The structure of a set of positive solutions to Dirichlet BVPs with time and space singularities

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1 The rucure of a e of poiive oluion o Dirichle BVP wih ime and pace ingulariie Irena Rachůnková a, Alexander Spielauer b, Svaolav Saněk a and Ewa B. Weinmüller b a Deparmen of Mahemaical Analyi, Faculy of Science, Palacký Univeriy, 17. liopadu 12, CZ Olomouc, Czech Republic irena.rachunkova@upol.cz vaolav.anek@upol.cz b Deparmen of Analyi and Scienific Compuing, Vienna Univeriy of Technology, Wiedner Haupraße 8-1, A-14 Wien, Auria alexander.pielauer@gmx.ne e.weinmueller@uwien.ac.a Abrac: The paper dicue he olvabiliy of he ingular Dirichle boundary value problem u ( + a u ( a 2u( = f(,u(,u (, u( =, u(t =. Here a (, 1 and f aifie he local Carahéodory condiion on [,T] D, where D = (, R. I i hown ha he cardinaliy of he e L of all poiive oluion o he problem i a coninuum. In addiion, he rucure and he properie of he e L are decribed. Applicaion and numerical imulaion of he reul are preened. Mahemaic Subjec Claificaion 21: 34B18, 34B16, 34A12 Key word: nonlinear ordinary differenial equaion of he econd order, ime and pace ingulariie, e of all poiive oluion, Leray-Schauder nonlinear alernaive. 1 Inroducion We conider he ingular Dirichle boundary value problem u ( + a u ( a 2u( = f(,u(,u (, (1.1a u( =, u(t =, (1.1b 1

2 where a (, 1. Here, f aifie he local Carahéodory condiion on [,T] D, where D = (, R. We recall ha a funcion h : [,T] A R, A R R, aifie he local Carahéodory condiion on [, T] A, if (i h(,x,y : [,T] R i meaurable for all (x,y A, (ii h(,, : A R i coninuou for a.e. [,T], (iii for each compac e U A here exi a funcion m U L 1 [,T] uch ha h(,x,y m U ( for a.e. [,T] and all (x,y U. For uch funcion we ue he noaion h Car([,T] A. We ee ha (,y D for each y R, and hence f(,x,y may be ingular (unbounded in our cae a x =. Equaion (1.1a ha a ime ingulariy a = due o he rucure of he differenial operaor on i lef hand ide. Thi operaor ha he equivalen form ( a ( a u and, afer he ubiuion v( = a u( i ake he form ( a v (. Therefore, reul derived for equaion (1.1a alo apply for he modified equaion ( a v ( = g(,v(,v (. Such ype of model arie in he udy of phae raniion of Van der Waal fluid [3], [8], [12], [14], [18], in populaion geneic, in model for he paial diribuion of he geneic compoiion of a populaion [6], [7], in he homogenenou nucleaion heory [1], in relaiviic comology in decripion of paricle which can be reaed a domain in he univere [15], and in he nonlinear field heory [9], in paricular, when decribing bubble generaed by calar field of he Higg ype in he Minkowki pace [5]. Problem (1.1, where f ha no ingulariy a x =, i.e. f aifie he local Carahéodory condiion on [, T] D, where D = [, R, ha been inveigaed in [16]. Thi paper provide a comprehenive udy of he e of all poiive oluion of problem (1.1. Syem of he form u ( A 1 u ( A u( = 2 f(,u(,u (, (,T], (1.2a G(u(,u (,u(1,u (1 =, u C 1 [, 1], (1.2b where, A 1 and A are real valued n n marice, f : (, 1] R n R n R n and G : R n R n R n R n R m are mooh funcion, m 2n, have been udied in [19]. The miing 2n m condiion have o be formulaed in uch a way ha he requiremen u C 1 [, 1] i aified. The main aim in [19] wa o inveigae he rucure of boundary condiion which yield a well-poed boundary value problem. Moreover, in linear cae, he exience and uniquene heory wa provided and he moohne of u wa udied. In he nonlinear cae, ufficien condiion for u o be iolaed, or locally unique, have been pecified. The approach aken in [19] i baed on a echnique developed in [1]. Inead of inveigaing direcly he econd order yem (1.2a, i fir order 2

3 form obained afer he o called Euler ranformaion y( = (y 1 (,y 2 ( T := (u(,u ( T i analyzed, y ( M y( = F(,y(, (,T], (1.3 where M = ( ( I, F(,y( = A A 1 + I f(,y 1 (, y 2(. I urn ou ha he eigenvalue of M play crucial role in decribing he oluion rucure and herefore, he rucure of boundary condiion neceary for he oluion o be coninuou on [, 1]. Thi i clear, becaue he fundamenal marix oluion read Y ( = e M ln. In cae of he homogeneou differenial equaion (1.1a, we have y ( M y( =, (,T], M = ( 1 a a + 1, (1.4 and he eigenvalue of M are λ 1 = a and λ 2 = 1. By decoupling (1.4, we conclude ha he general oluion of he homogeneou problem (1.1a i u( = c 1 λ 1 + c 2 λ 2 = c 1 a + c 2 wih arbirary conan 1 c 1,c 2 R. Since boh eigenvalue are poiive, i follow immediaely from [19] ha he problem (1.1 i no well-poed and ha infiniely many oluion. By precribing finial condiion, u(t =, u (T = c, inead of (1.1b, he problem become well-poed and can be olved numerically, cf. Secion 6. Since we are inereed in poiive oluion, we chooe c. The aim of hi paper i o exend reul from [16] and [19] o problem (1.1 having pace ingulariie. We dicu i olvabiliy and decribe he rucure of he e L of all i poiive oluion. The exience reul are proved by he combinaion of regularizaion and equenial echnique wih he Leray-Schauder nonlinear alernaive. We alo how he inereing reul aing ha for each c here exi a funcion u L uch ha u (T = c, and hence, he cardinaliy of he e L i a coninuum. Finally, by mean of hree nonlinear e example, we illurae he heoreical finding. Thee example are olved uing a Malab code bvpuie [13] baed on collocaion. We ar by inroducing he neceary noion. 1 Noe, ha we obain he ame oluion if in u ( + a u ( a 2 u( = he ubiuion u( := λ i made. Clearly, in he calar cae, he roo of he o called characeriic polynomial λ(λ 1 + aλ a = coincide wih he eigenvalue of M. 3

