Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. Part I. Direct (variational) approximation

Călin-Ioan Gheorghiu
(original), paper

Abstract

Publisher Name
We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, f(u) =u3, the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct (variational) methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method.

Authors

Calin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

Damian Trif
Babes-Bolyai University, Faculty of Mathematics and Computer Science

Keywords

nonlinear reaction-diffusion; positive solution; conserved integral; bifurcation; variational formulation; Lagrange multiplier; finite elements method.
[1] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 , pp. 349–381, 1973.
[2] Aronson, D. G. and Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Lecture Notes in Math., 446, Springer-Verlag, 1975.
[3] Crandal, M. G. and Rabinowitz, P. H., Nonlinear Sturm-Liouvil le eigenvalue problems and topological degree, J. Math. Mech., 19 , pp. 1083–1102, 1970.
[4] Crandal, M. G. and Rabinowitz, P. H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 52, pp. 161–180, 1973.
[5] Elsgolts, L., Differential Equations and the Calculus of Variations , Mir Publishers, Moscow, 1980.
[6] Gheorghiu, C. I., Solution to problem 97-8 by Ph. Korman, SIAM Review, 39, 1997, SIAM Review, 40 , no. 2, pp. 382–385, 1998.
[7] Gheorghiu, C. I.and Trif, D., On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem , Studia UBB, Mathemat- ica, XLX, pp. 29–37, 2000.
[8] Grindrod, P., The Theory and Applications of Reaction-Diffusion Equations; Patern and Waves, Second Edition, Clarendon Press, Oxford, 1996.
[9] Keller, H. B. and Cohen, D. S., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, pp. 1361–1376, 1967.
[10] Korman, Ph., Average temperature in a reaction-diffusion process, Problem 97-8, SIAM Review, 39 , p. 318, 1997.
[11] Laetsch, Th., The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20, pp. 1–13, 1970.
[12] Matkowsky, B. J., A simple nonlinear dynamic stability problem, Bull. Amer. Math.Soc., 76, pp. 620–625, 1970.
[13] Moore, R. A. and Nehari, Z., Nonoscil lation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 , pp. 30–52, 1959.
[14] Sattinger, D. H., Topics in Stability and Bifurcation Theory, Springer-Verlag, 1973.
[15] Sattinger, D. H., Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rat. Mech. Anal., 43, pp. 155–165, 1970.
[16] Sattinger, D. H., Monotone methods in nonlinear el liptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, pp. 979–1000, 1972.
[17] Schwind, W. J., Ji, J. and Koditschek, D. E., A physically motivated further note on the mean value theorem for integrals, Amer. Math. Monthly, 126 , pp. 559–564, 1999.
[18] Simpson, B. R. and Cohen, D. S., Positive solutions of nonlinear el liptic eigenvalue problems, J. Math. Mech., 19, pp. 895–910, 1970.
[19] Turner, R. E. I., Nonlinear eigenvalue problems with nonlocal operators, Comm. Pure Appl. Math., 23, pp. 963–972, 1970.
[20] Wollkind, D. J., Monoranjan, V. S. and Zhang, L., Weakly nonlinear stability analysis of prototype reaction-diffusion model equations , SIAM Review, 36 , no. 2, pp. 176– 214, 1994.

Cite this paper as

C.I. Gheorghiu, D. Trif, Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. Part I. Direct (variational) approximation, Rev. Anal. Numér. Théor. Approx. 31 (2002) 61-70, https://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art8/

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[1] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, pp. 349-381, 1973.

[2] Aronson, D. G. and Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Lecture Notes in Math., 446, Springer-Verlag, 1975.

[3] Crandal, M. G. and Rabinowitz, P. H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19, pp. 1083-1102, 1970.

[4] Crandal, M. G. and Rabinowitz, P. H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 52, pp. 161-180, 1973.

[5] Elsgolts, L., Differential Equations and the Calculus of Variations, Mir Publishers, Moscow, 1980.

[6] Gheorghiu, C. I., Solution to problem 97-8 by Ph. Korman, SIAM Review, 39, 1997, SIAM Review, 40, no. 2, pp. 382-385, 1998.

[7] Gheorghiu, C. I. and Trif, D., On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem, Studia UBB, Mathematica, XLV, pp. 29-37, 2000.

[8] Grindrod, P., The Theory and Applications of Reaction-Diffusion Equations; Patern and Waves, Second Edition, Clarendon Press, Oxford, 1996.

[9] Keller, H. B. and Cohen, D. S., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, pp. 1361-1376, 1967.

[10] Korman, Ph., Average temperature in a reaction-diffusion process, Problem 97-8, SIAM Review, 39, pp.318, 1997.

[11] Laetsch, Th., The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20, pp. 1-13, 1970.

[12] Matkowsky, B. J., A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc., 76, pp. 620-625, 1970.

[13] Moore, R. A. and Nehari, Z., Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93, pp. 30-52, 1959.

[14] Sattinger, D. H., Topics in Stability and Bifurcation Theory, Springer-Verlag, 1973.

[15] Sattinger, D. H., Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rat. Mech. Anal., 43, pp. 155-165, 1970.

[16] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, pp. 979-1000, 1972.

[17] Schwind, W. J., Ji, J. and Koditschek, D. E., A physically motivated further note on the mean value theorem for integrals, Amer. Math. Monthly, 126, pp. 559-564, 1999.

[18] Simpson, B. R. and Cohen, D. S., Positive solutions of nonlinear elliptic eigenvalue problems, J. Math. Mech., 19, pp. 895-910, 1970.

[19] Turner, R. E. I., Nonlinear eigenvalue problems with nonlocal operators, Comm. Pure Appl. Math., 23, pp. 963-972, 1970.

[20] Wollkind, D. J., Monoranjan, V. S. and Zhang, L., Weakly nonlinear stability analysis of prototype reaction-diffusion model equations, SIAM Review, 36, no. 2, pp. 176-214, 1994.

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DIRECT AND INDIRECT APPROXIMATIONS TO POSITIVE SOLUTION FOR A NONLINEAR REACTION-DIFFUSION PROBLEM I. DIRECT (VARIATIONAL)

CǍLIN IOAN GHEORGHIU* and DAMIAN TRIF † † ^(†){ }^{\dagger}†

Abstract

We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, f ( u ) = u 3 f ( u ) = u 3 f(u)=u^(3)f(u)=u^{3}f(u)=u3, the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct (variational) methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method.

MSC 2000. 34B18, 34C23, 65L10, 65L60.
Keywords. nonlinear reaction-diffusion, positive solution, conserved integral, bifurcation, variational formulation, Lagrange multiplier, finite elements method.

