Journal of Mathematical Analysis and Applications 255, 678᎐697Ž. 2001 doi:10.1006rjmaa.2000.7308, available online at http:rrwww.idealibrary.com on A Singular Perturbation Problem of Carrier and Pearson Walter G. Kelley View metadata, citation and similar papers at core.ac.uk brought to you by CORE Uni¨ersity of Oklahoma, Norman, Oklahoma 73019 provided by Elsevier - Publisher Connector Submitted by Robert O’Malley Received October 5, 1999 1. INTRODUCTION The use of perturbation methods to study singularly perturbed differen- tial equations has enjoyed spectacular success since its origination in the work of Prandtlwx 19 on viscous incompressible flow past an object for a large Reynolds number. During the nearly 100-year history of this area, a large number of formal approximation methods have been developed and refined, including the method of averaging, boundary layer methods, matched asymptotic expansions, and multiple scales. The textbooks of Nayfehwx 15 , O’Malley wx 18 , Kevorkian and Cole wx 11 , Bender and Orszag wx1 , Smith w 20 x , and Holmeswx 5 contain discussions of some or all of these methods with many examples that illustrate the fascinating variety of behavior exhibited by these problems. On the other hand, examples are known for which the formal methods fail, either because the expansions contain terms that cannot be deter- mined uniquelyŽ. so that the method leads to ambiguous results or because the computed expansions are spuriousŽ i.e., do not approximate actual solutions of the problem. One of the most famous examples of the failure of the formal methods is a problem introduced in a textbook of Carrier and Pearsonwx 3 , ⑀ 2 uЉ q u2 y 1 s 0, uŽ.y1 su Ž.1 s 0. 678 0022-247Xr01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. A SINGULAR PERTURBATION PROBLEM 679 They constructed an approximate solution q y 1 xxx0 AxŽ., ⑀ sy1 q 3 sech22q lnŽ.'' 2 q 3 q 3 sech ''2 ⑀ 2 ⑀ 1 y x q 3 sech2 q lnŽ.'' 2 q 3, '2 ⑀ y where x0 is an arbitrary number in Ž.1, 1 . This approximation is near y s " s 1, except for boundary layers at x 1 and a spike layer at x x0. One can show that AxŽ., ⑀ satisfies the boundary value problem except for exponentially small remainder terms. However, since the exact solutions of the problem can be expressed in terms of elliptic functions, which are s periodic, only the choice x0 0 corresponds to an actual solution. In fact, O’Malleywx 17 used phase plane analysis to show that any spike layers for autonomous problems of this type must occur at evenly-spaced points in the interval. The Carrier᎐Pearson example has been discussed in many papers. Langewx 12 used a formal matching procedure that was special to the problem under consideration and included exponentially small terms to resolve the ambiguity. Variational methodsŽ see Kath et al.wx 8. lead to a similar conclusion but do not provide detailed approximations. Wardw 21, 22x demonstrated that the indeterminacy can be resolved for this and related problems by an asymptotic projection method. Most recently, MacGillivraywx 13 presented a systematic matching scheme involving expo- nentially small terms that determined the position of the spike as a product of the matching process. In addition, there is a large body of work on the existence and location of spikes for positive solutions of semilinear elliptic boundary value prob- lems. For the case where the nonlinearity has three zeros, Kelley and Ko wx9 and Jang wx 6 provided a number of existence results for the Dirichlet problem. Ni and Weiwx 16 and Wei wx 23Ž see also the many other works cited in these papers. used variational methods to investigate the existence and location of single or multiple spikes for certain Dirichlet problems in which the nonlinearity has a pair of zeros. The Neumann problem has received even more attention because of its importance as a mathematical model for morphogenesisŽ see Gui and Weiwx 4 and the many references therein. In this paper, we reconsider the Carrier᎐Pearson example from a different point of view. Working with a general autonomous problem, we show that the assumption of a symmetric location of the spike allows the construction of uniform approximate solutions to all exponentially small