Linear singular perturbations of hyperbolic-parabolic type

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1 BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 4, 3, Pages ISSN Linear singular perurbaions of hyperbolic-parabolic ype Perjan A. Absrac. We sudy he behavior of soluions of he problem u u Au = f, u = u, u = u 1 in he Hilber space H as, where A is a linear, symmeric, srong posiive operaor. Mahemaics subjec classificaion: 35B5, 35K15, 35L15, 34G1. Keywords and phrases: singular perurbaions, hyperbolic equaion, parabolic equaion, boundary funcion. 1 Inroducion Le V and H be he real Hilber spaces endowed wih he norm and, respecively, such ha V H, where he embedding is defined densely and coninuously. By, denoe he scalar prodac in H. Le A : V H be a linear, closed, symmeric operaor and Au,u ω u, u V, ω >. 1 In his paper we shall sudy he behavior of he soluions of he problem u u Au = f, >, u = u, u = u 1 P as, where is a small posiive parameer. Our aim is o show ha u v as, where v is he soluion of he problem v Av = f, > P v = u. The main ool of our approach is he relaion beween he soluions of he problems P and P. For k N,p [1, and a,b, we denoe by W k,p a,b;h he usual Sobolev spaces of vecorial disribuions W k,p a,b;h = f D a,b;h; f l L p a,b;h,l =,1,...,k wih he norm c 3 Perjan A. k f W k,p a,b;h = f l p L p a,b;h 1/p. l= 95

2 96 PERJAN A. For each k N, W k, a,b;h is he Banach space equipped wih he norm f W k, a,b;h = max l k fl L a,b;h For s R, k N and p [1, ] we denoe he following Banach space a,b;h = f : a,b H;f l e s L p a,b;h wih he norm W k,p s f W k,p s a,b;h = max l k fl e s L p a,b;h. A priori esimaes for soluions of he problem P In his secion we shall prove he a priori esimaes for he soluions of he problem P which are uniform relaive o he small values of parameer. Firs of all we shall remind he exisence heorems for he soluions of he problems P and P. Theorem A. [1] For any T > suppose ha f W 1,1,T;H,u,u 1 V and he operaor A saisfies he condiion 1. Then here exiss a unique funcion u C,T;H L,T;V saisfying he problem P and he condiions: Au L,T;H, u L,T;V, u L,T;H. Theorem B. [1] If f W 1,1,T;H,u V and A saisfies he condiion 1, hen here exiss a unique srong soluion v W 1,,T;H of he problem P and esimaes v e ω u e ωτ fτ dτ, are rue for T. v e ω Au f e ωτ f τ dτ Before o prove he esimaes for soluions of problem P we recall he following well-known lemma. Lemma A. [] Le ψ L 1 a,b < a < b < wih ψ a. e. on a,b and le c be a fixed real consan. If h C[a,b] verifies 1 h 1 c a ψshsds, [a,b], hen also holds. h c a ψsds, [a,b]

3 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 97 Denoe by 1/ E 1 u, = u u Au,u 1/. Auτ,uτ dτ u τ dτ 1/ Lemma 1. Suppose ha for any T > f W 1,1,T;H,u,u 1 V and he operaor A saisfies he condiion 1. Then here exis posiive consans γ and C depending on ω such ha for he soluions of he problem P he following esimaes are rue. Proof. Denoe by E 1 u, C E 1 u, E 1 u, C E 1 u, fτ dτ Eu, = u 1 u Au,u u,u, T, f τ dτ, T 3 Auτ,uτ dτ. u τ dτ The direc compuaions show ha for every soluion of he problem P he following equaliy d d Eu, = f,u u 4 is fulfilled. From 4 i follows ha d d Eu, f u u. 5 As Eu, and u u CEu, 1/, hen from 5 we have d Eu, d Inegraing he las inequaliy we obain 1 Eu, 1 Eu, C 1/. Cf Eu, 1/ fτ dτ. Eu,τ From he las inequaliy using Lemma A we ge he esimae 1/ [ 1/ Eu, C Eu, fτ dτ ]. 6