1. INTRODUCTION

We are concerned with the existence, uniqueness and numerical approximations of positive solutions for the semilinear parabolic problem
(1) u t = u x x + u p , 0 < x < L , t > 0 , u ( 0 , t ) = u ( L , t ) = 0 , t > 0 , u ( x , 0 ) = u 0 ( x ) , 0 < x < L , (1) u t = u x x + u p , 0 < x < L , t > 0 , u ( 0 , t ) = u ( L , t ) = 0 , t > 0 , u ( x , 0 ) = u 0 ( x ) , 0 < x < L , {:(1){:[u_(t)=u_(xx)+u^(p)",",0 < x < L","t > 0","],[u(0","t)=u(L","t)=0",",t > 0","],[u(x","0)=u_(0)(x)",",0 < x < L","]:}:}\begin{array}{ll} u_{t}=u_{x x}+u^{p}, & 0<x<L, t>0, \\ u(0, t)=u(L, t)=0, & t>0, \tag{1}\\ u(x, 0)=u_{0}(x), & 0<x<L, \end{array}(1)ut=uxx+up,0<x<L,t>0,u(0,t)=u(L,t)=0,t>0,u(x,0)=u0(x),0<x<L,
where u 0 ( x ) u 0 ( x ) u_(0)(x)u_{0}(x)u0(x) satisfies the compatibility condition u 0 ( 0 ) = u 0 ( L ) = 0 u 0 ( 0 ) = u 0 ( L ) = 0 u_(0)(0)=u_(0)(L)=0u_{0}(0)=u_{0}(L)=0u0(0)=u0(L)=0, with any real p , p > 1 p , p > 1 p,p > 1p, p>1p,p>1 and L < ∞ L < ∞ L < ooL<\inftyL<∞. We further assume that (1) has a stationary positive solution u ¯ u ¯ bar(u)\bar{u}u¯, i.e. u ¯ u ¯ bar(u)\bar{u}u¯ is a solution of boundary value problem
u x x + u p = 0 , 0 < x < L (2) u ( 0 ) = u ( L ) = 0 u x x + u p = 0 , 0 < x < L (2) u ( 0 ) = u ( L ) = 0 {:[u_(xx)+u^(p)=0","quad0 < x < L],[(2)u(0)=u(L)=0]:}\begin{align*} & u_{x x}+u^{p}=0, \quad 0<x<L \\ & u(0)=u(L)=0 \tag{2} \end{align*}uxx+up=0,0<x<L(2)u(0)=u(L)=0
The stationary problem (2) is in fact the problem 97 − 8 97 − 8 97-897-897−8 by Ph. Korman [10]. He invokes a phase-plane analysis to observe that the positive solution u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x)
is unique ( u ¯ ( x ) > 0 ( u ¯ ( x ) > 0 ( bar(u)(x) > 0(\bar{u}(x)>0(u¯(x)>0 for x ∈ ( 0 , L ) ) x ∈ ( 0 , L ) ) x in(0,L))x \in(0, L))x∈(0,L)) and asks to show that for p = 3 p = 3 p=3p=3p=3 and for any L L LLL this solution satisfies
(3) ∫ 0 L u ¯ ( x ) d x = π 2 (3) ∫ 0 L   u ¯ ( x ) d x = π 2 {:(3)int_(0)^(L) bar(u)(x)dx=(pi)/(sqrt2):}\begin{equation*} \int_{0}^{L} \bar{u}(x) \mathrm{d} x=\frac{\pi}{\sqrt{2}} \tag{3} \end{equation*}(3)∫0Lu¯(x)dx=π2
which means that the integral of u ¯ u ¯ bar(u)\bar{u}u¯ is conserved independently on L L LLL.
We have proved in [6] that the solution u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x) of (2) satisfies
(4) ∫ 0 L u ( x ) d x = 2 ( p + 1 ) u max 3 − p 2 ( F ( 1 ) − F ( 0 ) ) (4) ∫ 0 L   u ( x ) d x = 2 ( p + 1 ) u max 3 − p 2 ( F ( 1 ) − F ( 0 ) ) {:(4)int_(0)^(L)u(x)dx=sqrt(2(p+1))u_(max)^((3-p)/(2))(F(1)-F(0)):}\begin{equation*} \int_{0}^{L} u(x) \mathrm{d} x=\sqrt{2(p+1)} u_{\max }^{\frac{3-p}{2}}(F(1)-F(0)) \tag{4} \end{equation*}(4)∫0Lu(x)dx=2(p+1)umax3−p2(F(1)−F(0))
where F ( v ) F ( v ) F(v)F(v)F(v) is the primitive function of v / 1 − v p + 1 , v = u / u max v / 1 − v p + 1 , v = u / u max  v//sqrt(1-v^(p+1)),v=u//u_("max ")v / \sqrt{1-v^{p+1}}, v=u / u_{\text {max }}v/1−vp+1,v=u/umax  and u max u max  u_("max ")u_{\text {max }}umax  is the maximum value of u u uuu on [ 0 , L ] [ 0 , L ] [0,L][0, L][0,L].
The condition (3) will play a key role in our analysis. This independence of the length of the domain condition can be used successfully to approximate, by a direct or indirect method, the solutions of the problems (1) and (2).
The interest in such stationary solutions, sometimes called dissipative structures, has been occasioned by their possible role in reflecting the corresponding phenomena of pattern formation in developing organisms and in ecological communities. Problems of type (1), in one or more spatial dimensions, have been used also to model some biological phenomenon or technological processes, but we do not go into much detail about applications in the various fields mentioned above (see for example [8]).
The first goal of the present paper is an analysis of bifurcation issue for problem (2). The particular form of reaction term in (2) make impossible that analysis in the terms of Stuart-Watson method or two-time technique (Matkowsky) as they are presented in [20]. Consequently, we try to find a functional relationship between u max u max  u_("max ")u_{\text {max }}umax  (the maximum value of u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x) on ( 0 , L 0 , L 0,L0, L0,L )) and L L LLL. This turn out to be of the form L u max = c o n s t L u max = c o n s t Lu_(max)=constL u_{\max }=c o n s tLumax=const., which shows that indeed the bifurcation appears at infinity from the null solution. In fact the problem (2) is autonomous, and when one attempts to solve it in a closed form, encounters an integrand of the form 1 / 1 − t 4 1 / 1 − t 4 1//sqrt(1-t^(4))1 / \sqrt{1-t^{4}}1/1−t4. We estimate subsequent integrals using a generalized mean value theorem suggested in [17].
The second aim of our paper is a direct approximation of the positive solution u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x) of (2) which satisfies (3). In this respect we built up a functional defined on H 0 1 ( 0 , L ) H 0 1 ( 0 , L ) H_(0)^(1)(0,L)H_{0}^{1}(0, L)H01(0,L) which has a positive lower bound. This functional is then augmented introducing the restriction (3) by means of Lagrange's multiplier method. Eventually, we determine the corresponding Euler's equations for this new functional and use them to obtain the finite elements approximation to u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x). Thus, it is underlined the importance of the conserved integral condition (3) in the numerical analysis of reaction-diffusion problem (2).
The content of the paper is as follows: in Section 2 we display some information on the existence and uniqueness of the positive solution u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x) of (2). We obtain the condition (4) and review some properties of u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x), including the bifurcation from infinity.
In section 3 we give the variational characterization of the positive solution and obtain the numerical approximation u ¯ h ( x ) u ¯ h ( x ) bar(u)_(h)(x)\bar{u}_{h}(x)u¯h(x) of u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x) by piecewise linear finite elements. We proceed as usually in f.e.m., write down the discrete analogous (18) of (2) and solve the resulting nonlinear algebraic system by Newton's method with initial guess satisfying (3).