4 Le u denoe by L 1 [,T] he e of funcion which are Lebegue inegrable on [,T] equipped wih he norm x 1 = x( d. Moreover, le u denoe by C[,T] and C 1 [,T] he e of funcion being coninuou on [,T], and having coninuou fir derivaive on [,T], repecively. The norm on C[,T] and C 1 [,T] i defined a x = max [,T] x( and x C 1 = x + x, repecively. Finally, we denoe by AC 1 [,T] he e of funcion which have aboluely coninuou fir derivaive on [,T], while ACloc 1 (,T] i he e of funcion having aboluely coninuou derivaive on each compac ubinerval of (,T]. We ay ha u : [,T] R i a poiive oluion of problem (1.1 if u AC 1 [,T], u > on (,T, u aifie he boundary condiion (1.1b and (1.1a hold for a.e. [,T]. We work wih he following condiion on f in (1.1a. (H 1 f Car([,T] D, where D = (, R. (H 2 There exi > uch ha f(,x,y for a.e. [,T] and all (x,y D. (H 3 For a.e. [,T] and all (x,y D he eimae f(,x,y h(,x, y + g(x, hold, where h Car([,T] A, A = [, [, and g C(, are poiive, h(, x, y i nondecreaing in he variable x, y, g i nonincreaing, and 1 T 1 lim h(,x,x d =, g( 2 d <. x x Remark 1.1 Le g aify he condiion given in (H 3. Then b g(c2 d < for each b,c (,, and i follow from he inequaliy T 2 2, [ ], T 2, ha (T 2 T 2 (T 2, [ T 2,T], g ( c(t 2 d < for each c (,. 4

5 The paper i organized a follow. Secion 2 conain inequaliie which we will require in he nex hree ecion. Secion 3 i devoed o he udy of limi properie a + of oluion o equaion of he following ype: u ( + u ( a 2u( = r(,u(,u (, where he funcion r aifie he global Carahéodory condiion on [, T] R 2. In Secion 4, we inveigae auxiliary regular problem aociaed wih he ingular problem (1.1. We how heir olvabiliy and properie of heir oluion. Exience reul for ingular problem (1.1 are given in Secion 5. Here, in addiion, he properie of he e L of all poiive oluion o he problem are derived ogeher wih ome applicaion. Finally, in Secion 6, we illurae he heoreical finding by mean of numerical experimen. Throughou he paper a (, 1. 2 Preliminarie Thi ecion conain inequaliie required for he proof in Secion 3 o 5. Lemma 2.1 Le p L 1 [,T]. Then he inequaliie a 1 a+1 p( d p( d, (2.1 hold for [,T]. ( a 2 ξ a+1 p(ξ dξ d 1 a + 1 p( d (2.2 Proof. See [16, Lemma 1]. Lemma 2.2 The inequaliy a 2 ( ξ a+1 dξ d (T 2, a [ 3, 1, 2T (T 2 2T(a + 2 2, a (, 3 (2.3 hold for [,T]. 5

6 Proof. Le a [ 2, 1, hen In paricular, a 2 ( ξ a+1 dξ d T a+1 (T a 2 d 1 T (T d = (T 2. 2T a 2 ( ξ a+1 dξ d (T 2 2T for [,T] and a [ 2, 1. (2.4 Le a (, 2. Then a 2 ( ξ a+1 dξ d = 1 a + 2 = T a ( ( a 2 1 d T /T (1 a 2 d (2.5 for [,T]. Chooe p(x := 1 x β β(1 x for x [, 1], where β (, 1. Then p( = 1 β >, p(1 =, and ince p (x = β ( 1 x β 1 < for x (, 1, we have p > on [, 1. Conequenly, 1 x β β(1 x for x [, 1] and β (, 1]. Thi give for β = a 2, 1 x a 2 a + 2 (1 x for x [, 1] and a [ 3, 2. Hence, by (2.5, he relaion a 2 ( 1 ξ a+1 dξ d T (1 d = /T (T 2 2T (2.6 i aified for [, 1] and a [ 3, 2. In order o verify (2.3 for a (, 3, le r(x := 1 x γ 1 x γ for x [, 1], where γ > 1. Then r( = 1 1 >, r(1 =, γ r (x = γx γ 1 + 1, and γ r (x = γ(γ 1x γ 2. Hence, r < on (, 1], and ince r ( = 1 > and γ r (1 = γ + 1 <, we conclude ha r on [, 1]. Tha i 1 γ xγ 1 x for γ x [, 1] and γ > 1. Therefore, for γ = a 2, 1 x a 2 1 x a + 2 for x [, 1] and a (, 3, 6

7 and o, by (2.5, a 2 ( ξ a+1 dξ d T 1 (1 d = (a /T (T 2 2T(a + 2 2, (2.7 for [,T] and a (, 3. Inequaliy (2.3 now follow from (2.4, (2.6 and ( Limi properie of oluion In hi ecion we conider he differenial equaion u ( + a u ( a 2u( = r(,u(,u (, (,T], (3.1 where r aifie he global Carahéodory condiion on [,T] R 2, ha i, (H 4 r(,x,y : [,T] R i meaurable for all (x,y R R, r(,, : R R R i coninuou for a.e. [,T], and here exi µ L 1 [,T] uch ha r(,x,y µ( for a.e. [,T] and all (x,y R 2. (3.2 We now decribe he analyical form and he aympoic behavior for + of funcion u aifying (3.1 a.e. on [,T]. Lemma 3.1 Le condiion (H 4 hold. Le he funcion u ACloc 1 (,T] aify (3.1 for a.e. [,T]. Then u can be exended on [,T] wih u AC 1 [,T] and he repreenaion ( u( = c 1 + c 2 a + a 2 ξ a+1 r(ξ,u(ξ,u (ξ dξ d, (3.3 where c 1,c 2 R, hold for [,T]. Proof. Keeping in mind ha u i fixed, conider he Euler linear differenial equaion v ( + a v ( a 2v( = r(,u(,u (. (3.4 Each funcion v ACloc 1 (,T] aifying (3.4 a.e. on [,T] ha he form ( v( = c 1 + c 2 a + a 2 ξ a+1 r(ξ,u(ξ,u (ξ dξ d, 7

