On a Boundary Value Problem for a Quasi-linear. Elliptic Equation Degenerating into a Parabolic. Equation in an Infinite Strip

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1 Nonl Analss and Dfferental Equatons Vol 3 no - 4 HIKARI Ltd wwwm-harcom On a Boundar Value Problem for a Quas-lnear Ellptc Equaton Degeneratng nto a Parabolc Equaton n an Infnte Strp Mahr M Sabzalev Department of Mathematcs Azerbaan State Ol Academ Аz Azadlg av Bau Azerbaan sabzalevm@malru Coprght 3 Mahr M Sabzalev Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n an medum provded the orgnal wor s properl cted Аbstract In ths paper we construct complete asmptotcs on the small parameter of the soluton of a sngularl perturbed boundar value problem for a quas-lnear ellptc equaton degeneratng n an nfnte strp nto the parabolc equaton and the remander term s estmated Kewords: Asmptotcs Boundar laer functon Remander term Introducton Whle studng numerous real phenomena wth non-unform transtons from one phscal characterstcs to another ones we have to nvestgate sngularl perturbed boundar value problems A lot of papers have been devoted to the asmptotcs of the soluton of dfferent boundar value problems for nonlnear ellptc equatons wth a small parameter at hgher dervatves In a great number of papers on nonlnear sngularl perturbed ellptc equatons the nput equatons degenerate for a zero value of the small parameter nto functonal equatons (see []-[4] [6] Besdes n all these papers wth the excepton of the paper [4] the dervatves of the desred functon enter lnearl to the equaton onl the desred functon tself enters nto the equaton nonlnearl All these and other problems nown to us are consdered onl n fnte domans In the present paper n an nfnte strp Π {( x x < < } we consder the followng boundar value problem

2 M M Sabzalev P P p U U U U Lε U ε + εδu + + au f ( x ( x U U ( < < ; lm U ( x ( x x where ε > s a small parameter p + s an arbtrar natural number Δ s a Laplace operator a > s a constant f ( x s a gven smooth functon The goal of the paper s to construct the asmptotc expanson of the generalzed soluton of problem ( ( from the class W p+ ( Π The frst teratve process In the frst teratve process we ll loo for the approxmate soluton of equaton ( n the form W W W n + ε + + ε Wn (3 and the functons W ( x wll be chosen so that ( ε Lε W (4 Substtutng (3 n (4 expandng the nonlnear terms n powers of ε and equatng the terms wth the same powers of ε for determnng the functon W ; n we get the followng recurrentl connected equatons: W W + aw f ( x n (5 f f x f x ΔW for ; f ( x ΔW + where + ( W W + s s g s W s for s + n and the functons g s are dependent polnomall on the frst and second dervatves of W W W s We ll solve equatons (5 under the followng boundar condtons: W ( < < ; lm W ( x n (6 x For problem (5 (6 s sad to be a degenerated problem correspondng to problem ( ( The followng lemma s vald Lemma Let f ( x be a functon gven n Π havng contnuous dervatves wth respect to x to the ( n + -th order nclusvel be nfntel dfferentable wth respect to and satsf the condton sup l f ( x ( + Cl < (7

3 BVP for quas-lnear ellptc equaton 3 where l s a nonnegatve number + n + s arbtrar ( C l > Then the functon W ( x beng the soluton of problem (5 (6 for n Π has contnuous dervatves wth respect to x to the ( n + -th order nclusvel s nfntel dfferentable wth respect to and satsfes the condton l W ( x + Cl sup < (8 where n + C l > Proof Applng the Fourer transformaton wth respect to problem (5 (6 for s reduced to the problem dw + a + W ( f x W (9 dx x Here W ( x W ( x e d f ( x f ( x e d π π The soluton of problem (9 s of the form x ( a+ ( х W ( x e f ( d ( W ( x s found as the nverse Fourer transformaton of the functon W ( x from the followng formula: W ( x W ( x e d ( π From condton (7 t follows that the functon f ( x and all ts dervatves wth respect to x to the ( n + -th order nclusvel wth respect to the varable belong to the S LSchwarts space (n the sequel we ll denote t b S Obvousl for provng lemma t suffces to show that the functon W that s the soluton of problem (9 and all ts dervatves wth respect to ( x x to the ( n + -th order nclusvel belong to S B the mathematcal nducton method we can prove the valdt of the followng formula: x W f ( ( a+ ( x a x e d ( a x s a polnomal wth respect to and x more exactl Here a r r ( x Cr ( x ( C moreover the coeffcents C r are real r numbers and some of them ma equal zero From belongness of the functon f ( x to the space S t follows that

