S-shaped bifurcation curves of nonlinear elliptic boundary value problems. (English) Zbl 0544.35015

Wiebers, Henning

Math. Ann. 270, 555-570 (1985).

The paper is concerned with the study of the elliptic boundary value problem \((1)_{\lambda}\quad Lu=\lambda f(u)\) in \(\Omega\), \(Bu=0\) on \(\partial \Omega\) where \(\lambda\) is a nonnegative parameter and (L,B) satisfies the strong maximum principle. The nonlinearity \(f:{\mathbb{R}}_+\to {\mathbb{R}}_+\) is assumed to satisfy \(f(0)>0\), \(f'(x)>0\) for all \(x\geq 0\) and \(f(x)-f'(x)x\geq \theta>0\) for x sufficiently large. For example \(f(x)=\exp(x/(1+\epsilon x))\) with \(\epsilon>0\) which arises in the theory of combustion.
It is shown that the solution set \(\Sigma:=\{(\lambda,u)\in {\mathbb{R}}_+\times C^ 2({\bar \Omega})|\) (\(\lambda\),u) satisfies \((1)_{\lambda}\}\) of \((1)_{\lambda}\) has the same shape as the solution set \(\Sigma_ s:=\{(\nu,x)\in {\mathbb{R}}_+\times {\mathbb{R}}_ x| \nu f(x)=x\}\) of the scalar equation \((2)_{\nu} \nu f(x)=x\), the so-called Semenov approximation. More precisely, if \((2)_{\nu}\) is uniquely solvable for large values of \(\nu\), the same is true for \((1)_{\lambda}\). \((2)_{\nu}\) and \((1)_{\lambda}\) are uniquely solvable for small parameter values. Moreover, either the solution of \((1)_{\lambda}\) is unique for each \(\lambda\) or \(\Sigma\) is ”S-shaped”, i.e. \(\Sigma\) contains a subcontinuum which looks like an ”S”. If the ”S- structure” of \(\Sigma_ s\) is strong enough, it carries over to \(\Sigma\).

MSC:

35B32	Bifurcations in context of PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

S-shaped bifurcation curves; combustion; Semenov approximation

References:

[1]	Amann, H.: Multiple positive fixed points of asymptotically linear maps. J. Funct. Anal.17, 174-213 (1974) · Zbl 0287.47037
[2]	Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev.18, 620-709 (1976) · Zbl 0345.47044
[3]	Aris, R.: The mathematical theory of siffusion and reaction engeneering 1. Oxford: Claredon Press 1975
[4]	Boddington, T., Gray, P.F.R.S., Robinson, C.: Thermal explosions and the disappearance of criticality at small activation energies: exact results for the slab. Proc. Roy. Soc. London Ser. A368, 441-461 (1979)
[5]	Boddington, T., Gray, P.F.R.S., Wake, G.C.: Criteria for thermal explosions with and without reactant consumption. Proc. Roy. Soc. London Ser. A357, 403-422 (1977)
[6]	Brown, K.J., Ibrahim, M.N.A., Shivaji, R.:S-shaped bifurcation curves. Nonlinear Anal.5, 475-486 (1981) · Zbl 0458.35036
[7]	Cohen, D.S., Laetsch, T.W.: Nonlinear boundary value problems suggested by chemical reactor theory. J. Differ. Equations7, 217-226 (1970) · Zbl 0201.43102
[8]	Frank-Kamenetskii, D.A.: Diffusion and heat transfer in chemical kinetics. New York, London: Plenum Press 1969
[9]	Legget, R.W., Williams, L.R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J.28, 673-688 (1979) · Zbl 0421.47033
[10]	Legget, R.W., Williams, L.R.: Multiple fixed point theorems for problems in chemical reactor theory. J. Math. Anal. Appl.69, 180-193 (1979) · Zbl 0416.47026
[11]	Lions, P.L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev.24, 441-467 (1982) · Zbl 0511.35033
[12]	Luss, D.: Sufficient conditions for uniqueness of the steady state solution in distributed parameter systems. Chem. Engrg. Science23, 1249-1255 (1968)
[13]	Parter, S.V.: Solutions of a differential equation in chemical reactor processes. SIAM J. Appl. Math.26, 687-715 (1974) · Zbl 0285.34013
[14]	Spence, A., Werner, B.: Nonsimple turning points and cusps. IMA J. Numer. Anal.2, 413-427 (1982) · Zbl 0539.65043
[15]	Vo?, H.: Positive solutions of superlinear boundary value problems. Proc. Roy. Soc. Edinburgh Sect. A88, 17-24 (1981) · Zbl 0447.34013
[16]	Vo?, H., Werner, B.: Ein Quotienteneinschlie?ungssatz f?r den kritischen Parameter nichtlinearer Randwertaufgaben. ISNM49, 147-158 (1979)
[17]	Wiebers, H.:S-f?rmige Verzweigungsdiagramme bei elliptischen Randwertaufgaben mit Anwendungen auf exotherme Reaktionen. PhD-thesis, Hamburg 1984 · Zbl 0618.35006