8 where c 1,c 2 R. Since, by he aumpion u ACloc 1 (,T] aifie (3.4 for a.e. [,T], here exi c 1,c 2 R uch ha equaliy (3.3 hold for (,T]. In order o prove ha u can be exended on [,T] a a funcion in AC 1 [,T], and conequenly, ha (3.3 i aified for [,T], we have o how ha By (3.3, a u ( a 2u( = a a 2 and hen uing (3.2 we obain a u ( a 2u( a a 2 Hence, by (2.2, u ( d <. (3.5 ( c 2 (a a+1 r(,u(,u ( d ( c 2 (a a+1 µ( d for (,T], for (,T]. (3.6 u ( d a u ( a 2u( d + r(,u(,u ( d ( ( a c 2 (a + 1 a 2 d + a 2 a+1 µ( d d + µ( d ac 2 T a 1 + (2a + 1 µ 1. a + 1 Conequenly, (3.5 hold and hi complee he proof. The following corollarie exend he aemen of Lemma 3.1 for r Car([,T] R 2, ha i for r aifying only he local Carahéodory condiion on [,T] R 2. Corollary 3.2 Le r Car([,T] R 2 and le u ACloc 1 (,T] aify (3.1 a.e. on [,T]. Aume alo ha L := up{ u( + u ( : (,T]} < hold. Then, he aerion of Lemma 3.1 i aified. Proof. Le L, z > L, ρ(z := z, z L, L, z < L, 8

9 and le r (,x,y := r(,ρ(x,ρ(y for a.e. [,T] and all (x,y R R. Then r aifie he global Carahéodory condiion on [,T] R 2 and he equaliy u ( + a u ( a 2u( = r (,u(,u ( hold for a.e. [,T]. The reul now follow from Lemma 3.1, where r i replaced by r in equaion (3.1. Corollary 3.3 Le r Car([,T] R 2 and le u AC 1 [,T] be a oluion of equaion (3.1. Then here exi c 1,c 2 R uch ha equaliy (3.3 i aified for [,T]. Proof. We can apply Corollary 3.2, ince u AC 1 [,T] yield up{ u( + u ( : [,T]} <. Remark 3.4 Corollary 3.3 how ha each oluion u AC 1 [,T] of equaion (3.1 wih r Car([,T] R 2 ha he form given in (3.3, where c 1,c 2 R, and herefore, i aifie u( =. Conequenly, when dicuing oluion u AC 1 [,T] of equaion (3.1 ogeher wih boundary condiion, epecially including he condiion u( = u, hen, necearily, u =. 4 Auxiliary regular problem Since equaion (1.1a i ingular, we ue he regularizaion and equenial echnique for olving problem (1.1. To hi end, we define f n : [,T] R 2 R, n N, by he formula f(,x,y, x 1 n, f n (,x,y = f (, 1n,y, x < 1 n. Under condiion (H 1 (H 3, f n Car([,T] R 2 and f n (,x,y for a.e. [,T] and all (x,y R 2, (4.1 } f n (,x,y h(, 1 + x, y + g( x for a.e. [,T] and all (x,y R R. (4.2 9

10 Here R = R\{}. Hence, } λf n (,x,y + (1 λ h(, 1 + x, y + g( x for a.e. [,T] and all (x,y R R, λ [, 1]. (4.3 and We conider he differenial equaion u ( + a u ( a 2u( = f n(,u(,u (, n N, (4.4 u ( + a u ( a 2u( = λf n(,u(,u ( + (1 λ, λ [, 1], n N. (4.5 A funcion u : [,T] R i called a oluion of (4.4 if u AC 1 [,T] and u aifie (4.4 for a.e. [,T]. Soluion of (4.5 are defined analogouly. Le u now define he boundary value problem (4.6 coniing of he differenial equaion pecified in (4.4 ubjec o he boundary condiion (1.1b, u ( + a u ( a 2u( = f n(,u(,u (, n N, (4.6a u( =, u(t =. (4.6b Lemma 4.1 Le condiion (H 1 hold. Then, all oluion u AC 1 [,T] of problem (4.6 form a one-parameer yem A, where { A = c 2 ( a 1 T a 1 ( } + a 2 ξ a+1 f n (ξ,u(ξ,u (ξ dξ d : c 2 R. Proof. Le u AC 1 [,T] be a oluion of problem (4.6. Since u i a oluion of (4.6a, he equaliy ( u( = c 1 + c 2 a + a 2 ξ a+1 f n (ξ,u(ξ,u (ξ dξ d (4.7 hold for [,T] by Corollary 3.3, where c 1,c 2 R. Then u( = and he condiion u(t = yield c 1 = c 2 T a 1. Hence, by (4.7, u A. Le u A, ha i, ( u( = c 2 ( a 1 T a 1 + a 2 ξ a+1 f n (ξ,u(ξ,u (ξ dξ d (4.8 1

11 for [,T], where c 2 R. Then u aifie condiion (4.6b and u ( = c 2 (a a 1 + T a 1 ( + a 2 ξ a+1 f n (ξ,u(ξ,u (ξ dξ d a 1 a+1 f n (,u(,u ( d, [,T], u ( = a(a + 1c 2 a 2 + a a 2 a+1 f n (,u(,u ( d +f n (,u(,u (, for a.e. [,T]. By (H 1, f n (,u(,u ( L 1 [,T] and conequenly, (2.2 implie (4.9 (4.1 a 2 a+1 f n (,u(,u ( d L 1 [,T]. A a reul, u AC 1 [,T]. Uing (4.8, (4.9 and (4.1, we can verify ha u aifie (4.6a for a.e. [,T]. In he following lemma, we dicu oluion u of he boundary value problem: u ( + a u ( a 2u( = f n(,u(,u (, n N, (4.11a u( =, u(t =, u (T = c, c. (4.11b In hi problem u aifie, beide he Dirichle condiion (4.6b, he addiional condiion u (T = c, (4.12 for a fixed c. Noe ha condiion (4.12 ogeher wih (4.9 yield in (4.8. c 2 = c a + 1 T a+1 (4.13 Lemma 4.2 Le (H 1 hold. Then a funcion u AC 1 [,T] i a oluion of problem (4.11 if and only if u i a oluion of he inegral equaion ct a+1 u( = a + 1 (T a 1 a 1 + ( a 2 ξ a+1 f n (ξ,u(ξ,u (ξ dξ d (4.14 in he e C 1 [,T]. 11