4 4 M M Sabzalev sup From ( and (3 we get l W x l sup + sup + l f ( x ( 3 + Cl < (3 x ( x f ( a x e sup x l f ( + b Denotng n the last nequalt sup d b x ( 3 ( 4 b C C l sup l l ( + we get l W ( x (4 + Cl f ( ( 3 d d b C (4 σ e W ( x S Whle obtanng (4 we used ( σ e b ;( σ a where ; b > are some numbers W ( x Now prove S ; n + It s eas to show that the dervatves of the functon W ( x wth respect to x of an order are expressed b the formula W ( x f [ ] [ ] ( x a + W + a + (5 The functons ϕ ( [ ( a + ] m m wth respect to have a polnomal W to the space growth Above we proved the belongness of the functon S Thus each summand contaned n the rght hand sde of (5 s the product of two functons one of them has a polnomal growth the another one enters nto W ( x the space S Therefore the relaton S s vald whence t follows W ( x that S ; n + Lemma s proved The remanng functons W W Wn contaned n expanson (3 wll be sequentall determned from boundar value problems (5 (6 for n From lemma t follows that the functons W beng the solutons of problems (5 (6 for n wll have contnuous dervatves wth respect to x to the ( n + th order and condton (8 for the functon W wll be satsfed for n + ; n x l

5 BVP for quas-lnear ellptc equaton 5 From (3 and (6 we get that the constructed functon W satsfes the followng boundar condtons: W ( < < ; lm W ( (6 x x The functon W doesn t satsf generall speang boundar condton ( for x For compensatng the mssed boundar condton t s necessar to construct a boundar laer tpe functon near the boundar x 3 The Second Iteratve Process-Constructon of Boundar Laer Functons Let s construct a boundar laer tpe functon near the boundar x The frst teratve process s conducted on the base of decomposton ( of the operator L ε For conductng the second teratve process b means of whch we ll construct a boundar laer functon near the boundar x t s necessar to wrte a new decomposton of the operator L ε near ths boundar We mae change of varables: x ε Let s consder the auxlar functon n ε r ( where r ( + r are some smooth functons determned near x Expanson of L ε ( r n powers of ε n the coordnates ( has the form: + r r r Lε r ε (7 ( r r + + r r ε Φ ( + r r r ε where Φ are the nown functons dependent on r r r and ther frst second dervatves We loo for a boundar laer tpe functon near the boundar x n the form: + V V + ε V + + ε n V (8 as the soluton of the equaton Lε ( W + V Lε W ( ε (9 Expandng each functon W ( ε ; n n Talor formula at the pont ( we get a new expanson of the functon W n powers of ε n the n the followng form: coordnates W n + ε ω ( ( + ( ε

6 6 M M Sabzalev ω are ndependent of and the remanng functons ω are determned from the formula W ( ω ( ; n + ( + Substtutng expressons (8 ( for the functons V W to (9 and tang nto account (7 for determnng V V V n + we get the followng equatons: where W ( + V + + ( Q (3 where Q are the nown functons dependent on V V V ω ω ω n + ther frst and second dervatves We can wrte the formulae for Q obvousl but the are of bul form Here we gve formulae onl for Q and Q : ( V ω Q + ( + av V ω Q ( + av ( + ( + ω +! The boundar condtons for equatons ( (3 are obtaned from the requrement that the sum W + V should satsf the boundar condton ( W + V x (4 Substtutng the expressons for W and V respectvel from (3 and (8 nto (4 tang nto account that we loo for V ; n + as a boundar laer tpe functon we have V ϕ lm V ; n + (5 W where ϕ for n; ϕ The followng lemma s vald problem ( (5 (for has a unque soluton that s nfntel dfferentable wth respect to both varables and And the followng estmaton s vald Lemma For each ( (

7 BVP for quas-lnear ellptc equaton 7 ( ( ϕ ( ϕ ( ϕ ( e ; + V ( G (6 where G t t t are some nown polnomals of ther own arguments ( + wth non-negatve coeffcents the free members of these polnomals equal zero and even one of other coeffcents s non-zero Proof Exstence and unqueness of the soluton of problem ( (5 for were proved n [5] (see theorem The soluton of problem ( (5 for n the parametrc form s as follows + t + ( t t + ln V ( t + t (7 t where t s a parameter t ( s a real root of the algebrac equaton + t + t + ϕ ( (8 Note that f ϕ ( for some ( then the correspondng real root t ( of algebrac equaton (8 also vanshes and the expresson for n (7 loses s sense For ϕ ( as the soluton V ( of problem ( (5 for we can tae V ( Thus the desred soluton of problem ( (5 for s gven n the parametrc form (7 f ϕ ( and s predetermned b an dentt zero f ϕ ( The nfnte dfferentablt of V ( wth respect to was also proved n [5] But there ϕ ( has contnuous dervatves wth respect to to defnte fnte order In connecton wth the fact that here ϕ ( S s nfntel dfferentable t ( also wll be an nfntel dfferentable functon Hence t follows an nfnte dfferentablt of V ( wth respect to Prove the valdt of estmaton (6 From the frst equalt of (7 we can get an estmaton of the form + t t ( exp t ( exp( (9 Havng transformed equaton (8 we have: t ( [ t ( + ] ϕ ( whence t + follows that t ( ϕ ( Hence t s seen that the functon exp t ( + s bounded e exp t ( C Consequentl from (9 we get the followng estmaton t C ϕ exp (3 (