4 98 PERJAN A. I is easy o see ha here exis posiive consans C,C 1 such ha 1/ 1/. C Eu, E1 u, C 1 Eu, 7 Using he inequaliy 7 from 6 we obain he esimae. To prove he esimae 3 le us denoe by E h u, = u h u Au h u,u h u 1 u h u u h u,u h u u τ h u τ dτ Auτ h uτ,uτ h uτ dτ, h >,. For any soluion of he problem P we have d d E hu, = u h u u h u,f h f,. Dividing he las equaliy by h and hen passing o he limi as h we ge d d Eu, = f,u u. 8 Since u = u 1,u = f u 1 Au, hen he esimae 3 follows from 8 in he same way as he esimae follows from 4. Lemma 1 is proved. 3 Relaion beween he soluions of he problems P and P In his secion we shall give he relaion beween he soluions of he problems P and P. This relaion was inspired by he work [3]. A firs we shall prove some properies of he kernel K,τ of ransformaion which realizes his connecion. For > denoe where and s = s e η dη. K,τ = 1 π K 1,τ 3K,τ K 3,τ, 3 τ K 1,τ = exp 4 3 6τ K,τ = exp 4 τ K 3,τ = exp τ τ τ, 9, 1, 11

5 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 99 Lemma. The funcion K,τ possesses he following properies: i K CR R C R R ; ii K,τ = K ττ,τ K τ,τ, >,τ > ; iii K τ, K, =, ; iv K,τ = 1 exp τ, τ ; v For each fixed >, here exis consans C 1, > and C > such ha K,τ C 1,exp C τ/, K,τ C 1,exp C τ/, K τ,τ C 1,exp C τ/, K ττ,τ C 1,exp C τ/ for τ > ; vi K,τ >,, τ ; vii For any ϕ : [, H coninuous on [, such ha ϕ M expc for, he relaion lim K,τϕτdτ = is valid in H for each fixed, < 1; viii K,τdτ = 1, ; e τ ϕτdτ ix Le ρ : [, R,ρ C 1 [,,ρ and ρ be increasing funcions and ρ Me c, ρ Me c, for [,. Then here exis posiive consans C 1 and C such ha K,τ ρ ρτ dτ C 1 e C, > ; x Le fe C,f e C L, ;H wih some C. Then here exis posiive consans C 1,C such ha f xi There exiss C > such ha K,τfτdτ C 1 e C f L H C, ;H,, < 1; Kτ,θexp θ dθdτ C,, >.

6 1 PERJAN A. Proof. The properies i-iv can be verified by direc calculaion. Proof v. From 9, 1 and 11 we have K,τ = 1 [ 8π 3K 1,τ9K,τ 6 exp ] τ, >,τ >, 1 4 K τ,τ = 1 [ ] 4π K 1,τ 9K,τ 4K 3,τ, >,τ >, 13 K ττ,τ = 1 [ 8π 3 K 1,τ 7K,τ 8K 3,τ 6 exp ] τ, >,τ >. 4 As s π for s R and exps s C for s [,, hen τ K 1,τ exp, τ >, >, 15 4 K,τ C exp >,τ >, 16 τ 4 τ K 3,τ C exp >,τ > Using 15, 16 and 17 from 1, 13 and 14 we ge he esimaes from propery v. The propery v is proved. Proof vi. We shall prove propery vi using he maximum principle for he soluions of equaion ii. I is easy o see ha We inend o prove ha K, = 1 [ 3 exp π ],. 18 K, >,. 19 To his end we consider he funcion fs = qs qs/, where qs = exps s,s [,. Because K, = π 1 exp /4f /, o prove 19 i is sufficien o show ha fs > for s [,. A firs we shall prove ha q s < for s [,. Since q s = sqs 1, q s = s 1qs s, q s = 8s 3 1sqs 4s 1 and lim s sqs = 1, hen q = 1 and lim s q s =. Suppose ha here exiss s 1, such ha q s 1 =, i. e. qs 1 = s 1 s As q s 1 = 4s 1 1 1, hen s 1 is he poin of maximum for q s, and q s 1 <,s 1 [, and consequenly he funcion qs is decreasing on,. Furher, we noe ha π f = q =, lim fs =. s

7 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 11 Suppose ha s 1, is any criical poin for funcion fs, i. e. f s 1 =, hen we have: 4s 1 qs 1 1 s 1 qs 1 / 3/ =, from which follows s1 fs 1 = qs 1 q = 3 6qs 1. 1 s 1 As q s < for s,, hen s 1 qs 1 < 1. Hence from 1 i follows ha fs 1 >. The las condiion and condiions permi us o conclude ha fs > for s [,, i. e. K, > for. Finally, from ii, iv, v and 18 i follows ha he funcion V,τ = exp τ/4k,τ is he bounded soluion of he problem V,τ = V ττ,τ, >,τ > V,τ = 1 exp τ, τ, V, = 1 π f,, P.V in Q T =,τ : τ, T, for any T >. Using he maximum principle for he soluions of problem P.V we conclude ha V,τ > and consequenly K, τ >. The propery vi is proved. Proof vii. For any fixed C > and for any fixed >, we ge K,τe Cτ dτ = 3 ] exp = 4 [ exp C1 C 1 C 3 C [ 3 C 1 exp 3 4 1C e η dη 1 exp C1 C 1 C ] = O,. If ϕ : [, H, and ϕ H Me C,, hen from we have K,τϕτdτ M K,τe Cτ dτ MC, < 1, 3 H for any fixed >. Similarly as was obained we ge K 3,τe Cτ dτ = = [ expc1 C 1 C 1 C 1 [ exp C1 C 1 1 C 1 C ] =