12 Proof. ( Le u fir aume ha u AC 1 [,T] i a oluion of problem (4.11. Then u A, ha i u aifie (4.8, wih c 2 given by (4.13. A a reul, u i a oluion of (4.14 in C 1 [,T]. ( Le now u C 1 [,T] be a oluion of (4.14. Then u aifie (4.8, (4.9 and (4.1 wih c 2 given by (4.13. Therefore, u aifie boundary condiion (4.11b. The ame reaoning a in he proof of Lemma 4.1 implie ha u AC 1 [,T] and u aifie equaion (4.11a for a.e. [,T]. Le M = 2T, a [ 3, 1, (4.15 2T(a + 2 2, a (, 3, wih pecified in condiion (H 2. Conider a fixed c. We now derive bound for oluion of he boundary value problem u ( + a u ( a 2u( = λf n(,u(,u ( + (1 λ, λ [, 1],(4.16a u( =, u(t =, u (T = λc. (4.16b Noe, ha hi ime u aifie addiionally, beide he Dirichle condiion (4.6b, he following condiion u (T = λc, λ [, 1]. (4.17 Lemma 4.3 Le condiion (H 1 (H 3 hold. Then, here exi a poiive conan S independen of n and λ uch ha for all oluion u of problem (4.16 he eimae u( M(T 2, [,], (4.18 hold. u < ST, u < S, (4.19 Proof. Le u be a oluion of problem (4.16 for ome n N and λ [, 1]. Applying Lemma 4.2 o hi problem we obain he equaliy ct a+1 u( = λ a + 1 (T a 1 a 1 + ( a 2 ξ a+1 [λf n (ξ,u(ξ,u (ξ + (1 λ ] dξ d (4.2 12

13 for [,T]. Since c, λ, we have due o (2.3, (4.1 and (4.15, ( u( a 2 ξ a+1 dξ d M(T 2 for [,T], and o (4.18 i rue. Furhermore, u > on (,T and g(u( g (M(T 2 for (,T ince g i nonincreaing on (, by (H 3. Hence, g(u( 1 g ( M(T 2 1 =: W 1, (4.21 where W 1 < by Remark 1.1. Noe ha he value of W 1 neiher depend on he choice of oluion u o problem (4.16 nor on n and λ. Since u ( = λct a+1 a + 1 (T a 1 + a a 1 ( + a 2 ξ a+1 [λf n (ξ,u(ξ,u (ξ + (1 λ ] dξ d a 1 a+1 [λf n (,u(,u ( + (1 λ ] d, [,T], (4.22 i follow from (2.1, (2.2, (4.3 and (4.21 ha he relaion ( 1 u ( c a ( + a 2 ξ a+1 [h(ξ, 1 + u(ξ, u (ξ + g(u(ξ] dξ d + a 1 a+1 [h(, 1 + u(, u ( + g(u(] d ( ( 1 T a h(, 1 + u(, u ( d + W 1 + c ( ( 1 T a h(, 1 + u, u d + W i aified for [,T] and W := W 1 + c. In paricular, ( ( 1 T u a h(, 1 + u, u d + W. Since u( = u ( d, we have u T u, (

14 and herefore u ( ( 1 T a h(, 1 + T u, u d + W. ( By (H 3, lim x h(, 1+Tx,x d =, and conequenly, here exi S > x uch ha ( ( 1 T a h(, 1 + Tx,x d + W < x for all x S. Now we conclude from he la relaion and from (4.24 ha u < S, and herefore, by (4.23, u < ST. Hence (4.19 hold and hi complee he proof. We are now in he poiion o how he exience of a oluion of problem (4.11. Thi reul i proved by he following nonlinear alernaive of Leray- Schauder ype which follow, for example, from [2, Theorem 5.1]. Lemma 4.4 Le X be a Banach pace, Ω an open bounded ube of X and p Ω. Aume ha F : Ω X i a compac operaor. Then, eiher (i F ha a fixed poin in Ω, or (ii There exi a u Ω and λ (, 1 uch ha u = λfu + (1 λp. Theorem 4.5 Le (H 1 (H 3 hold. Le S be he poiive conan from Lemma 4.3. Then problem (4.11 are olvable. If u i a oluion of (4.11 for ome n N, hen u aifie (4.18 and (4.19 wih he poiive conan M given in (4.15. Proof. Le Ω := {x C 1 [,T] : x < ST, x < S}. Then Ω i an open bounded ube of he Banach pace C 1 [,T]. Chooe n N and conider an operaor K : [, 1] Ω C 1 [,T], where ct a+1 (Fx( = a + 1 (T a 1 a 1 + K(λ,x = λfx + (1 λp, (4.25 ( a 2 ξ a+1 f n (ξ,x(ξ,x (ξ dξ ( p = a 2 ξ a+1 dξ d. 14 d,