2. SOME PROPERTIES OF U ¯ ( X ) U ¯ ( X ) bar(U)(X)\bar{U}(X)U¯(X)

In his paper [1], Laetsch considers a problem of type (2) with a more general reaction term. Specifically, he puts instead of u p , λ f ( u ) u p , λ f ( u ) u^(p),lambda f(u)u^{p}, \lambda f(u)up,λf(u), with f ( 0 ) = 0 f ( 0 ) = 0 f(0)=0f(0)=0f(0)=0 and f ( w ) / w f ( w ) / w f(w)//wf(w) / wf(w)/w is a non-decreasing function of w w www for w > 0 w > 0 w > 0w>0w>0. Then the problem (2) has exactly one positive solution for λ > 0 λ > 0 lambda > 0\lambda>0λ>0 and as λ λ lambda\lambdaλ increases from 0 , the norms of the corresponding solutions decreases from + ∞ + ∞ +oo+\infty+∞ to 0 . He reduced the problem of solving (2) to a quadrature and observed that all positive solutions of (2), for λ > 0 λ > 0 lambda > 0\lambda>0λ>0 are strictly positive and have exactly one maximum on ( 0 , L ) ( 0 , L ) (0,L)(0, L)(0,L). They are also symmetric about the point x = L / 2 x = L / 2 x=L//2x=L / 2x=L/2.
With these we can prove easy 6 the conditions (3) and (4). Multiplying the differential equation in (2) with p = 3 p = 3 p=3p=3p=3 by 2 u ′ ( x ) 2 u ′ ( x ) 2u^(')(x)2 u^{\prime}(x)2u′(x), we write it in the form
(5) [ ( u ′ ( x ) ) 2 ] ′ = − 1 2 ( u 4 ( x ) ) ′ (5) u ′ ( x ) 2 ′ = − 1 2 u 4 ( x ) ′ {:(5)[(u^(')(x))^(2)]^(')=-(1)/(2)(u^(4)(x))^('):}\begin{equation*} \left[\left(u^{\prime}(x)\right)^{2}\right]^{\prime}=-\frac{1}{2}\left(u^{4}(x)\right)^{\prime} \tag{5} \end{equation*}(5)[(u′(x))2]′=−12(u4(x))′
If we integrate (5) from 0 to x , 0 < x < L x , 0 < x < L x,0 < x < Lx, 0<x<Lx,0<x<L, we get the first integral
u ′ ( x ) 2 − u ′ ( 0 ) 2 = − 1 2 u ( x ) 4 u ′ ( x ) 2 − u ′ ( 0 ) 2 = − 1 2 u ( x ) 4 u^(')(x)^(2)-u^(')(0)^(2)=-(1)/(2)u(x)^(4)u^{\prime}(x)^{2}-u^{\prime}(0)^{2}=-\frac{1}{2} u(x)^{4}u′(x)2−u′(0)2=−12u(x)4
or explicitly
(6) u ′ ( x ) = ± 1 2 ( 2 u ′ ( 0 ) ) 2 − u ( x ) 4 . (6) u ′ ( x ) = ± 1 2 2 u ′ ( 0 ) 2 − u ( x ) 4 . {:(6)u^(')(x)=+-(1)/(sqrt2)sqrt((sqrt2u^(')(0))^(2)-u(x)^(4)).:}\begin{equation*} u^{\prime}(x)= \pm \frac{1}{\sqrt{2}} \sqrt{\left(\sqrt{2} u^{\prime}(0)\right)^{2}-u(x)^{4}} . \tag{6} \end{equation*}(6)u′(x)=±12(2u′(0))2−u(x)4.
For the maximum value u max u max  u_("max ")u_{\text {max }}umax  of u ( x ) u ( x ) u(x)u(x)u(x) on ( 0 , L ) ( 0 , L ) (0,L)(0, L)(0,L) we have from the first integral
(7) u max 2 = 2 u ′ ( 0 ) (7) u max 2 = 2 u ′ ( 0 ) {:(7)u_(max)^(2)=sqrt2u^(')(0):}\begin{equation*} u_{\max }^{2}=\sqrt{2} u^{\prime}(0) \tag{7} \end{equation*}(7)umax2=2u′(0)
From (6) and (7) we deduce
∫ 0 L u ( x ) d x = ∫ 0 L u d u d u d x = 2 ⋅ 1 2 ∫ 0 u max 2 d u 2 ( 2 u ′ ( 0 ) ) 2 − ( u 2 ) 2 = 2 arcsin ⁡ u 2 2 u ′ ( 0 ) | 0 u max = π 2 ∫ 0 L   u ( x ) d x = ∫ 0 L   u d u d u d x = 2 ⋅ 1 2 ∫ 0 u max   2 d u 2 2 u ′ ( 0 ) 2 − u 2 2 = 2 arcsin ⁡ u 2 2 u ′ ( 0 ) 0 u max = π 2 {:[int_(0)^(L)u(x)dx=int_(0)^(L)(u(d)u)/(((d)u)/((d)x))=2*(1)/(2)int_(0)^(u_(max))(sqrt2(d)u^(2))/(sqrt((sqrt2u^(')(0))^(2)-(u^(2))^(2)))],[=sqrt2arcsin((u^(2))/(sqrt2u^(')(0)))|_(0)^(u_(max))=(pi)/(sqrt2)]:}\begin{aligned} \int_{0}^{L} u(x) \mathrm{d} x & =\int_{0}^{L} \frac{u \mathrm{~d} u}{\frac{\mathrm{~d} u}{\mathrm{~d} x}}=2 \cdot \frac{1}{2} \int_{0}^{u_{\max }} \frac{\sqrt{2} \mathrm{~d} u^{2}}{\sqrt{\left(\sqrt{2} u^{\prime}(0)\right)^{2}-\left(u^{2}\right)^{2}}} \\ & =\left.\sqrt{2} \arcsin \frac{u^{2}}{\sqrt{2} u^{\prime}(0)}\right|_{0} ^{u_{\max }}=\frac{\pi}{\sqrt{2}} \end{aligned}∫0Lu(x)dx=∫0Lu du du dx=2⋅12∫0umax2 du2(2u′(0))2−(u2)2=2arcsin⁡u22u′(0)|0umax=π2
We can apply the same strategy for the reaction term f ( u ) = u p , p > 1 f ( u ) = u p , p > 1 f(u)=u^(p),p > 1f(u)=u^{p}, p>1f(u)=up,p>1, to obtain (4).
The nondimensional form of problem (2) for p = 3 p = 3 p=3p=3p=3 reads as follows:
(8) { θ ′ ′ + λ 2 θ 3 = 0 , 0 < t < 1 θ ( 0 ) = θ ( 1 ) = 0 , (8) θ ′ ′ + λ 2 θ 3 = 0 , 0 < t < 1 θ ( 0 ) = θ ( 1 ) = 0 , {:(8){[theta^('')+lambda^(2)theta^(3)=0","quad0 < t < 1],[theta(0)=theta(1)=0","]:}:}\left\{\begin{array}{l} \theta^{\prime \prime}+\lambda^{2} \theta^{3}=0, \quad 0<t<1 \tag{8}\\ \theta(0)=\theta(1)=0, \end{array}\right.(8){θ′′+λ2θ3=0,0<t<1θ(0)=θ(1)=0,
where u m := max x ∈ [ 0 , L ] u ( x ) , λ := L u m u m := max x ∈ [ 0 , L ]   u ( x ) , λ := L u m u_(m):=max_(x in[0,L])u(x),lambda:=Lu_(m)u_{m}:=\max _{x \in[0, L]} u(x), \lambda:=L u_{m}um:=maxx∈[0,L]u(x),λ:=Lum and t := x / L t := x / L t:=x//Lt:=x / Lt:=x/L.
As the differential equation in (8) is autonomous, usual manipulations and boundary condition in 0 imply:
(9) t = 2 λ ∫ 0 θ d s 1 − s 4 , t ∈ ( 0 , 1 2 ) , θ ∈ ( 0 , 1 ) (9) t = 2 λ ∫ 0 θ   d s 1 − s 4 , t ∈ 0 , 1 2 , θ ∈ ( 0 , 1 ) {:(9)t=(sqrt2)/(lambda)int_(0)^(theta)(ds)/(sqrt(1-s^(4)))","quad t in(0,(1)/(2))","theta in(0","1):}\begin{equation*} t=\frac{\sqrt{2}}{\lambda} \int_{0}^{\theta} \frac{\mathrm{d} s}{\sqrt{1-s^{4}}}, \quad t \in\left(0, \frac{1}{2}\right), \theta \in(0,1) \tag{9} \end{equation*}(9)t=2λ∫0θds1−s4,t∈(0,12),θ∈(0,1)
Here we used tacitly the symmetry of the positive solution about the middle of the interval. To approximate the integral in (9) we use the extension of the mean value theorem for integral suggested in [17]. Thus, there exists θ t θ t theta_(t)\theta_{t}θt, such that
(10) t = 2 λ θ 1 − θ t 4 (10) t = 2 λ θ 1 − θ t 4 {:(10)t=(sqrt2)/(lambda)(theta)/(sqrt(1-theta_(t)^(4))):}\begin{equation*} t=\frac{\sqrt{2}}{\lambda} \frac{\theta}{\sqrt{1-\theta_{t}^{4}}} \tag{10} \end{equation*}(10)t=2λθ1−θt4
where 0 < θ t < θ , lim θ → 0 θ t / θ = 1 / ( r + 1 ) 1 / r 0 < θ t < θ , lim θ → 0   θ t / θ = 1 / ( r + 1 ) 1 / r 0 < theta_(t) < theta,lim_(theta rarr0)theta_(t)//theta=1//(r+1)^(1//r)0<\theta_{t}<\theta, \lim _{\theta \rightarrow 0} \theta_{t} / \theta=1 /(r+1)^{1 / r}0<θt<θ,limθ→0θt/θ=1/(r+1)1/r, for − 1 < r < 0 − 1 < r < 0 -1 < r < 0-1<r<0−1<r<0 for which lim s → 0 1 / ( s r 1 − s 4 ) = 0 lim s → 0   1 / s r 1 − s 4 = 0 lim_(s rarr0)1//(s^(r)sqrt(1-s^(4)))=0\lim _{s \rightarrow 0} 1 /\left(s^{r} \sqrt{1-s^{4}}\right)=0lims→01/(sr1−s4)=0. and observe that θ t = O ( θ ) θ t = O ( θ ) theta_(t)=O(theta)\theta_{t}=\mathcal{O}(\theta)θt=O(θ), as θ → 0 θ → 0 theta rarr0\theta \rightarrow 0θ→0 and 0 < R < 1 0 < R < 1 0 < R < 10< R<10<R<1. If, motivated by the above asymptotics, we substitute R θ R θ R thetaR \thetaRθ for θ t θ t theta_(t)\theta_{t}θt in (10), we obtain the following approximation of the positive (and negative) solution of (8):
(11) θ ( t ) = ± t 1 λ 2 + 1 λ 4 + t 4 R 4 (11) θ ( t ) = ± t 1 λ 2 + 1 λ 4 + t 4 R 4 {:(11)theta(t)=(+-t)/(sqrt((1)/(lambda^(2))+sqrt((1)/(lambda^(4))+t^(4)R^(4)))):}\begin{equation*} \theta(t)=\frac{ \pm t}{\sqrt{\frac{1}{\lambda^{2}}+\sqrt{\frac{1}{\lambda^{4}}+t^{4} R^{4}}}} \tag{11} \end{equation*}(11)θ(t)=±t1λ2+1λ4+t4R4
in an arbitrary small neighborhood of t = 0 t = 0 t=0t=0t=0.
Notice that the representation (11) retains all particularities of the exact solution of (8): the symmetry, the smoothness and the asymptotics properties near the boundary points. This entitles us to assimilate the behavior of θ ′ ( 0 ) θ ′ ( 0 ) theta^(')(0)\theta^{\prime}(0)θ′(0) obtained from (11) with that corresponding to exact solution. In fact we get
(12) L u m = θ ′ ( 0 ) 2 (12) L u m = θ ′ ( 0 ) 2 {:(12)Lu_(m)=theta^(')(0)sqrt2:}\begin{equation*} L u_{m}=\theta^{\prime}(0) \sqrt{2} \tag{12} \end{equation*}(12)Lum=θ′(0)2
But from [6], as an intermediate result, θ ′ ( 0 ) θ ′ ( 0 ) theta^(')(0)\theta^{\prime}(0)θ′(0) does not depend on L L LLL. Thus (12) means our bifurcation relationship. Thus, in the lack of a coherent strategy to study bifurcation from infinity we have considered the above ad-hoc method.
Remark 1. There do exist some literature gathered around the subject of existence, uniqueness, bifurcation and stability of solutions of (1). Thus, in his work [12], Matkowsky considers the stability of the null solution by asymptotic methods. He imposes on the reaction term f ( u ) f ( u ) f(u)f(u)f(u) some technical conditions and avoids computational difficulties when takes into account only this solution. In the works of Ambrosetti and Rabinowitz [1], Aronson and Weinberger [2], Crandal and Rabinowitz [3],4], Keller and Cohen [9], Simpson and Cohen [18, Sattinger [14, [15], [16] and Turner [19], to quote but a few, the authors take the advantage of a linear term in the full reaction term f ( u ) f ( u ) f(u)f(u)f(u). Consequently, these analyses are useless for our purposes.

