## Abstract

A one-dimensional reaction-diffusion problem, with the reaction term of the form \(u^{p}\), \(p>1\) is considered. For \(p=3\), we deduce an independence on the domain (integral) condition and we use this to approximate the positive solution of the problem by a completely discrete scheme (finite elements – finite differences).

We show that actually this positive solution and its numerical approximation are respectively the stationary unstable hyperbolic points of the continuous and discrete dynamical systems. Straightforward direct (based on Newton’s method) and indirect approaches of this positive steady state proved to be unsatisfactory. Thus, we suggest a modified indirect algorithm, which takes into account the integral conditions, and prove its convergence.

## Authors

Călin-Ioan **Gheorghiu
**Tiberiu Popoviciu Institute of Numerical Analysis

Damian **Trif
**Babeș-Bolyai University

## Keywords

semilinear parabolic problem; independence of domain; weak formulation; Galerkin method; discrete problem; Newton method; unstable hyperbolic solution; indirect algorithm; convergence;

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## Paper coordinates

C.I. Gheorghiu, D. Trif, *The numerical approximation to positive solution for some reaction-diffusion problems*, PU.M.A., **11 **(2000) no. 2, 243-253.

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## References

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

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

[3] W. J. Beyn, * On the numerical approximation of phase portraits near stationary points, * SIAM J. Numer, Anal. 24 pp. 1095-1113, 1987

[4] L. Cesari, *Functional analysis and Galerkin’s methodI, *Mich. Math. J., 11, pp. 383-414, 1964

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

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

[7] P. C. Fife, *Stationary patterns for reaction – diffusion equations, *MRC Technical Summary Report # 1709, Wisconsin -Madison, 1976.

[8] C. I. Gheorghiu, *The solution to problem 97-8 by Ph. Korman, SIAM Review, 39 (1997), *SIAM Review, 40, 2, pp. 382-385, 1998.

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

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

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

[12] S. Larsson and J. M. Sanz-Serna, IThe behaviour of finite element soluitons of semilinear parabolic *problems near stationary point*, SIAM J. Numer. Math. Anal. 31, PP. 1000-1018, 1994.

[13] H. A. Levine, *The role of critical exponents in blowup theorems, *SIAM Review, 32, pp. 262-288, 1990.

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

[15] S. Rosenblat, S. H. Davis, *Bifurcation from infinity*, SIAM J. Appl. Math., 37, pp. 1-19, 1979.

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

[17] D. H. Sattinger, *Topics in stability and bifurcation theory, *Springer-Verlag, 1973.

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

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

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