15 By Lemma 4.2, any fixed poin of he operaor K(1, = F i a oluion of problem (4.11. Hence, o how he reul we need o prove ha K(1, ha a fixed poin. Applying Lemma 4.4 for X = C 1 [,T], we have o how ha (i K(1, : Ω C 1 [,T] i a compac operaor, and (ii K(λ,x x for each λ (, 1 and x Ω. We begin by proving he coninuiy of K(1,. To hi end le {x m } Ω be a convergen equence and le lim m x m = x. Le r m ( := f n (,x m (,x m( f n (,x(,x ( for a.e. [,T]. Then (2.1 and (2.2 yield K(1,x m ( K(1,x( = K(1,x m ( K(1,x ( = ( a 2 ( a 2 ξ a+1 r m (ξ dξ ξ a+1 r m (ξ dξ a 1 a+1 r m ( d ( 1 a r m 1, d T r m 1 a + 1, d for [,T] and m N. Here, K(1,x = d K(1,x. In paricular, d K(1,x m K(1,x T r m 1 a + 1, ( 1 K(1,x m K(1,x a r m 1, for m N. If we how ha lim m r m 1 =, hen he above inequaliie guaranee ha K(1, i a coninuou operaor. From lim f n(,x m (,x m( = f n (,x(,x ( for a.e. [,T], m and from he fac ha f n Car([,T] R 2 and {x m } i bounded in C 1 [,T], i follow ha f n (,x m (,x m( ρ( for a.e. [,T] and all m N, where ρ L 1 [,T]. Finally, lim m r m 1 = follow by he Lebegue dominaed convergence heorem. Now, we how ha he e K(1, Ω i relaively compac in C 1 [,T]. From f n Car([,T] R 2 we conclude ha f n (,x(,x ( µ( for a.e. [,T] and all x Ω, (

16 where µ L 1 [,T]. Then, by (2.1, (2.2, (4.25 and (4.26, K(1,x( ct ( a a 2 ξ a+1 µ(ξ dξ d T a + 1 (c + µ 1, K(1,x ( ct a+1 a + 1 (T a 1 + a a 1 ( + a 2 ξ a+1 µ(ξ dξ d + a 1 a+1 µ( d ( 1 a (c + µ 1 for [,T] and x Ω. Therefore, he e K(1, Ω i bounded in C 1 [,T]. We now how ha he e {K(1,x : x Ω} i equiconinuou on [,T]. For a.e. [,T] and all x Ω we have, by (4.26, K(1,x ( = act a+1 a 2 + a a 2 a+1 f n (,x(,x ( d + f n (,x(,x ( Hence, by (2.1, a ct a+1 a 2 + a a 2 a+1 µ( d + µ(. K(1,x ( d a a + 1 c + a a + 1 µ 1 + µ 1 for all x Ω, which guaranee he equiconinuiy of he e {K(1,x : x Ω} on [,T]. Therefore, he e K(1, Ω i relaively compac in C 1 [,T] by he Arzelà-Acoli heorem and conequenly, K(1, i a compac operaor and (i follow. I remain o prove (ii, ha i, K(λ,x x for each λ (, 1 and x Ω. Le u be a fixed poin of K(λ, for ome λ (, 1. Then equaliie (4.2 and (4.22 hold for [,T]. We ee ha u aifie (4.16b and, a in he proof of Lemma 4.1, we conclude ha u i a oluion of equaion (4.16a. Therefore, Lemma 4.3 guaranee ha K(λ,x x for λ (, 1 and x Ω. By Lemma 4.4, problem (4.11 ha a oluion u Ω. Lemma 4.3 guaranee ha u aifie (4.18 and (4.19. The following reul provide an imporan propery of oluion of problem (4.11 which will be ued in he proof of Theorem 5.1 in Secion 5. Lemma 4.6 Le (H 1 (H 3 hold. Le u n be a oluion of problem (4.11. Then, he equence {u n} i equiconinuou on [,T]. 16

17 Proof. By Theorem 4.5, and u n ( M(T 2 for [,T] and n N, (4.27 u n < ST, u n < S for n N, (4.28 where S i a poiive conan and M i given in (4.15. Since u n i a fixed poin of K(1,, cf. (4.25, he equaliy u n( = act a+1 a 2 + a a 2 a+1 f n (,u n (,u n( d + f n (,u n (,u n( hold for a.e. [,T] and all n N. Owing o (4.1, (4.2, (4.27 and (4.28 we have f n (,u n (,u n( h(, 1 + ST,S + ρ( for a.e. [,T] and all n N, where ρ( = g (M(T 2 for (,T. Le u chooe χ( := h(, 1+ST,S+ ( for a.e. [,T]. By (H 3 and Remark 1.1, χ i poiive and χ L 1 [,T]. Hence, u n( a ct a+1 a 2 + a a 2 a+1 χ( d + χ( for a.e. [,T] and all n N. Conequenly, by (2.2, he inequaliy u n( d a a + 1 c + a a + 1 χ 1 + χ 1 hold for all n N, which mean ha ha he equence {u n} i equiconinuou on [,T]. 5 Analyical properie of oluion o problem (1.1 In hi ecion, we denoe by L he e of all poiive oluion of he ingular Dirichle problem (1.1. For c, we denoe by S c he e of all poiive oluion of problem (1.1 aifying condiion (4.12. Our aim i o decribe he rucure of L. In paricular, we how ha L i a one parameer e. 17

18 Theorem 5.1 Le (H 1 (H 3 hold. Then, for each c, he e S c i nonempy and L = c S c. Hence he cardinaliy of he e L i a coninuum. Moreover, wih M given in (4.15, hold for each u L. u( M(T 2, [,T], (5.1 Proof. Le u fix c. Theorem 4.5 guaranee ha he regular problem (4.11 ha a oluion u n aifying inequaliie (4.27 and (4.28, where S i a poiive conan and M i given in (4.15. Furhermore, {u n} i equiconinuou on [,T] by Lemma 4.6. Conequenly, by Arzelà-Acoli heorem, here i a ubequence {u ln } of {u n } converging in C 1 [,T]. Denoe lim n u ln =: u. Then u aifie he boundary condiion (4.11b and aking he limi n in (4.27 and (4.28, wih u n replaced by u ln, we obain u ST, u S and u( M(T 2 for [,T]. Hence u > on (,T and lim n f l n (,uln (,u l n ( = f(,u(,u ( for a.e. [,T]. By (H 3 and Remark 1.1, f ln (,uln (,u l n ( h(, 1 + ST,S + g ( M(T 2 L 1 [,T]. Therefore, f(,u(,u ( L 1 [,T] and ( lim fln,uln (,u l n ( f(,u(,u ( 1 = (5.2 n by he Lebegue dominaed convergence heorem. I follow from he inequaliy (2.2, ha i hold ( a 2 ξ [ ( a+1 f ln ξ,uln (ξ,u l n (ξ f(ξ,u(ξ,u (ξ ] dξ d T ( f ln,uln (,u l a + 1 n ( f(,u(,u ( 1. Now, from (5.2 we conclude ha lim n ( ( a 2 ξ a+1 f ln ξ,uln (ξ,u l n (ξ dξ = a 2 ( d ξ a+1 f(ξ,u(ξ,u (ξ dξ d 18