3. THE VARIATIONAL CHARACTERIZATION OF THE POSITIVE SOLUTION

As is apparent from [6], some solvers contributed to a considerable insight into the nature of the solutions of (2). They used exclusively direct manipulation of the equation. With respect to the positive solution, a more penetrating discussion requires to study of an appropriate variational problem whose solution must satisfy (2) and (3).
The obvious choice, namely, the variational problem
(V1) find u ∈ H 0 1 ( 0 , L ) , ∫ 0 L ( u ′ v ′ − u 3 v ) d x = 0 , ∀ v ∈ H 0 1 ( 0 , L ) (V1)  find  u ∈ H 0 1 ( 0 , L ) , ∫ 0 L   u ′ v ′ − u 3 v d x = 0 , ∀ v ∈ H 0 1 ( 0 , L ) {:(V1)" find "u inH_(0)^(1)(0","L)","quadint_(0)^(L)(u^(')v^(')-u^(3)v)dx=0","quad AA v inH_(0)^(1)(0","L):}\begin{equation*} \text { find } u \in H_{0}^{1}(0, L), \quad \int_{0}^{L}\left(u^{\prime} v^{\prime}-u^{3} v\right) \mathrm{d} x=0, \quad \forall v \in H_{0}^{1}(0, L) \tag{V1} \end{equation*}(V1) find u∈H01(0,L),∫0L(u′v′−u3v)dx=0,∀v∈H01(0,L)
of which (2) represents the Euler's equation, proves to be utterly unsuitable for our purposes. The main reason is that, the family of extremals of this problem which pass through a point ( 0 , 0 ) ( 0 , 0 ) (0,0)(0,0)(0,0) for definiteness, do not form a field, and consequently, the classical sufficient criteria for the existence of extrema, due to Jacobi, become unapplicable (see [5, ch. 8]). We have to notice at this point that we have failed in our attempt to show that the functional
J 1 ( v ) := ∫ 0 L ( ( v ′ ) 2 − v 4 ) d x , J 1 : H 0 1 ( 0 , L ) → R J 1 ( v ) := ∫ 0 L   v ′ 2 − v 4 d x , J 1 : H 0 1 ( 0 , L ) → R J_(1)(v):=int_(0)^(L)((v^('))^(2)-v^(4))dx,quadJ_(1):H_(0)^(1)(0,L)rarrRJ_{1}(v):=\int_{0}^{L}\left(\left(v^{\prime}\right)^{2}-v^{4}\right) \mathrm{d} x, \quad J_{1}: H_{0}^{1}(0, L) \rightarrow \mathbb{R}J1(v):=∫0L((v′)2−v4)dx,J1:H01(0,L)→R
has a positive minimum.
Instead, using an idea from [13], we introduce the following generalized Rayleigh quotient (functional)
(13) J : H 0 1 ( 0 , L ) → R , J ( v ) := ( ∫ 0 L ( v ′ ) 2 d x ) 2 / ∫ 0 L v 4 d x (13) J : H 0 1 ( 0 , L ) → R , J ( v ) := ∫ 0 L   v ′ 2 d x 2 / ∫ 0 L   v 4 d x {:(13)J:H_(0)^(1)(0","L)rarrR","quad J(v):=(int_(0)^(L)(v^('))^(2)(d)x)^(2)//int_(0)^(L)v^(4)dx:}\begin{equation*} J: H_{0}^{1}(0, L) \rightarrow \mathbb{R}, \quad J(v):=\left(\int_{0}^{L}\left(v^{\prime}\right)^{2} \mathrm{~d} x\right)^{2} / \int_{0}^{L} v^{4} \mathrm{~d} x \tag{13} \end{equation*}(13)J:H01(0,L)→R,J(v):=(∫0L(v′)2 dx)2/∫0Lv4 dx
and prove the following result.
Lemma 1. The functional J ( v ) J ( v ) J(v)J(v)J(v) defined in (13) has a positive lower bound on H 0 1 ( 0 , L ) H 0 1 ( 0 , L ) H_(0)^(1)(0,L)H_{0}^{1}(0, L)H01(0,L).
Proof. Recall that for any y ∈ H 0 1 ( 0 , L ) y ∈ H 0 1 ( 0 , L ) y inH_(0)^(1)(0,L)y \in H_{0}^{1}(0, L)y∈H01(0,L) the Poincaré's inequality affirms that
π 2 ∫ 0 L y 2 d x ≤ L 2 ∫ 0 L ( y ′ ) 2 d x π 2 ∫ 0 L   y 2 d x ≤ L 2 ∫ 0 L   y ′ 2 d x pi^(2)int_(0)^(L)y^(2)dx <= L^(2)int_(0)^(L)(y^('))^(2)dx\pi^{2} \int_{0}^{L} y^{2} \mathrm{~d} x \leq L^{2} \int_{0}^{L}\left(y^{\prime}\right)^{2} \mathrm{~d} xπ2∫0Ly2 dx≤L2∫0L(y′)2 dx
For y ( x ) = v 2 ( x ) y ( x ) = v 2 ( x ) y(x)=v^(2)(x)y(x)=v^{2}(x)y(x)=v2(x), this implies another useful inequality
(14) π 2 ∫ 0 L v 4 d x ≤ ( 2 L ) 2 ∫ 0 L ( v v ′ ) 2 d x (14) π 2 ∫ 0 L   v 4 d x ≤ ( 2 L ) 2 ∫ 0 L   v v ′ 2 d x {:(14)pi^(2)int_(0)^(L)v^(4)dx <= (2L)^(2)int_(0)^(L)(vv^('))^(2)dx:}\begin{equation*} \pi^{2} \int_{0}^{L} v^{4} \mathrm{~d} x \leq(2 L)^{2} \int_{0}^{L}\left(v v^{\prime}\right)^{2} \mathrm{~d} x \tag{14} \end{equation*}(14)π2∫0Lv4 dx≤(2L)2∫0L(vv′)2 dx
Cauchy-Schwarz inequality, the left hand side boundary condition, and the inequality (14), enable one to write successively:
v 2 ( x ) = ( ∫ 0 x v ′ ( t ) d t ) 2 ≤ x ∫ 0 x ( v ′ ) 2 d t < L ∫ 0 L ( v ′ ) 2 d t ( π 2 L ) 2 ∫ 0 L v 4 d x ≤ L ∫ 0 L ( v ′ ) 2 ( ∫ 0 L ( v ′ ) 2 d t ) d x = L ( ∫ 0 L ( v ′ ) 2 d x ) 2 v 2 ( x ) = ∫ 0 x   v ′ ( t ) d t 2 ≤ x ∫ 0 x   v ′ 2 d t < L ∫ 0 L   v ′ 2 d t π 2 L 2 ∫ 0 L   v 4 d x ≤ L ∫ 0 L   v ′ 2 ∫ 0 L   v ′ 2 d t d x = L ∫ 0 L   v ′ 2 d x 2 {:[v^(2)(x)=(int_(0)^(x)v^(')(t)dt)^(2) <= xint_(0)^(x)(v^('))^(2)dt < Lint_(0)^(L)(v^('))^(2)dt],[((pi)/(2L))^(2)int_(0)^(L)v^(4)dx <= Lint_(0)^(L)(v^('))^(2)(int_(0)^(L)(v^('))^(2)(d)t)dx=L(int_(0)^(L)(v^('))^(2)(d)x)^(2)]:}\begin{gathered} v^{2}(x)=\left(\int_{0}^{x} v^{\prime}(t) \mathrm{d} t\right)^{2} \leq x \int_{0}^{x}\left(v^{\prime}\right)^{2} \mathrm{~d} t<L \int_{0}^{L}\left(v^{\prime}\right)^{2} \mathrm{~d} t \\ \left(\frac{\pi}{2 L}\right)^{2} \int_{0}^{L} v^{4} \mathrm{~d} x \leq L \int_{0}^{L}\left(v^{\prime}\right)^{2}\left(\int_{0}^{L}\left(v^{\prime}\right)^{2} \mathrm{~d} t\right) \mathrm{d} x=L\left(\int_{0}^{L}\left(v^{\prime}\right)^{2} \mathrm{~d} x\right)^{2} \end{gathered}v2(x)=(∫0xv′(t)dt)2≤x∫0x(v′)2 dt<L∫0L(v′)2 dt(π2L)2∫0Lv4 dx≤L∫0L(v′)2(∫0L(v′)2 dt)dx=L(∫0L(v′)2 dx)2
This means that
J ( v ) ≥ ( π 2 ) 2 1 L 3 , ∀ v ∈ H 0 1 ( 0 , L ) J ( v ) ≥ π 2 2 1 L 3 , ∀ v ∈ H 0 1 ( 0 , L ) J(v) >= ((pi)/(2))^(2)(1)/(L^(3)),quad AA v inH_(0)^(1)(0,L)J(v) \geq\left(\frac{\pi}{2}\right)^{2} \frac{1}{L^{3}}, \quad \forall v \in H_{0}^{1}(0, L)J(v)≥(π2)21L3,∀v∈H01(0,L)
Our main result is concentrated in the following theorem.
Theorem 2. Given condition (3), a function u ( x ) u ( x ) u(x)u(x)u(x) that extremizes the functional J ( ⋅ ) J ( ⋅ ) J(*)J(\cdot)J(⋅), defined by (13), satisfies-for an appropriate choice of multiplier μ μ mu\muμ-the Euler's equation corresponding to the functional
J ∗ ∗ : H 0 1 ( 0 , L ) → R , J ∗ ∗ ( v ) := J ( v ) + μ ∫ 0 L v d x J ∗ ∗ : H 0 1 ( 0 , L ) → R , J ∗ ∗ ( v ) := J ( v ) + μ ∫ 0 L   v d x J^(****):H_(0)^(1)(0,L)rarrR,quadJ^(****)(v):=J(v)+muint_(0)^(L)vdxJ^{* *}: H_{0}^{1}(0, L) \rightarrow \mathbb{R}, \quad J^{* *}(v):=J(v)+\mu \int_{0}^{L} v \mathrm{~d} xJ∗∗:H01(0,L)→R,J∗∗(v):=J(v)+μ∫0Lv dx
Thus, the function u ( x ) u ( x ) u(x)u(x)u(x) and the multiplier μ μ mu\muμ can be determined from the system of equations
(15) { ∫ 0 L ( u ′ v ′ − u 3 v ) d x = μ 4 ∫ 0 L v d x , ∀ v ∈ H 0 1 ( 0 , L ) ∫ 0 L u d x = π 2 (15) ∫ 0 L   u ′ v ′ − u 3 v d x = μ 4 ∫ 0 L   v d x , ∀ v ∈ H 0 1 ( 0 , L ) ∫ 0 L   u d x = π 2 {:(15){[int_(0)^(L)(u^(')v^(')-u^(3)v)dx=(mu)/(4)int_(0)^(L)vdx","quad AA v inH_(0)^(1)(0","L)],[int_(0)^(L)udx=(pi)/(sqrt2)]:}:}\left\{\begin{array}{l} \int_{0}^{L}\left(u^{\prime} v^{\prime}-u^{3} v\right) \mathrm{d} x=\frac{\mu}{4} \int_{0}^{L} v \mathrm{~d} x, \quad \forall v \in H_{0}^{1}(0, L) \tag{15}\\ \int_{0}^{L} u \mathrm{~d} x=\frac{\pi}{\sqrt{2}} \end{array}\right.(15){∫0L(u′v′−u3v)dx=μ4∫0Lv dx,∀v∈H01(0,L)∫0Lu dx=π2
Proof. First, we observe that a function u u uuu that minimizes J ( ⋅ ) J ( ⋅ ) J(*)J(\cdot)J(⋅) satisfies the necessary condition of extremum
(16) ∫ 0 L ( ( u ′ ) 2 − u 4 ) d x = 0 (16) ∫ 0 L   u ′ 2 − u 4 d x = 0 {:(16)int_(0)^(L)((u^('))^(2)-u^(4))dx=0:}\begin{equation*} \int_{0}^{L}\left(\left(u^{\prime}\right)^{2}-u^{4}\right) \mathrm{d} x=0 \tag{16} \end{equation*}(16)∫0L((u′)2−u4)dx=0
In order to handle the condition (3) we have to introduce a new dependent variable z ( x ) z ( x ) z(x)z(x)z(x) by z ( x ) := ∫ 0 x u ( s ) d s , z ( 0 ) = 0 , z ( L ) = π 2 z ( x ) := ∫ 0 x   u ( s ) d s , z ( 0 ) = 0 , z ( L ) = π 2 z(x):=int_(0)^(x)u(s)ds,z(0)=0,z(L)=(pi)/(sqrt2)z(x):=\int_{0}^{x} u(s) \mathrm{d} s, z(0)=0, z(L)=\frac{\pi}{\sqrt{2}}z(x):=∫0xu(s)ds,z(0)=0,z(L)=π2 and z ′ ( x ) = u ( x ) z ′ ( x ) = u ( x ) z^(')(x)=u(x)z^{\prime}(x)=u(x)z′(x)=u(x). With this, consider the functional
J ∗ : H 0 1 ( 0 , L ) × H 1 ( 0 , L ) → R , J ∗ ( v ) := J ( v ) + ∫ 0 L μ ( x ) ( v ( x ) − z ′ ( x ) ) d x J ∗ : H 0 1 ( 0 , L ) × H 1 ( 0 , L ) → R , J ∗ ( v ) := J ( v ) + ∫ 0 L   μ ( x ) v ( x ) − z ′ ( x ) d x J^(**):H_(0)^(1)(0,L)xxH^(1)(0,L)rarrR,quadJ^(**)(v):=J(v)+int_(0)^(L)mu(x)(v(x)-z^(')(x))dxJ^{*}: H_{0}^{1}(0, L) \times H^{1}(0, L) \rightarrow \mathbb{R}, \quad J^{*}(v):=J(v)+\int_{0}^{L} \mu(x)\left(v(x)-z^{\prime}(x)\right) \mathrm{d} xJ∗:H01(0,L)×H1(0,L)→R,J∗(v):=J(v)+∫0Lμ(x)(v(x)−z′(x))dx
for a sufficiently regular function μ ( x ) μ ( x ) mu(x)\mu(x)μ(x).
The necessary conditions of extremum for J ∗ J ∗ J^(**)J^{*}J∗ are
