×

Invariant sets and existence theorems for semilinear parabolic and elliptic systems. (English) Zbl 0387.35038


MSC:

35K55 Nonlinear parabolic equations
35J60 Nonlinear elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Amann, H., Periodic solutions of semi-linear parabolic equations, (Nonlinear Analysis: A Collection of Papers in Honor of Erich Rothe (1978), Academic Press: Academic Press New York), 1-29 · Zbl 0464.35050
[3] Amann, H., Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150, 281-295 (1976) · Zbl 0331.35026
[4] Bony, J.-M, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénères, Ann. Inst. Fourier, 19, 277-304 (1969) · Zbl 0176.09703
[5] Brezis, H.; Browder, F. E., A general principle on ordered sets in nonlinear functional analysis, Advances in Math., 21, 355-364 (1976) · Zbl 0339.47030
[6] Deimling, K., Ordinary Differential Equations in Banach Spaces, (Lecture Notes in Mathematics, Vol. 596 (1977), Springer Verlag: Springer Verlag Berlin) · Zbl 0555.60036
[7] Deuel, J., Nichtlineare parabolische Randwertprobleme mit Unter- und Oberlösungen, (Diss. No. 5750 (1976), ETH: ETH Zürich)
[8] J. Deuel and P. Hess; J. Deuel and P. Hess · Zbl 0372.35045
[9] Dugundji, J., Topology (1970), Allyn & Bacon: Allyn & Bacon Boston
[10] Dunford, N.; Schwartz, J. T., “Linear Operators,” Part I (1957), Interscience: Interscience New York
[11] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0144.34903
[12] Hess, P., A second order nonlinear elliptic boundary value problem, (Nonlinear Analysis: A Collection of Papers in Honor of Erich Rothe (1978), Academic Press: Academic Press New York), 99-108 · Zbl 0464.35041
[13] Hille, E.; Phillips, R. S., Functional Analysis and Semi-Groups (1957), Amer. Math. Soc: Amer. Math. Soc Providence, R.I · Zbl 0078.10004
[14] Kolesov, J. S., Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc., 21, 114-146 (1970) · Zbl 0226.35040
[15] Krasnosel’skii, M. A.; Zabreiko, P. P.; Pustylnik, E. I.; Sobolevskii, P. E., Integral Operators in Spaces of Summable Functions (1976), Noordhoff: Noordhoff Leyden · Zbl 0312.47041
[16] Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and Quasi-linear Equations of Parabolic Type, (Math. Monographs, Vol. 23 (1968), Amer. Math. Soc: Amer. Math. Soc Providence, R.I), Transl. of · Zbl 0174.15403
[17] Lightbourne, J. H.; Martin, R. H., Relatively continuous nonlinear perturbations of analytic semigroups, Nonlinear Analysis, Theory, Methods & Applications, 1, 277-292 (1977) · Zbl 0356.34073
[18] Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603
[19] Lions, J. L.; Magenes, E., Problemi ai limiti non omogenei, III, Ann. Sci. Norm. Sup. Pisa Ser. III, 15, 41-103 (1961) · Zbl 0115.31401
[20] Martin, R. H., Nonlinear perturbations of uncoupled systems of elliptic operators, Math. Ann., 211, 155-169 (1974) · Zbl 0297.35027
[21] Martin, R. H., Invariant sets for perturbed semigroups of linear operators, Ann. Mat. Pura Appl. IV, 105, 221-239 (1975) · Zbl 0315.34074
[22] Martin, R. H., Nonlinear Operators and Differential Equations in Banach Spaces (1976), Wiley: Wiley New York · Zbl 0333.47023
[23] McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, 837-842 (1934) · Zbl 0010.34606
[24] Nagumo, M., Über die Lage der Integralkurven gewöhnlicher Differential-gleichungen, (Proc. Phys.-Math. Soc. Japan, 24 (1942)), 551-559 · Zbl 0061.17204
[25] Pazy, A., Semi-Groups of Linear Operators and Applications to Partial Differential Equations, University of Maryland Lecture Notes, No. 19 (1974) · Zbl 0516.47023
[26] Prodi, G., Soluzioni periodiche di equazioni alle derivate parziali di tipo parabolico e non lineari, Riv. Mat. Univ. Parma, 3, 265-290 (1952) · Zbl 0049.07502
[27] Prodi, G., Teoremi di esistenza par equazioni alle derivate parziali non lineari di tipo parabolico, II, Rend. Ist. Lombardo Cl. Sci., 84, 27-47 (1953) · Zbl 0052.32602
[28] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0153.13602
[29] Puel, J. P., Existence, comportement á l’infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d’ordre 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV, 3, 89-119 (1976) · Zbl 0331.35027
[30] Redheffer, R. M., The theorems of Bony and Brézis on flow-invariant sets, Amer. Math. Monthly, 79, 740-747 (1972) · Zbl 0278.34039
[31] Schmitt, K.; Thompson, T., Boundary value problems for infinite systems of second-order differential equations, J. Differential Equations, 18, 277-295 (1975) · Zbl 0302.34081
[32] Sobolevskii, P. E., (Equations of Parabolic Type in a Banach Space, Ser. 2, Vol. 49 (1966), Amer. Math. Soc. Transl: Amer. Math. Soc. Transl Providence, R.I), 1-62 · Zbl 0178.50301
[33] Stastnova, V.; Vejvoda, O., Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation, Apl. Mat., 13, 466-477 (1968) · Zbl 0165.44302
[34] Tanabe, H., On the equations of evolution in a Banach space, Osaka Math. J., 12, 363-376 (1960) · Zbl 0098.31301
[35] Tomi, F., Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111, 350-366 (1969) · Zbl 0179.43602
[36] Vidossich, G., On the structure of periodic solutions of differential equations, J. Differential Equations, 21, 263-278 (1976) · Zbl 0357.34029
[37] Webb, G. F., Continuous nonlinear perturbations of linear accretive operators in Banach spaces, J. Functional Analysis, 10, 191-203 (1972) · Zbl 0245.47052
[38] Weinberger, H. F., Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat., 8, 295-310 (1975) · Zbl 0312.35043
[39] Bebernes, J. W.; Schmitt, K., Invariant sets and the Hukuhara-Kneser property for systems of parabolic partial differential equations, Rocky Mountain J. Math., 7, 557-567 (1977) · Zbl 0377.35035
[40] Chueh, K. N.; Conley, C. C.; Smoller, J. A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26, 373-392 (1977) · Zbl 0368.35040
[41] Redheffer, R.; Walter, W., Invariant sets for systems of partial differential equations. I. Parabolic equations, Arch. Rat. Mech. Anal., 67, 41-52 (1978) · Zbl 0377.35038
[42] Schmitt, K., Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal., Theory, Methods & Appl., 2, 263-309 (1978) · Zbl 0378.35022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.