19 i aified for [,T]. Leing n in ct a+1 u ln ( = a + 1 (T a 1 a 1 + yield a 2 ( ( ξ a+1 f ln ξ,uln (ξ,u l n (ξ dξ d ξ a+1 f(ξ,u(ξ,u (ξ dξ d a+1 ( ct T u( = a + 1 (T a 1 a 1 + a 2 (5.3 for [,T]. A direc compuaion how ha u i a oluion of (1.1a. Thi mean ha u i a poiive oluion of problem (1.1 and (4.12, ha i u S c. Conequenly S c. Since for each poiive oluion u of problem (1.1 here exi c = c(u uch ha u (T = c, we ee ha L = c S c i he e of all poiive oluion of problem (1.1. For K, le u denoe Then we have he following heorem. L K := c K Theorem 5.2 Le (H 1 (H 3 hold. Then, for each K, he e L K i compac in C 1 [,T]. Proof. Le u chooe K. Then inequaliy (5.1 wih M given in (4.15 hold for each u L K. Conider an arbirary u L K. Then (5.3 i aified for ome c = c(u [,K] and [,T]. Therefore, u ( = ct a+1 S c. a + 1 (T a 1 + a a 1 ( + a 2 ξ a+1 f(ξ,u(ξ,u (ξ dξ d a 1 a+1 f(,u(,u ( d, I follow from (2.1 and (2.2, ( ( 1 1 T u ( K a a [,T]. f(,u(,u ( d, [,T]. Hence, by (H 3 and u T u, ince u( = u ( d, we have ( ( 1 T u ( a K + (h(,u(, u ( + g(u( d ( ( 1 T a K + h(,t u, u d + W, [,T]. 19

20 Noe ha by Remark 1.1, g(u( d g(m(t 2 d =: W <. (5.4 In paricular, 1 1 ( ( 1 T u a K + h(,t u, u d + W. (5.5 Due o (H 3, and o lim w lim w 1 T h(ξ,tw,wdξ =, w ( ( 1 1 T w a K + h(,tw,w d + W =, which implie ha here exi > uch ha ( ( 1 1 T w a K + h(,tw,w d + W < 1 for each w. Thi ogeher wih (5.5 reul in u <, u < T for each u L K and herefore, L K i bounded in C 1 [,T]. We now verify ha he e {u : u L K } i equiconinuou on [,T]. For any u L K we have Therefore, u ( = act a+1 a 2 + a a 2 a+1 f(,u(,u ( d + f(,u(,u ( for a.e. [,T] and c = c(u [,K]. u ( a KT a+1 a 2 + a a 2 a+1 m( d + m( for a.e. [,T], where m( = h(,t, +g(m(t 2. Since, cf. (5.4 and (H 3, m( d h(,t, d + W <, 2

21 we ee ha m L 1 [,T]. Therefore, by (2.2, ( a 2 a+1 m( d L 1 [,T]. Conequenly, here exi a majoran funcion p L 1 [,T] aifying u ( p ( for a.e. [,T] and all u L K. A a reul he e {u : u L K } i equiconinuou on [,T]. In order o complee he proof, we need o how ha he e L K i cloed in C 1 [,T]. To hi end, we conider a equence {u n } L K converging in C 1 [,T] o a funcion u C 1 [,T]. Therefore, here exi a equence {c n } [,K] uch ha, due o (5.3, u n ( = c nt a+1 a + 1 (T a 1 a 1 + ( a 2 ξ a+1 f(ξ,u n (ξ,u n(ξ dξ d for [,T] and n N. Since u n(t = c n, we ee ha {c n } i convergen. Le u define lim n c n =: c [,K] and le n in he above equaliy for u n (. Then, by he Lebegue dominaed convergence heorem, arguing a in he proof of Theorem 5.1, we obain ct a+1 u( = a + 1 (T a 1 a 1 + ( a 2 ξ a+1 f(ξ,u(ξ,u (ξ dξ Therefore u aifie (1.1 and (4.12 and hence u S c L K. d for [,T]. (5.6 Remark 5.3 I follow from he proof of Theorem 5.2 ha for each fixed c he e S c i compac in C 1 [,T]. Remark 5.4 Conider a equence {u n } L. Since L = c S c, we ee ha u n S cn for ome c n and u n(t = c n, n N. We now can how ha lim n u n C 1 = lim n c n = (5.7 hold. Fir, aume ha lim u n C 1 =, (5.8 n 21

22 and furher, aume in a conrary, ha here exi K > and a ubequence {c mn } {c n } uch ha c mn K for n N. Since L K = c K S c, we have {u mn } L K. By Theorem 5.2, L K i compac in C 1 [,T] which implie ha he equence {u mn } i bounded in C 1 [,T], which conradic (5.8. If we aume ha lim n c n =, hen (5.8 immediaely follow, becaue u n C 1 u n c n, n N. Remark 5.5 I follow from (5.7 ha he e L i unbounded in C 1 [,T]. In paricular, if u L, hen u S c for ome c and, keeping in mind ha f i poiive, we ge by (5.6, Tha i u cl, where Hence, u ( T 2 L = Tc ( a a 1 ( T 1 1 >. (5.9 2 a a 1 up{ u : u L} up{cl : c [, } =, and Theorem 5.2 implie he following reul concerning minimal value of funcional defined on he e L of all poiive oluion o problem (1.1. Le M be he e of coninuou funcional Φ : C 1 [,T] [,, which are coercive on L, ha i, lim x L, x C 1 Φ(x =. (5.1 Theorem 5.6 Le (H 1 (H 3 hold and le Φ M. Then here exi a poiive oluion u of problem (1.1 uch ha min{φ(x : x L} = Φ(u. (5.11 Proof. Chooe u S and A := Φ(u. By (5.1, here exi B > uch ha x L, x C 1 > B Φ(x > A. (