8 8 M M Sabzalev Tang nto account (3 n the second equalt of (7 we have V ( C ϕ exp( C > (3 Recallng that the parametrc form of the soluton of problem ( (5 for was obtaned b means of substtuton q from (3 we get an estmaton for C ϕ ( exp( C > (3 V We can represent the functon n the form V V B ( (33 B denotes the followng functon: where B( ( + + (34 V < B ( from (3 (33 we get an estmaton for For obtanng estmatons for the dervatves V ( wth respect to we dfferentate sequentall the both parts of (33 wth respect to and each tme tae nto account the estmatons of prevous dervatves These estmatons wll V ( 3 ; C > Now prove the estmatons for the dervatves V ( wth respect to be of the form (3 e C ϕ exp( and for mxed dervatves The functon ψ satsfes the equaton n varatons that s obtaned from equaton ( b dfferentatng wth respect to : ψ ψ B ( + (35 For from (5 we get that the functon ψ should satsf the boundar condtons ψ ϕ ( lm ψ (36 The soluton of problem (35 (36 s of the form ψ ϕ ( exp B ( ξ dξ (37

9 BVP for quas-lnear ellptc equaton 9 exp B ξ dξ n the followng Usng (34 and estmaton (3 we estmate wa: exp dξ exp B ( ξ ( + C dξ exp ( ξ + exp exp(ξ ( + C + exp ( ξ dξ exp ln ( + C + exp ξ ( ξ ξ [( + C + ] C e [( + C + exp( ] where C ϕ ( C C [( + C + ] Hence and from (37 we get the estmaton ψ C ϕ exp( ( C > (38 ψ From (37 t follows that B ( ψ Tang nto account (38 hence we get an estmaton for the mxed dervatve ψ V C ϕ exp( ( (39 V Now we can get an estmaton also for (37 wth respect to we have Dfferentatng the both sdes of [ ] + B ξ dξ ψ ϕ exp B ( ξ dξ ψ (4 From (34 t follows that B ( [ ] ( + B ( V Obvousl < B for an natural number Knowng estmaton (3 for V and estmaton (39 for we estmate [ B ( ] : B C ϕ ( ϕ ( exp (4 [ ] Tang nto account (38 and (4 n (4 we have

10 M M Sabzalev ψ V [ C ϕ ( ϕ ( + C ϕ ( ] exp( The valdt of estmaton (6 for subsequent dervatves s proved n the same wa Lemma s proved B theorem whose proof s gven n [5] (see theorem 3 there exsts a unque soluton of each problem ( (5 for n + and these solutons are represented b the followng formula: ν z V ( ϕ ( B ( z e Q ( ξ dξ dz exp[ ν ( ] (4 z ν denotes the functon Here ( B ( ξ ν dξ (43 Unle the estmatons n the paper [5] here t s necessar to get such estmatons for V V V n + that could enable to stud the behavor of functons not onl as and also as Substtutng n (4 we get a formula for V ( Usng the obvous expressons for the functons Q ( and ( at for ω and tang nto V account the nown estmatons for V also the belongness of W ( the functons W ( and to the space S we get Q ( ( q exp( (44 where q ( s a nown functon from the space S Followng (44 from (4 for we can get the estmaton exp( C ( C ϕ ( + q ( V > (45 Dfferentatng the both sdes of (4 for wth respect to we have B ( V + Q ξ dξ (46 Usng estmates (44 (45 n (46 we get an estmaton for The estmates for hgher dervatves wth respect to are obtaned from formulae obtaned b sequental dfferentaton of both sdes of (46 and from the estmatons for prevous dervatves of V ( Note that these estmatons are of the form

