Einschliessungsaussagen bei Systemen semilinearer parabolischer Differentialgleichungen
Applications of Mathematics, Tome 36 (1991) no. 2, pp. 96-122 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Für die Lösungen seminlinearer parabolischer Differentialgleichungen werden Einschliessungsaussagen hergeleitet. Hierbei werden Aussagen zur Stabilität von Lösungen ermittelt. Die Resultate werden am Beispiel der Fitzhugh-Nagumo Gleichungen diskutiert.
Für die Lösungen seminlinearer parabolischer Differentialgleichungen werden Einschliessungsaussagen hergeleitet. Hierbei werden Aussagen zur Stabilität von Lösungen ermittelt. Die Resultate werden am Beispiel der Fitzhugh-Nagumo Gleichungen diskutiert.
DOI : 10.21136/AM.1991.104448
Classification : 35A40, 35B35, 35K15, 35K20, 35K40, 35K60, 92C20, 92C40
Mots-clés : semilinear; asymptotic stability; invariance; biochemical reaction; model for the nerve membrane
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Heinrichs, Wilhelm. Einschliessungsaussagen bei Systemen semilinearer parabolischer Differentialgleichungen. Applications of Mathematics, Tome 36 (1991) no. 2, pp. 96-122. doi: 10.21136/AM.1991.104448

[1] E. Abrahams T. Tsuneto: Time variation of the Ginzberg-Landau order parameter. Phys. Rev. 152 (1968), 416-432. | DOI

[2] N. Alikakos: Remarks on invariance in reaction-diffusion equations. Nonlinear Analysis 5 (1981), 593-614. | DOI | MR | Zbl

[3] H. Amann: Invariant sets and existence theorems for semilinear parabolic and elliptic systems. J. Math. Anal. App. 65 (1978), 432-467. | DOI | MR | Zbl

[4] N. R. Amundson: Nonlinear problems in chemical reactor theory. SIAM-AMS Proc. 8 (1974), 59-84. | Zbl

[5] J. W. Bebernes K. Schmitt: Invariant sets and the Hukuhara-Kneser property for systems of parabolic partial differential equations. Rocky Mountain J. Math. 7 (1977), 575-567. | MR

[6] J. A. Boa: Multiple steady states in a model biochemical reaction. Studies in appl. Math. 54 (1975), 9-15. | DOI | MR | Zbl

[7] R. G. Casten C. J. Holland: Stability properties of solutions to systems of reaction-diffusion equations. SIAM J. Appl. Math. 33 (1977), 353-364. | DOI | MR

[8] E. Conway D. Hoff J. Smoller: Large time behaviour of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 35 (1978), 1-16. | DOI | MR

[9] R. Courant K. O. Friedrichs: Supersonic flow and shock waves. Interscience Publishers, Inc., New York, 1948. | MR

[10] R. Courant D. Hilbert: Methods of mathematical physics. Vol. I. Interscience Publishers, Inc. New York, 1953. | MR

[11] K. N. Chueh C. C. Conley J. A. Smoller: Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26 (1977), 373-392. | DOI | MR

[12] R. Fitzhugh: Impulses and physiological states in theoretical models of nerve membran. Biophys. J. 1 (1961), 445-466. | DOI

[13] G. R. Gavalas: Nonlinear differential equations of chemically reacting systems. Springer, New York, 1968. | MR | Zbl

[14] C. Georgakis R. L. San: On the stability of the steady state in systems of coupled diffusion and reaction. Arch. Rat. Mech. Anal. 52 (1973), 266-296. | DOI | MR

[15] A. L. Hodgkin A. F. Huxley: A quantitative description of membran current and its application to conduction and excitation in nerves. J. Physiol. 117 (1952), 500-544. | DOI

[16] W. E. Kastenberg: On the asymptotic stability of non-linear distributed parameter energy systems. Internat. J. Control 19 (1974), 73 - 79. | DOI | MR

[17] R. Lemmert: Über die Invarianz einer konvexen Menge in bezug auf Systemen gewöhnlichen, parabolischen und elliptischen Differentialgleichungen. Math. Ann. 230 (1977), 43-56. | DOI | MR

[18] R. Lemmert: Über die Invarianz konvexer Mengen eines normierten Raumes in bezug auf elliptische Differentialgleichungen. Comm. Partial Differential Equations 3 (1978), 297-318. | DOI | MR

[19] R. R. Martin, Jr.: Nonlinear perturbations of uncoupled systems of elliptic operators. Math. Ann. 211 (1974), 155-169. | DOI | MR | Zbl

[20] J. Nagumo S. Arimoto S. Yoshizawa: Am active pulse transmission line simulating nerve axon. Proc. IRE 50 (1964), 2061 - 2070.

[21] J. Rauch J. A. Smoller: Qualitative theory of the Fitzhugh-Nagumo equations. Advances in Math. 27 (1978), 12-44. | DOI | MR

[22] R. Redheffer W. Walter: Invariant sets for systems of partial differential equations. I. parabolic equations. Arch. Rat. Mech. Anal. 67 (1978), 41 - 52. | DOI | MR

[23] R. Redheffer W. Walter: Invariant sets for systems of partial differential equations, II. First-order and elliptic equations. Arch. Rat. Mech. Anal. 73 (1980), 19-29. | MR

[24] D. H. Sattinger: Stability of nonlinear parabolic systems. J. Math. Anal. and Appl. 24 (1968), 241-245. | DOI | MR | Zbl

[25] C. Schazfer: Invariant sets and contractions for weakly coupled systems of parabolic differential equations. Rendiconti di Mathematica 13 (1980), 337-357. | MR

[26] K. Schmitt: Boundary value problems for quasilinear second order elliptic equations. Nonlinear Anal. 2 (1978), 263-309. | DOI | MR | Zbl

[27] J. Schröder: Snaps-invariant bounds and more general estimates for vector-valued elliptic-parabolic problems. J. of Diff. Equ. 45 (1982), 431-460. | DOI | MR

[28] J. Schröder: Operator inequalities. Academic Press, New York, 1980. | MR

[29] H. Triebel: Höhere Analysis. VEB Dsutscher Verlag der Wissenschaften, Berlin, 1972. | MR | Zbl

[30] V. S. Vladimirov: Equations of mathematical physics. Marcel Dekker, Inc., New York, 1971. | MR | Zbl

[31] W. Walter: Differential and integral inequalities. Ergebnisse der Mathematik und ihrer Grenzgebiete 55. Springer, Berlin, Heidelberg, New York, 1970. | MR | Zbl

[32] G. N. Watson: A treatise on the theory of Bessel functions. Cambridge at the University Press, 1958. | MR

[33] H. F. Weinberger: Invariants sets for weakly coupled parabolic and elliptic systems. Rendiconti di Mathematica 8, Serie VI (1975), 295-310. | MR

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