23 Le K := B/(L + 1, wih L > from (5.9. According o Theorem 5.2, L K i compac in C 1 [,T] and conequenly, he coninuiy of Φ implie he exience of u L K uch ha min{φ(x : x L K } = Φ(u A. (5.13 Now, aume ha c > K. Then, for each x S c, we have x x (T = c and, by Remark 5.5, x cl. Therefore, x C 1 c(l + 1 > K(L + 1 = B. Thi ogeher wih (5.12 yield Φ(x > A. Conequenly, Φ(x > A for each x L \ L K and (5.11 follow from (5.13. We now preen applicaion of Theorem 5.6. Aume (H 1 (H 3 o hold. Chooe α (, and conider funcional Φ 1, Φ 2 : C 1 [,T] [, given by Φ 1 (x = x( α d, Φ 2 (x = 1 + x 2 (d. (5.14 Then, Φ 1, Φ 2 are coninuou on C 1 [,T]. Le u now how ha Φ 1 and Φ 2 aify (5.1, where L i he e of all poiive oluion of problem (1.1. Chooe an arbirary equence {u n } L uch ha lim n u n C 1 =. For c n = u n(t, n N, we obain uing (5.7, lim n c n =. Since u n ( = u n(d, i follow from Remark 5.5 ha c n L u n u n( d Φ 2 (u n, n N. Conequenly, Φ 2 i coercive on L. Furhermore, from (5.6 and he poiiviy of f, we have where u n ( c n ϕ(, [,T], ϕ( := T a+1 a + 1 (T a 1 a 1 >, (,T. According o (5.14, Φ 1 (u n = u n ( α d c α n ϕ α (d = c α nm, n N, where M := ϕα (d >. Hence, we have hown ha lim n Φ 1 (u n = and herefore, Φ 1 i coercive on L. Conequenly, Theorem 5.6 i applicable o boh, Φ 1 and Φ 2. We can eaily ee a geomerical meaning of hi reul. For example, dealing wih Φ 2, we ge ha among all poiive oluion of (1.1 here exi a oluion having a graph wih he hore lengh. Noe, ha value of Φ 1 wih α = 1 are dicued in Example 3. 23

24 6 Numerical imulaion For he numerical imulaion, we ue an alernaive formulaion of problem (1.1, u ( + a u ( a 2u( = f(,u(,u (, [, 1], (6.1a u(1 =, u (1 = c, (6.1b where c. According o [19], he above boundary value problem (6.1 i well-poed and herefore i i uiable for he numerical reamen. For all example, he calculaion have been carried for he value of a = 2 and c =,.1,.2,.3,.5, 1, 2, 5, 1, Malab Code bvpuie To illurae he analyical reul dicued in he previou ecion, we olved numerically Example (6.2, (6.4 and (6.5 uing a MATLAB TM ofware package bvpuie deigned o olve boundary value problem in ordinary differenial equaion and differenial algebraic equaion. The olver rouine i baed on a cla of collocaion mehod whoe order may vary from 2 o 8. Collocaion ha been inveigaed in conex of ingular differenial equaion of fir and econd order in [11] and [2], repecively. Thi mehod could be hown o be robu wih repec o ingulariie in ime and for reain i high convergence order in cae ha he analyical oluion i appropriaely mooh. The code alo provide an aympoically correc eimae for he global error of he numerical approximaion. To enhance he efficiency of he mehod, a meh adapaion raegy i implemened, which aemp o chooe grid relaed o he oluion behavior, in uch a way ha he olerance i aified wih he lea poible effor. Error eimae procedure and he meh adapaion work dependably provided ha he oluion of he problem and i global error are appropriaely mooh 2. The code and he manual can be downloaded from hp:// ewa. For furher informaion ee [13]. Thi ofware i ueful for he approximaion of numerou ingular boundary value problem imporan for applicaion, ee e.g. [4], [9], [12], [17]. 6.2 Example 1 We fir inveigae he following problem: u ( + a u ( a 2u( = u( , [, 1], (6.2a u(1 =, u (1 = c, (6.2b 2 The required moohne of higher derivaive i relaed o he order of he ued collocaion mehod. 24

25 where a = 2. In Figure 1 and 2, oluion o problem (6.2 for differen value of c are hown c= c=.1 c=.2 c=.3 c=.5 c= Figure 1: Problem (6.2: Parameer a = c=1 c=2 c=5 c=1 c= Figure 2: Problem (6.2: Parameer a = 2. 25

26 For a given c denoe by u c a oluion of problem (6.2. We can ee in Figure 1 and 2 ha graph of oluion u c are ordered, ha i c 1 < c 2 = u c1 ( < u c2 (, (, 1. (6.3 Thi correpond o he heory in Secion 4 of [16], where he pecial cae of problem (1.1 wih f Car([, T] [, and f(, u increaing in u, ha been inveigaed. 6.3 Example 2 Here, we udy he influence of u in f(,u,u = u u The boundary value problem ha now he form u ( + a u ( a 2u( = u(2 3 + u ( , [, 1], (6.4a u(1 =, u (1 = c, (6.4b where a = 2. Noe ha ince u ( may become negaive, we replace u ( 2 3 by u ( u ( 1 3 in numerical imulaion. The oluion of he boundary value problem (6.4 for differen value of c, can be found in Figure 3 and c= c=.1 c=.2 c=.3 c=.5 c= Figure 3: Problem (6.4: Parameer a = 2. 26

27 c= c=1 c=2 c=5 c=1 Figure 4: Problem (6.4: Parameer a = 2. Figure 3 how ordered graph of oluion u c of problem (6.4 wih c changing from o 1 bu Figure 4 demonrae ha, for c having value from 1 o 1, he graph of oluion u c do no keep order (6.3. The oluion of problem (6.4 for a = 3 and a = 1 how imilar behavior a for a = 2 and hence, hey are no diplayed here. All reul for he above cla have been obained uing he ame aring gue: he numerical oluion for c = obained wih he piecewie ha funcion a an iniial profile, ee Figure 5. 27