11 BVP for quas-lnear ellptc equaton V ( ( q ( + q ( exp( ; where ( q ( q S q S nown functons moreover Now we get estmatons for the dervatves of V ( q are the wth respect to and for mxed dervatves We can defne the functon as the soluton of a boundar value problem for the equaton n varatons that s obtaned from (3 for b dfferentatng wth respect We can note that the functon s also determned b formula (4 onl n ths formula the functon ϕ ( should be replaced b ϕ and functon Q ( ξ dξ z z b the followng functon: ( z Q ( d B ξ ξ + ( z z Consequentl ths tme b obtanng estmatons nstead of (44 we use the estmaton z Q As a result for ( dξ + B ( z ( z ξ ( q ( + q ( z exp( z we get the estmaton: z ( q ( + q ( + q ( exp( 3 If we dfferentate the both sdes of the formula for we can get the followng estmaton: V q( + q q where ; 3 ( ( + q ( exp( 3 S wth respect to It should be noted that at each dfferentaton of V ( wth respect to the power of the polnomal wth respect to standng at the rght sde of the estmaton ncreases b a unt The estmaton for V ( n the general case has the form V ( + ( q( + q( + + q ( exp( + where q S ; are the nown functons ( +

12 M M Sabzalev Usng the obvous form for Q ( and tang nto account the nown estmatons for V V and ther dervatves we can show that Q ( ( q( + q ( + q3( exp( (47 Havng put n (4 and tang nto attenton estmaton (47 the valdt of the followng estmatons s proved n the same wa as above V ( + 3 ( q( + q( + + q 3( exp( + Contnung ths process and each tme tang nto account the obvous form of the rght sde of the equaton for V we get the estmaton V ( + s q s ( exp( ; n + (48 s where q s S are the nown functons Multpl all the functons V ; n + b a smoothng multpler and leave prevous denotaton for the obtaned new functons At the expense of smoothng multplers all the functons V ; n + vansh for x Therefore t follows from (6 that the constructed sum U W + V n addton to boundar condton (4 also satsfes the condton ( W + V (49 x From (6 and (5 we get that ths sum satsfes the followng boundar condton as well lm ( W + V (5 Havng denoted U U z we get the followng asmptotc expanson n small parameter of the soluton of problem ( (: U n where z s a remander term Now estmate the remander term ε W + ε V + z (5 4 Estmaton of Remander Term Puttng together (4 and (9 we get that U satsfes the equaton L U ( ε ε (5 Subtractng (5 from ( we have p p p u u p u u p p ε ε ε Δz +

13 BVP for quas-lnear ellptc equaton 3 where z z n + + az ε + F( ε x (53 F(ε x C for an ε ε and C > s ndependent of ε L (Π [ From ( (4 (49 (5 and (5 t follows that z satsfes the boundar condtons: z z lm z (54 x x Multplng the both sdes of (53 b z U U and ntegratng b parts allowng for boundar condtons (54 after some transformatons we get the estmaton p+ p+ p z z z z z ε + dxd + ε + dxd + dx + Π Π Π (55 + C Π z dxd C ε ( where C > C are the constants ndependent of ε > 5 Concluson Combnng the obtaned results we arrve at the followng statement Theorem Let f ( x be a functon gven n Π have contnuous dervatves wth respect to x to the ( n + -th order nclusvel be nfntel dfferentable wth respect to and satsf equaton (7 Then for the generalzed soluton of problem ( ( t holds asmptotc representaton (5 where the functons W are determned b the frst teratve process V s a boundar laer tpe functon near the boundar x z s a remander term and estmaton (55 s vald for t References [] Beon Jaeoung Sngularl perturbed nonlnear Drchlet problems wth a general nonlneart Trans Amer Math Soc 36 (4 ( 98- [] Cheng Yan Asmptotc soluton of a boundar problem for the semlnear sngularl perturbed ellptcal equaton Shuxue Zazh J Math (5 5-9

14 4 M M Sabzalev [3] Del Pno Manuel Felmer Patrco Localzng spelaer patterns sngularl perturbed ellptc problems Tohou Math Publ 8 ( [4] MMSabzalev The asmptotc form of the soluton of boundar value problem for sngular perturbed quaslnear parabolc dfferental equaton Proceedngs of Mathematcs and Mechancs of NAS Azerbaan ( [5] MMSabzalev The asmptotc form of the soluton of boundar value problem for sngular perturbed quaslnear parabolc dfferental equaton Proceedngs of Mathematcs and Mechancs of NAS Azerbaan ( [6] ZHU Zhen-bo NI Mng-ang Drchlet problem of semlnear sngular perturbaton ellptc equatons Huadong shfan daxue xuebao Zran Kexue Ban I East Chna Norm Unv Sc 5 ( [7] ZHU Zhen-bo NI Mng-ang Drchlet problem of semlnear sngular perturbaton ellptc equatons Huadong shfan daxue xuebao Zran Kexue Ban I East Chna Norm Unv Sc 5 ( [8] VYu Lunn On asmptotcs of solutons of the frst boundar problem for quaslnear ellptc equatons Vestn Mosow Unv 3 ( Receved: September