d J ∗ ( u + ε v , z ) d ε | ε = 0 = 0 and d J ∗ ( u , z + η y ) d η | η = 0 = 0 . d J ∗ ( u + ε v , z ) d ε ε = 0 = 0  and  d J ∗ ( u , z + η y ) d η η = 0 = 0 . (dJ^(**)(u+epsi v,z))/(depsi)|_(epsi=0)=0quad" and " quad(dJ^(**)(u,z+eta y))/(deta)|_(eta=0)=0.\left.\frac{\mathrm{d} J^{*}(u+\varepsilon v, z)}{\mathrm{d} \varepsilon}\right|_{\varepsilon=0}=0 \quad \text { and }\left.\quad \frac{\mathrm{d} J^{*}(u, z+\eta y)}{\mathrm{d} \eta}\right|_{\eta=0}=0 .dJ∗(u+εv,z)dε|ε=0=0 and dJ∗(u,z+ηy)dη|η=0=0.
The first one, in combination with (16), implies
(17) ∫ 0 L ( u ′ v ′ − u 3 v ) d x = 1 4 ∫ 0 L μ ( x ) v ( x ) d x , ∀ v ∈ H 0 1 ( 0 , L ) (17) ∫ 0 L   u ′ v ′ − u 3 v d x = 1 4 ∫ 0 L   μ ( x ) v ( x ) d x , ∀ v ∈ H 0 1 ( 0 , L ) {:(17)int_(0)^(L)(u^(')v^(')-u^(3)v)dx=(1)/(4)int_(0)^(L)mu(x)v(x)dx","quad AA v inH_(0)^(1)(0","L):}\begin{equation*} \int_{0}^{L}\left(u^{\prime} v^{\prime}-u^{3} v\right) \mathrm{d} x=\frac{1}{4} \int_{0}^{L} \mu(x) v(x) \mathrm{d} x, \quad \forall v \in H_{0}^{1}(0, L) \tag{17} \end{equation*}(17)∫0L(u′v′−u3v)dx=14∫0Lμ(x)v(x)dx,∀v∈H01(0,L)
and the second one leads to
∫ 0 L μ ( x ) y ′ ( x ) d x = 0 , ∀ y ∈ H 1 ( 0 , L ) ∫ 0 L   μ ( x ) y ′ ( x ) d x = 0 , ∀ y ∈ H 1 ( 0 , L ) int_(0)^(L)mu(x)y^(')(x)dx=0,quad AA y inH^(1)(0,L)\int_{0}^{L} \mu(x) y^{\prime}(x) \mathrm{d} x=0, \quad \forall y \in H^{1}(0, L)∫0Lμ(x)y′(x)dx=0,∀y∈H1(0,L)
For sufficiently smooth y ∈ H 0 1 ( 0 , L ) y ∈ H 0 1 ( 0 , L ) y inH_(0)^(1)(0,L)y \in H_{0}^{1}(0, L)y∈H01(0,L), such that the fundamental lemma of variational calculus apply, the last integral equality ensures that μ ′ ( x ) = 0 μ ′ ( x ) = 0 mu^(')(x)=0\mu^{\prime}(x)=0μ′(x)=0. Consequently, the Lagrange's multiplier μ μ mu\muμ reduces to a real parameter.
Thus, (17) and (3) imply (15). More than that,
d J ∗ ( u + ε v , z ) d ε | ε = 0 = 0 d J ∗ ( u + ε v , z ) d ε ε = 0 = 0 (dJ^(**)(u+epsi v,z))/(depsi)|_(epsi=0)=0\left.\frac{\mathrm{d} J^{*}(u+\varepsilon v, z)}{\mathrm{d} \varepsilon}\right|_{\varepsilon=0}=0dJ∗(u+εv,z)dε|ε=0=0
means the first equation in the system (15). This completes the proof.
We have to underline that all problems of extremum we have encountered are meant on the whole Sobolev space H 0 1 ( 0 , L ) H 0 1 ( 0 , L ) H_(0)^(1)(0,L)H_{0}^{1}(0, L)H01(0,L) equipped with the usual norm.
The positive solution u ( x ) u ( x ) u(x)u(x)u(x) of the system (15) is a weak approximation to the positive solution u ¯ ( x ) u ¯ ( x ) bar(u)(x)\bar{u}(x)u¯(x) of (2) and (3). To find numerically this approximation we discretize the equations in (15) using classical f.e.m.
The positive solution is approximated by
u h ( x ) = ∑ k = 1 N c k φ k ( x ) , u h ( x ) = ∑ k = 1 N   c k φ k ( x ) , u_(h)(x)=sum_(k=1)^(N)c_(k)varphi_(k)(x),u_{h}(x)=\sum_{k=1}^{N} c_{k} \varphi_{k}(x),uh(x)=∑k=1Nckφk(x),
where the piecewise linear function φ k ( x ) φ k ( x ) varphi_(k)(x)\varphi_{k}(x)φk(x) satisfies φ k ( x j ) = 0 φ k x j = 0 varphi_(k)(x_(j))=0\varphi_{k}\left(x_{j}\right)=0φk(xj)=0 for k ≠ j k ≠ j k!=jk \neq jk≠j and φ k ( x k ) = 1 , x k = k h φ k x k = 1 , x k = k h varphi_(k)(x_(k))=1,x_(k)=kh\varphi_{k}\left(x_{k}\right)=1, x_{k}=k hφk(xk)=1,xk=kh for k = 0 , 1 , … , N + 1 k = 0 , 1 , … , N + 1 k=0,1,dots,N+1k=0,1, \ldots, N+1k=0,1,…,N+1 and h = L / ( N + 1 ) h = L / ( N + 1 ) h=L//(N+1)h=L /(N+1)h=L/(N+1).
For each N , u h N , u h N,u_(h)N, u_{h}N,uh must be a solution of the discrete analogous of (15),
(18) { ∫ 0 L ( u h ′ φ k ′ − u h 3 φ k ) d x = μ 4 ∫ 0 L φ k d x , for k = 1 , … , N ∫ 0 L u h d x = π 2 (18) ∫ 0 L   u h ′ φ k ′ − u h 3 φ k d x = μ 4 ∫ 0 L   φ k d x ,  for  k = 1 , … , N ∫ 0 L   u h d x = π 2 {:(18){[int_(0)^(L)(u_(h)^(')varphi_(k)^(')-u_(h)^(3)varphi_(k))dx=(mu)/(4)int_(0)^(L)varphi_(k)dx","quad" for "k=1","dots","N],[int_(0)^(L)u_(h)dx=(pi)/(sqrt2)]:}:}\left\{\begin{array}{l} \int_{0}^{L}\left(u_{h}^{\prime} \varphi_{k}^{\prime}-u_{h}^{3} \varphi_{k}\right) \mathrm{d} x=\frac{\mu}{4} \int_{0}^{L} \varphi_{k} \mathrm{~d} x, \quad \text { for } k=1, \ldots, N \tag{18}\\ \int_{0}^{L} u_{h} \mathrm{~d} x=\frac{\pi}{\sqrt{2}} \end{array}\right.(18){∫0L(uh′φk′−uh3φk)dx=μ4∫0Lφk dx, for k=1,…,N∫0Luh dx=π2