28 Fir Gue ued gue Figure 5: Saring gue for c ha been obained by olving he problem for c = and he iniial profile hown above. 6.4 Example 3 In order o dicu he influence of a poible pace ingulariy in f, we pu f(,u = u and look a he following problem: u ( + a u ( a 2u( = u( , [, 1], (6.5a u(1 =, u (1 = c, (6.5b where a = 2. The oluion o he above boundary value problem can be found in Figure 6 and 7. 28

29 c= c=.1 c=.2 c=.3 c=.5 c= Figure 6: Problem (6.5: Parameer a = c= c=1 c=2 c=5 c=1 Figure 7: Problem (6.5: Parameer a = 2. For he value of c 1 he aring gue menioned above ha been ued. For larger value of c providing alernaive aring guee wa neceary. For 29

30 example, for c = 5 he earlier compued oluion for c = 2 ha been ued. Figure 6 illurae ha oluion u c of problem (6.5 wih c changing from o 1 do no fulfil he order given by (6.3. Thi ogeher wih Figure 4 lead o he hypohei ha he order (6.3 canno be proved for oluion of (1.1, where f(,x,y depend on y or f ha a ingulariy a x =. Now, conider he funcional Φ 1 of (5.14 wih α = 1 and T = 1, ha i Φ 1 (x = 1 x( d for x C1 [, 1]. Le L be he e of all poiive oluion of problem (6.5. Uing Theorem 5.6, we have proved ha here exi a poiive oluion of problem (6.5 giving a minimal value of Φ 1 on L. To illurae hi reul, we have approximaed he value of he inegral Φ 1 (u c = 1 u c (d. Here, we pu a = 2 and by u c we denoe a poiive oluion of (6.5 for a pecific nonnegaive value of c. In order o approximae Φ 1 (u c ], we inroduce a pariion of he inerval [, 1] ino equidian ubinerval of lengh 1 2. A a quadraure formula, we ue he compoed Gauian rule wih five evaluaion poin in each ubinerval of [, 1]. The reul can be found in Table 1 below. c Φ 1 (u c Table 1: Problem (6.5. Table 1 how ha Φ 1 (u c i no monoonou for c [,, ha i he inequaliy c 1 < c 2 need no imply Φ 1 (u c1 Φ 1 (u c2. Bu we know ha here exi a lea one c [, uch ha Φ 1 (u c reache i minimum a u c. Acknowledgemen Thi reearch wa uppored by he gran Maemaické modely a rukury, PrF

31 Reference [1] F. F. Abraham, Homogeneou Nucleaion Theory, Acad. Pre, New York [2] R. P. Agarwal, M. Meehan, and D. O Regan, Fixed Poin Theory and Applicaion, Cambridge Univeriy Pre, 21. [3] V. Bongiorno, L. E. Scriven, and H. T. Davi, Molecular heory of fluid inerface, J. Colloid and Inerface Science 57 (1967, [4] C. Budd, O. Koch, and E. Weinmüller, From nonlinear PDE o ingular ODE, Appl. Num. Mah. 56 (26, [5] G. H. Derrick, Commen on nonlinear wave equaion a model for elemenary paricle, J. Mah. Phyic 5 (1965, [6] P. C. Fife, Mahemaical apec of reacing and diffuing yem, Lecure noe in Biomahemaic 28, Springer [7] R. A. Ficher, The wave of advance of advanegeou gene, J. Eugenic 7 (1937, [8] H. Gouin and G. Rooli, An analyical approximaion of deniy profile and urface enion of microcopic bubble for Van der Waal fluid, Mech. Reearch Communic. 24 (1997, [9] R. Hammerling, O. Koch, C. Simon, and E. Weinmüller, Numerical Soluion of Eigenvalue Problem in Elecronic Srucure Compuaion, J. Comp. Phy. 181 (21, [1] F. de Hoog, and R. Wei, Difference mehod for boundary value problem wih a ingulariy of he fir kind, SIAM J. Numer. Anal. 13 (1976, [11] F. de Hoog and R. Wei. Collocaion mehod for ingular boundary value problem, SIAM J. Numer. Anal. 15 (1978, [12] G. Kizhofer, O. Koch, P. Lima, and E. Weinmüller, Efficien Numerical Soluion of he Deniy Profile Equaion in Hydrodynamic, J. Sci. Comp. 32 (27, [13] G. Kizhofer, G. Pulverer, C. Simon, O. Koch, and E. Weinmüller, The New Malab Solver BVPSUITE for he Soluion of Singular Implici BVP, JNAIAM, J. Numer. Anal. Ind. Appl. Mah. 5 (21,

32 [14] N. B. Konyukhova, P. M. Lima, M. L. Morgado, and M. B. Sololviev, Bubble and drople in nonlinear phyic model: analyi and numerical imulaion of ingular nonlinear boundary value problem, Comp. Mah. Mah. Phy. 48 (28, [15] A. P. Linde, Paricle Phyic and Inflaionary Comology, Harwood Academic, Chur, Swizerland 199. [16] I. Rachůnková and S. Saněk, Properie of he e of poiive oluion o Dirichle boundary value problem wih ime ingulariie, Cenral European J. Mah., o appear. [17] I. Rachůnková, G. Pulverer, and E. Weinmüller, A unified approach o ingular problem ariing in he membrane heory, Applicaion of Mahemaic 55 (21, [18] J. D. van der Waal and R. Kohnamm, Lehrbuch der Thermodynamik, Vol. 1, Leipzig 198. [19] E. B. Weinmüller, On he Boundary Value Problem of Ordinary Second Order Differenial Equaion wih a Singulariy of he Fir Kind, SIAM J. Mah. Anal. 15 (1984, [2] E. B. Weinmüller, Collocaion for ingular boundary value problem of econd order, SIAM J. Numer. Anal. 23 (1986,