8 1 PERJAN A. 1 ] e η dη = O,, 4 1C for any fixed >. If ϕ : [, H, and ϕ H M expc,, hen from 4 i follows ha K 3,τϕτdτ M K 3,τexpCτdτ CM 5 H for < 1. For ϕ : [, H, ϕ C, ;H and ϕ H M expc,, we have 3 K 1,τϕτdτ = exp exp τ [ τ 4 τ ] ϕτdτ 3 exp 1 4 exp τ exp τ τ τ ϕτdτ ϕτdτ = I 1 I I 3. 6 Le us evaluae he inegrals I i, i = 1,,3, from 6. For any fixed < < C 1 we have I 1 H M exp 3 4 exp τ τ 4 Cτ τ exp η dηdτ M C exp C, < 1, 4 and 3 I H M exp 1 π 4 C, < 1. exp τ Cτ dτ 8 A las, le us invesigae he behaviour of inegral I 3 as. I 3 can be represened in he form I 3 = exp τ [ τ ] π ϕτdτ π exp τ ϕτdτ. 9 The firs erm of he righ side of 9 can be evaluaed as follows exp τ [ τ ] π ϕτdτ H

9 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 13 M = M [ exp 1 C exp τ τ Cτ dτ = = M [ 1 exp 1 C 1 C 4 1 C 4 1 C ] = 1 C exp 4 1 C From 9 and 3 follows he esimae I 3 π Hence due o 6, 7, 8 and 31 we have K 1,τϕτdτ π exp η ] dη C, < 1. 3 exp τ ϕτdτ C, < ǫ H e τ ϕτdτ C, < 1, 3 H for any fixed, < 1. Finally, from 3, 5 and 3 we ge he proof of he propery vii. Proof viii. Inegraing by pars we have [ K 1,τdτ = exp K,τdτ = 3 [ [ K 3,τdτ = 1 from which follows he proof of he propery viii. 1 3 exp 4 1 ], ], Proof ix. As ρ is increasing and ρ M expc, hen inegraing by pars and using he propery v we ge K 1,τ ρ ρτ dτ = exp 3 4 [ exp τ τ ], ρ ρτ dτ

10 14 PERJAN A. exp τ τ exp ] 3 ρτ ρ dτ = ρ ρ exp 4 τ 4 exp τ ρ τ Similarly can be obained he equaliies ρ dτ 3 ρτ exp 4 τ sign τdτ. 33 K,τ ρ ρτ dτ = 3 ρ ρ exp exp 4 exp exp τ ρ dτ ρτ 4 3τ ρ τ τ sign τdτ, 34 and 1 K 3,τ ρ ρτ dτ = ρ ρ 1 exp τ ρ dτ ρτ 4 τ τ exp ρ τ sign τdτ, 35 As a consequence from 33, 34 and 35 we ge K,τ ρ ρτ dτ = 1 1 [ ρ ρ π 1 exp [ 3 6τ ρ τ exp 4 τ ρ dτ ρτ 4 τ 3 τ exp 4 τ τ τ ] ] exp sign τdτ, 36

11 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 15 Since ρ is increasing and ρ M expc, hen i follows ha 1 1 ρ ρ C 1 exp M expc 4 C C 1 expc,, 1 8C. 37 Furher we have exp τ ρ ρτ dτ 4 M = 4M 1 τ exp C max,τ τ dτ = 4 η exp η C max, η dη = = 4M expc 1 η exp η dη η exp η C η dη As sexps C, for s, hen we have M exp 3 4 exp = C 1 expc 3 4 C 1 expc,. 38 3τ ρ τ exp exp Cτ 3τ τ 1 Similarly we ge he esimaes exp 3 4 exp τ τ dτ C 1 dτ exp Cτ τ = 4τ exp C η η dη C 1 exp C,. 39 τ ρ τ dτ C 1 exp C,, 4 and τ exp τ ρ τ dτ C 1 exp C,. 41