which becomes a nonlinear algebraic system, F ( c ) = 0 F ( c ) = 0 F(c)=0F(c)=0F(c)=0, for the unknowns c = ( c 1 , … , c N , μ ) c = c 1 , … , c N , μ c=(c_(1),dots,c_(N),mu)c=\left(c_{1}, \ldots, c_{N}, \mu\right)c=(c1,…,cN,μ). Here F = ( F 1 , … , F N , F N + 1 ) F = F 1 , … , F N , F N + 1 F=(F_(1),dots,F_(N),F_(N+1))F=\left(F_{1}, \ldots, F_{N}, F_{N+1}\right)F=(F1,…,FN,FN+1) and
(19) F n ( c ) = c n − 1 h − 2 c n h + c n + 1 h + c n − 1 3 h 20 + c n − 1 2 c n h 10 + 3 c n − 1 c n 2 h 20 + 2 c n 3 h 5 + 3 c n 2 c n + 1 h 20 + c n c n + 1 2 h 10 + c n + 1 3 h 20 + μ 4 h (19) F n ( c ) = c n − 1 h − 2 c n h + c n + 1 h + c n − 1 3 h 20 + c n − 1 2 c n h 10 + 3 c n − 1 c n 2 h 20 + 2 c n 3 h 5 + 3 c n 2 c n + 1 h 20 + c n c n + 1 2 h 10 + c n + 1 3 h 20 + μ 4 h {:[(19)F_(n)(c)=(c_(n-1))/(h)-(2c_(n))/(h)+(c_(n+1))/(h)+(c_(n-1)^(3)h)/(20)+(c_(n-1)^(2)c_(n)h)/(10)+(3c_(n-1)c_(n)^(2)h)/(20)],[+(2c_(n)^(3)h)/(5)+(3c_(n)^(2)c_(n+1)h)/(20)+(c_(n)c_(n+1)^(2)h)/(10)+(c_(n+1)^(3)h)/(20)+(mu)/(4)h]:}\begin{align*} F_{n}(c)= & \frac{c_{n-1}}{h}-\frac{2 c_{n}}{h}+\frac{c_{n+1}}{h}+\frac{c_{n-1}^{3} h}{20}+\frac{c_{n-1}^{2} c_{n} h}{10}+\frac{3 c_{n-1} c_{n}^{2} h}{20} \tag{19}\\ & +\frac{2 c_{n}^{3} h}{5}+\frac{3 c_{n}^{2} c_{n+1} h}{20}+\frac{c_{n} c_{n+1}^{2} h}{10}+\frac{c_{n+1}^{3} h}{20}+\frac{\mu}{4} h \end{align*}(19)Fn(c)=cn−1h−2cnh+cn+1h+cn−13h20+cn−12cnh10+3cn−1cn2h20+2cn3h5+3cn2cn+1h20+cncn+12h10+cn+13h20+μ4h
for n = 1 , … , N n = 1 , … , N n=1,dots,Nn=1, \ldots, Nn=1,…,N, where we put c 0 = c N + 1 = 0 c 0 = c N + 1 = 0 c_(0)=c_(N+1)=0c_{0}=c_{N+1}=0c0=cN+1=0, in order to impose the boundary conditions and
(20) F N + 1 ( c ) = h ∑ n = 1 N c n − π 2 (20) F N + 1 ( c ) = h ∑ n = 1 N   c n − π 2 {:(20)F_(N+1)(c)=hsum_(n=1)^(N)c_(n)-(pi)/(sqrt2):}\begin{equation*} F_{N+1}(c)=h \sum_{n=1}^{N} c_{n}-\frac{\pi}{\sqrt{2}} \tag{20} \end{equation*}(20)FN+1(c)=h∑n=1Ncn−π2
We solve this nonlinear system by Newton's method. Starting with an initial guess of the form c k 0 = x k ( L − x k ) c k 0 = x k L − x k c_(k)^(0)=x_(k)(L-x_(k))c_{k}^{0}=x_{k}\left(L-x_{k}\right)ck0=xk(L−xk), for k = 1 , … , N k = 1 , … , N k=1,dots,Nk=1, \ldots, Nk=1,…,N, and μ = 10 μ = 10 mu=10\mu=10μ=10, Newton's method implies the following sequence of iterations by solving the sequence of linear systems
F ′ ( c α ) ( c α + 1 − c α ) = − F ( c α ) , for α = 0 , 1 , … F ′ c α c α + 1 − c α = − F c α ,  for  α = 0 , 1 , … F^(')(c^(alpha))(c^(alpha+1)-c^(alpha))=-F(c^(alpha))," for "alpha=0,1,dotsF^{\prime}\left(c^{\alpha}\right)\left(c^{\alpha+1}-c^{\alpha}\right)=-F\left(c^{\alpha}\right), \text { for } \alpha=0,1, \ldotsF′(cα)(cα+1−cα)=−F(cα), for α=0,1,…
This means that, for every α α alpha\alphaα, we have to solve a linear algebraic system, until for a given ε , ‖ c α + 1 − c α ‖ < ε ε , c α + 1 − c α < ε epsi,||c^(alpha+1)-c^(alpha)|| < epsi\varepsilon,\left\|c^{\alpha+1}-c^{\alpha}\right\|<\varepsilonε,‖cα+1−cα‖<ε. In spite of that, the method is not so expensive because of the sparsity of the Jacobian F ′ ( c α ) = ( ∂ F i ( c α ) ∂ c k ) i , k = 1 , … , N + 1 F ′ c α = ∂ F i c α ∂ c k i , k = 1 , … , N + 1 F^(')(c^(alpha))=((delF_(i)(c^(alpha)))/(delc_(k)))_(i,k=1,dots,N+1)F^{\prime}\left(c^{\alpha}\right)=\left(\frac{\partial F_{i}\left(c^{\alpha}\right)}{\partial c_{k}}\right)_{i, k=1, \ldots, N+1}F′(cα)=(∂Fi(cα)∂ck)i,k=1,…,N+1. We remark that whenever the f.e.m. is applied directly to the problem (2), i.e. the integral condition (3) is ignored, the attracting basin of the positive solution becomes very narrow. Consequently, Newton's method converges to the positive solution only for initial approximation u h 0 ( x ) u h 0 ( x ) u_(h)^(0)(x)u_{h}^{0}(x)uh0(x) sufficiently close to this solution. That is why we have incorporated the integral condition (3) in the above algorithm. The numerical experiments with this (most of them
reported in [7] confirm the fact that the attracting basin of the positive solution becomes larger. This underlines the importance of the integral condition (3) and furnishes an effective algorithm for the numerical computation of the positive solution to problem (2).

