is a bouded domai with C 2. We will search for the solutios u i the base sace W 1, 0 (D T ). Our aim is to set [1, ) such that all solutios are cotiuous, i.e. W 1, (D T ) C µ (D T ) for some µ (0, 1). Let us take 0 = µ. The we see that = µ > 0, ad from Sobolev s embeddig theorem we coclude that W 1, (D T ) C µ (D T ). Sice u W 1, (D T ) we kow that all derivatives u t, u xi for i {1,..., } belog to the sace L (D T ) ad usig equatios (4.2) we claim that u = u t L (D T ). Now, our urose is to show that u belogs to W 2, (D T ). Kowig that u W 1, 0 () ad usig the Calderó-Zygmud tye estimatios (see [5, Theorems 8.12 ad 8.13, ]) we get that u W 2, () ad fulfills a estimate u W 2, () C( u L () + u L ()) = C( u t L () + u L ()), for some costat C > 0 deedig o,. The, for every multiidex α with α = 2 we have D α u L (), so there exist costats M α such that D α u L () M α <, ( α = 2). We have u W 2, (D T ) = α =2 D α u L (D T ) + u W 1, (D T ),

9 A alicatio of the theory of scale of aach saces 59 thus we obtai the estimatio: u W 2, (D T ) = α =2 α =2 = α =2 ( T 0 ( T Mα dt 0 D α u dx dt) 1/ + u W 1, (D T ) ) 1 + u W 1, (D T ) M α T 1 + u W 1, (D T ) <. This examle showed that, uder our assumtios ad whe > +1 1 µ, every solutio u of (4.2) which belogs to W 1, 0 (D T ), is also a elemet of the sace C µ (D T ) W 2, (D T ). Cosequetly, the above examle lead us to the coclusio. Theorem 4.4. Let R be a bouded domai with C 2, T > 0, ad [1, ). Assume that u W 1, 0 (D T ) is a solutio of the roblem (4.2). If > 0 = +1 1 µ the the fuctio u belogs to the sace Cµ (D T ) W 2, (D T ). Refereces [1] Adams R.A., Sobolev saces, Academic Press, New York Sa Fracisco Lodo, [2] Cholewa J.W., Dłotko T., Global attractors i abstract arabolic roblems, Cambridge Uiversity Press, Cambridge, [3] Dawidowski Ł., Scales of aach saces, theory of iterolatio ad their alicatios, Wydawictwo Uiwersytetu Śląskiego, Katowice, [4] Evas L.C., Partial differetial equatios, Graduate Studies i Mathematics 19, America Mathematical Society, Providece, Rhode Islad, [5] Gilbarg D., Trudiger N., Ellitic artial differetial equatios of secod order, Rerit of the 1998 editio, Sriger-Verlag, erli, [6] Hery D., Geometric theory of semiliear arabolic equatios, Lecture Notes i Mathematics 840, Sriger-Verlag, erli, [7] Komatsu H., Fractioal owers of oerators, Pacific J. Math. 19 (1966), [8] Yagi A., Abstract arabolic evolutio equatios ad their alicatios, Sriger Moograhs i Mathematics, Sriger-Verlag, erli, Istitute of Mathematics Uiversity of Silesia akowa Katowice Polad lukasz.dawidowski@us.edu.l