12 16 PERJAN A. Finally from 36 and he esimaes follows he esimae from propery ix. Proof x. From he properies viii and ix i follows ha f K,τ K,τfτdτ H τ f θ H dθ M K,τ f fτ H dτ C 1 e C f L C, ;H, for, 1. Propery x is proved. K,τ e Cτ e C dτ Proof xi. Denoe by K,τ = K,τ =1, K i,τ = K i,τ =1,i = 1,,3. Then I = Kτ,θexp θ dθdτ = Kτ, θ exp θdθdτ = = I 1 3I I 3. 4 As < K i τ,θ C exp I i τ θ 4τ For I 1 we have he esimae I 1 = = 1 3 K 1 τ,θe θ dθdτ = π,i =,3, hen τ θ exp dθdτ C,,i =, τ 3τ τ 1/ exp τdτ exp 9τ τ exp 3η τ ηdηdτ = 4 τ τ 1/ dτ C,. 44 From 4, 43 and 44 follows he propery xi. Lemma is proved. Now we are ready o esablish he relaion beween he soluions of he problem P and he corresponding soluions of he problem P. Theorem 1. Le A : DA H H be a linear and closed operaor, f W 1, C, ;H for some C. If u is a soluion of he problem P such ha u W, C, ;H wih some C, hen he funcion v which is defined by v = K,τuτdτ saisfies he following condiions: v Av = F,, >, v = ϕ, P.v

13 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 17 where F, = 1 3 [exp π 4 ϕ = 1 ] u 1 e τ uτdτ. K,τfτdτ, Proof. Inegraing by pars and using he properies i iii and v of Lemma we ge v = K,τuτdτ = K ττ,τ K τ,τ uτdτ = = K,τ u τ u τ dτ K,u 1 Av K,τfτdτ. Thus v saisfies he equaion from P.v. From propery viii of Lemma follows he validiy of he iniial condiion of P.v. Theorem 1 is proved. 4 The limi of he soluions of he problem P as In his secion we shall sudy he behavior of he soluions of he problem P as. Theorem. Suppose f W 1, C, ;H, wih some C, u,u 1 H,Au,Au 1 H and he operaor A saisfies he condiion 1. Then u v C 1 Me C,, 1, 45 where u and v are he soluions of he problems P and P.v, respecively, M = f u Au u 1 f L C, ;H, and C 1 and C are independen of M and. If u,au,u 1,f V,f W, C, ;H, wih some C, 46 hen u v hexp C1 M 1 e C,, 1, 47 where h = f u 1 Au, M 1 = f Ah f L C, ;H, and C 1 and C are independen of M 1 and. If u,au,au 1 V,Af W 1, C, ;H, wih some C, 48 hen u v C 1 M e C,, 1, 49 where M = Af Au Au 1 A u Af L C, ;H, and C 1 and C are independen of M and.

14 18 PERJAN A. Proof. Under he condiions of he heorem from 3 follows he esimae u CM,. 5 According o Theorem 1 he funcion w which is defined by w = K,τuτdτ is a soluion of he problem w Aw = F,, w = w, P.w where F, = 1 3 [exp π 4 F, = F, 1 K,τfτdτ, ] u 1, w = Using he propery x of Lemma and he esimae 5 we ge e τ uτdτ. u w C 1 Me c,. 51 Le us denoe R = v w, where v is he soluion of he problem P.v and w is he soluion of he problem P.w. Then R is he soluion of he problem R AR = F,,, where R = u w and R = R, As and hence F, = f K,τfτdτ F,. d d R = AR,R F,, R ω R F, R,, 1 R e ω 1 R hen using Lemma A we obain he esimae R e ω R Fτ, Rτ e ωτ dτ,, Fτ, e ωτ dτ,. 5

15 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 19 From 5 follows he esimae R e τ uτ u dτ τ e τ u s dsdτ CM 53 for < 1. Now le us esimae F,. Using he propery x of Lemma we have f K,τfτdτ C 1 M e C,. 54 As and hen 3τ exp 4 ωτ τ 3τ τ dτ = exp 4 ωτ dτ C e τ τ C,, < 1, 1 τ e ωτ dτ C,, < 1, e ωτ F τ, dτ C u 1 CM,, < From 54 and 55 follows he esimae e ωτ Fτ, dτ C 1 Me ω,, < From 5, using he esimaes 53 and 56 we ge R C 1 Me C,, < Finally from esimaes 51 and 57 we have u v u w R C 1 Me C,, < 1. The esimae 45 is proved. Le us prove he esimae 47. Denoe by z = u hexp. If u,u 1 and f saisfy he condiions 46 and A saisfies he condiion 1, hen z is a soluion of he problem z z Az = f exp h,, z = f Au, z =. According o Theorem 1 he funcion w 1 which is defined by w 1 = K,τzτdτ