REFERENCES

[1] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, pp. 349-381, 1973.
[2] Aronson, D. G. and Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Lecture Notes in Math., 446, Springer-Verlag, 1975.
[3] Crandal, M. G. and Rabinowitz, P. H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19, pp. 1083-1102, 1970.
[4] Crandal, M. G. and Rabinowitz, P. H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 52, pp. 161-180, 1973.
[5] Elsgolts, L., Differential Equations and the Calculus of Variations, Mir Publishers, Moscow, 1980.
[6] Gheorghiu, C. I., Solution to problem 97-8 by Ph. Korman, SIAM Review, 39, 1997, SIAM Review, 40, no. 2, pp. 382-385, 1998.
[7] Gheorghiu, C. I. and Trif, D., On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem, Studia UBB, Mathematica, XLV, pp. 29-37, 2000.
[8] Grindrod, P., The Theory and Applications of Reaction-Diffusion Equations; Patern and Waves, Second Edition, Clarendon Press, Oxford, 1996.
[9] Keller, H. B. and Cohen, D. S., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, pp. 1361-1376, 1967.
[10] Korman, Ph., Average temperature in a reaction-diffusion process, Problem 97-8, SIAM Review, 39, p. 318, 1997.
[11] Laetsch, Th., The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20, pp. 1-13, 1970.
[12] Matkowsky, B. J., A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc., 76, pp. 620-625, 1970.
[13] Moore, R. A. and Nehari, Z., Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93, pp. 30-52, 1959.
[14] Sattinger, D. H., Topics in Stability and Bifurcation Theory, Springer-Verlag, 1973.
[15] Sattinger, D. H., Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rat. Mech. Anal., 43, pp. 155-165, 1970.
[16] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, pp. 979-1000, 1972.
[17] Schwind, W. J., Ji, J. and Koditschek, D. E., A physically motivated further note on the mean value theorem for integrals, Amer. Math. Monthly, 126, pp. 559-564, 1999.
[18] Simpson, B. R. and Cohen, D. S., Positive solutions of nonlinear elliptic eigenvalue problems, J. Math. Mech., 19, pp. 895-910, 1970.
[19] Turner, R. E. I., Nonlinear eigenvalue problems with nonlocal operators, Comm. Pure Appl. Math., 23, pp. 963-972, 1970.
[20] Wollkind, D. J., Monoranjan, V. S. and Zhang, L., Weakly nonlinear stability analysis of prototype reaction-diffusion model equations, SIAM Review, 36, no. 2, pp. 176214, 1994.
Received by the editors: June 29, 2000.
  1. *"T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: ghcalin@ictp.acad.ro.
    † "Babeş-Bolyai" University, Faculty of Mathematics and Computer Science, Str. M. Kogǎlniceanu 1, 3400 Cluj-Napoca, Romania, e-mail: dtrif@math.ubbcluj.ro.
2002

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