orders. Moreover, the existence of these formal approximations permits 680 WALTER G. KELLEY their rigorous validation by means of a simple Green’s function argument. One of the interesting details of the argument is that the Green’s function is exponentially large, from which it follows that approximate solutions with exponentially small errors may be spurious if there is no procedure available for improving the approximation. Our analysis does not establish the nonexistence of solutions with spikes located elsewhere, but it will be clear that the assumption of symmetry is essential to our construction of higher order approximations. The final section of this paper contains some related examples. The first two deal with the issues of nonsymmetric boundary conditions and multi- ple spikes. The last one is an example from Carrierwx 2 involving a nonautonomous differential equation. This example has also been dis- cussed by Kathwx 7 , Bender and Orszag wx 1 , and most recently MacGillivray et al.wx 14 . Once again formal methods produce approximate solutions with spikes at arbitrary interior points. However, in this case the problem has solutions with spikes only at the endpoints and near the midpoint of the interval. 2. CONSTRUCTION OF AN APPROXIMATION We will construct an approximate solution with a spike near x s 0 for the following boundary value problem with a small, positive parameter ⑀ multiplying the highest derivative: ⑀ 2 uЉ q fuŽ., ⑀ s 0,Ž. 2.1 uŽ.y1 su Ž.1 s 0.Ž. 2.2 Our basic assumptions are that fuŽ.Ž, ⑀ s 1 q g Ž..Ž.⑀ Fu, where g Ž.⑀ ª 0 as ⑀ ª 0, and that F satisfies the following hypotheses. There are numbers a - 0 - b such that Ž.H1 Fa Ž.s 0, Fb Ž.) 0; Ž.H2 FЈ Ž.a - 0; u - - - b s Ž.H3 HaaFsdsŽ. 0 for a u b and H FsdsŽ. 0; Ž.H4 F has derivatives of all orders. It is well known thatŽ. H3 is a necessary condition for the existence of a solution with a spike extending from near u s a up to u s b Žsee O’Mal- leywx 17. The following elementary lemma will be useful in the construction. LEMMA 2.1. Let hŽ. x and j Ž. x be continuous and e¨en on Žyϱ, ϱ .. Then any solution yŽ. x of yЉ q hxy Ž.s j Ž. x with yЈ Ž.0 s 0 is also e¨en. A SINGULAR PERTURBATION PROBLEM 681 The construction of the initial approximation is fairly standard, so we give only a sketch. First, the approximation a q SxŽ.r⑀, ⑀ near the spike at x s 0 is given implicitly by << aqS d¨ x sy . H y ¨ ⑀ b ' 2Ha f One can show that a q SxŽ.r⑀, ⑀ is a smooth solution of Eq.Ž. 2.1 and even, and that yl Ž⑀ .< x < r⑀ F r⑀ ⑀ F yl Ž⑀ .< x < r⑀ Ce12SxŽ., Ce y F F ⑀ syЈ ⑀ for 1 x 1, where lŽ. ' f Ž.a, and C12, C are positive con- stants. Similarly, the boundary layer approximation a q LŽŽ1 q x .r⑀, ⑀ . at x sy1 is given implicitly by q d¨ 1 q x a L s H yy ¨ ⑀ 0 ' 2Ha f for y1 F x F 1if a - 0. If a ) 0, we simply change the sign on one side of the equation. Then a q L is a smooth solution of Eq.Ž. 2.1 , satisfies the left boundary condition, and yl Ž⑀ .Ž1qx.r⑀ F q r⑀ ⑀ F yl Ž⑀ .Ž1qx.r⑀ De12LŽ.Ž.1 x , De y F F for 1 x 1 and some positive constants D12, D . By symmetry, the approximation at x s 1is a q RŽ.Ž.Ž.1 y x r⑀ , ⑀ ' a q L Ž.1 y x r⑀ , ⑀ . Now the initial approximation is ⑀ s q r⑀ ⑀ q q r⑀ ⑀ q y r⑀ ⑀ Ax0 Ž., a Sx Ž, .LŽ.Ž. Ž.1 x , R Ž.1 x , , Ž.2.3 an even function. To determine the accuracy of A0 , consider ⑀ 2 Y q ⑀ s ⑀ 2 Љ q Љ q Љ q q q q ⑀ A00fAŽ.Ž, S L R .Žfa S L R, .. On the interval wxy1, 0 , we have ⑀ 2 RЉ, R s OŽeyl Ž⑀ .r⑀ .,as⑀ ª 0, so ⑀ 2 Y q ⑀ s q q ⑀ y q ⑀ y q ⑀ A00fAŽ.Ž, fa S L, .Žfa S, .Žfa L, . q O Ž.eyl r⑀ . 682 WALTER G. KELLEY The mean value theorem, together with the properties of S and L, implies that the expression in brackets is also OŽeyl r⑀ .. A similar argument on the intervalwx 0, 1 then yields ⑀ 2 Y q ⑀ s yl r⑀ A00fAŽ.Ž., O e wxy ⑀ ª uniformly on 1, 1 as 0. A0 also satisfies the boundary conditions except for exponentially small terms. If S0 were not centered at 0 then the error term would be larger, but still exponentially small. Next, we seek better approximations of the form N ⑀ s ⑀ q r⑀ ⑀ q q r⑀ ⑀ AxN Ž., Ax0 Ž., Ý Txkk Ž, .U Ž. Ž1 x ., ks1 eyklr⑀ qV 1 y x r⑀ , ⑀ .2.4 k Ž.Ž. ⑀ ky1 Ž. ŽSince all of our functions depend on ⑀, we will usually omit ⑀ in our functional notation.