16 11 PERJAN A. is a soluion of he problem w 1 Aw 1 = F 1,,, w 1 = exp τ zτdτ, where F 1, = [ K,τ f τ exp ] Ah dτ. Furher denoe by v 1 = v, where v is he soluion of he problem P.v. Then v 1 is he soluion of he problem v 1 Av 1 = f,, v 1 = f Au. Le R 1 = w 1 v 1. Then R 1 is he soluion of he problem R 1 AR = F 1, f,, R 1 = Using Theorem B we obain he esimae R 1 e ω R 1 Using he esimae 3 we ge z C 1 f Ah τ exp τ z θdθdτ. for. Then from 59 follows he esimae e ω F 1 τ, f τ dτ,. 58 f 1 exp Ahdτ C 1 e C M 1 59 R C 1, < 1. 6 Due o he propery x of Lemma we ge he esimae f K,τdτ C 1 e C f L C, ;H,, < Furher using he propery xi of Lemma we have Kτ,θexp θ Ahdθdτ CM 1,. 6 Using he esimaes 6, 61 and 6 from 58 follows he esimae R 1 C 1 e C M 1,, < 1. 63

17 LINEAR SINGULAR PERTURBATIONS OF HYPERBOLIC-PARABOLIC TYPE 111 From he propery xi of Lemma and he esimaes 59 we ge w 1 z K,τ τ z θdθdτ C 1 e C M 1,, < Finally, from he esimaes 63 and 64 we obain z v 1 z w 1 R 1 C 1 e C M 1,, < 1, i. e. he esimae 47. Le us prove he esimae 49. Denoe by y = Au, y 1 = Av. Then under condiions 48 y is he soluion of he problem y y Ay = Af,, y = Au, y = Au 1, and y 1 is he soluion of he problem From 45 follows he esimae As from 1 i follows ha y 1 Ay 1 = Af, y 1 = Au. Au Av C 1 e C M,, < Au Av ω u v, hen using 65 we obain he esimae 48. Theorem is proved. Remark 1. The relaion 47 shows ha he funcion u possesses he boundary funcion in he neighborhood of he line =. Bu, if h =, hen he funcion u like u does no have a boundary funcion. Finally le us give one simple example. Consider he following iniial boundary problems u x, u x, Lx, x ux, = fx,, x Ω, >, ux, = u x, u x, = u 1 x, x Ω, ux, =, x, on Ω [,, v x, Lx, x vx, = fx,, x Ω, >, vx, = u x, x Ω, ux, =, x, on Ω [,, 66 67

18 11 PERJAN A. where Ω R n is a bounded domain wih a smooh boundary Ω. The operaor Lx, x = n i,j=1 a ij x ax x i x j is uniformly ellipic in Ω, i.e. a,a ij : Ω R, a,a ij CΩ, a ij x = a ji x, and n a ij xξ i ξ j ω ξ, ξ R n,x Ω, i,j=1 where ω >,ax for x Ω. Le us pu H = L Ω,V = H 1 Ω. In his condiions he problems P and P.v represen he funcional analyical saemen of he problems 66 and 67 respecively, where A is he closure of he operaor L in L Ω. Under suiable condiions on he funcions u,u 1 and f which follow from condiions 46 and 48 from Theorem for he variaional soluions of he problems 66, 67 we ge u = v O in C,T;L Ω,, u = v hexp O in L,T;L Ω,, u = v O in L,T;H 1 Ω,, where hx = u 1 x Lx, x u x fx,. References [1] V.Barbu. Semigroups of nonlinear conracions in Banach spaces. Buchares, Ed. Acad. Rom., 1974 in Romanian. [] Gh.Moroşanu. Nonlinear Evoluion Equaions and Applicaions, Buchares, Ed. Acad. Rom., [3] M.M. Lavreniiev, K.G. Rezniskaia, B.G.Yahno. The inverse one-dimenional problems from mahemaecal physics. Nauka, Novosibirsk, 198 in Russian. Perjan A. Faculy of Mahemaics and Informaics, Moldova Sae Universiy, 6, Maeevici sr., Chşinau, 9, Republic of Moldova perjan@usm.md Received December 31,