zbMATH — the first resource for mathematics

Nonlinear interaction problems. (English) Zbl 0817.35035
From the introduction: In many fields of applied mathematics arises the necessity to consider mutual influences of evolution phenomena on a (possible infinite) family of domains. In this paper, we give a general concept for the treatment of many different kinds of problems of this type (called interaction problems). It uses the theory of maximal monotone operators, perturbation results, features on evolution equations, theorems on degenerate linear equations and a compact imbedding result for infinite products of Sobolev spaces. As applications we have existence results in the following situations: nonlocal interaction conditions, concurrence of populations, infinite networks (question of discreteness of the spectrum), two-dimensional disk and one- dimensional interval with overlap, semipermeable interfaces, mixed hyperbolic-parabolic or elliptic-parabolic problems, simultaneous treatment of nonlinear evolution and nonlinear interaction, and quasilinear problems. The concept of interaction problems generalizes the notion of ramified spaces (i.e. families of domains, where certain parts of the boundaries are identified, cf. G. Lumer [C. R. Acad. Sci., Paris. Sér. A 291, 627-630 (1980; Zbl 0449.35110)]). Most of the results of this paper are new, even in this important special case. Further, this concept gives the insight of the structural similarity of many different kinds of problems on an abstract level.

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
Full Text: DOI
[1] Kato, T., Accretive operators and nonlinear evolution operators in Banach spaces, (), 138-161
[2] Brezis, H., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, (), 101-156
[3] Brezis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, () · Zbl 0252.47055
[4] Showalter, R.E., Hilbert space methods for partial differential equations, () · Zbl 0991.35001
[5] Lumer, G., Espaces ramifiés et diffusions sur LES réseaux topologiques, C.R. acad. sci. Paris, 291, 627-630, (1980), Série A · Zbl 0449.35110
[6] Nicaise, S., Spectre des réseaux topologiques finis, Bull. sci. math., 111, 401-413, (1987), 2ème série · Zbl 0644.35076
[7] Nicaise, S., Problèmes de Cauchy posés en norme uniforme sur LES espaces ramifiés élémentaires, C.R. acad. sci. Paris, 303, 443-446, (1986), Série I · Zbl 0595.47035
[8] Nicaise, S., Le laplacien sur lesréseaux deux-dimensionnel polygonaux topologiques, J. math. pures appl., 67, 93-113, (1988) · Zbl 0698.35047
[9] Nicaise, S., Diffusion sur LES espaces ramifiés, ()
[10] von Below, J., Classical solvability of linear equations on networks, J. diff. eqns, 72, 316-337, (1988) · Zbl 0674.35039
[11] Gramsch, B., Zum einbettungssatz von Rellich bei sobolevräumen, Math. Z., 106, 81-87, (1968) · Zbl 0159.42001
[12] Ali Mehmeti, F., Problèmes de transmission pour des équations des ondes linéaires et quasilinéaires, (), 75-96 · Zbl 0603.35057
[13] Ali Mehmeti, F.; Ali Mehmeti, F., Global existence of solutions of semilinear equations with interaction, (), Teubner-texte zur math., 112, 11-23, (1989) · Zbl 0736.35069
[14] Ali Mehmeti, F., Lokale und globale Lösungen linearer und nichtlinearer hyperbolischer evolutionsgleichungen mit transmission, () · Zbl 0633.35052
[15] Ali Mehmeti, F., Regular solutions of transmission and interaction problems for wave equations, Math. meth. appl. sci., 11, 665-685, (1989) · Zbl 0722.35062
[16] Minty, G.J., On the monotonicity of the gradient of a convex function, Pacif. J. math., 14, 243-247, (1964) · Zbl 0123.10601
[17] Ali Mehmeti, F.; Nicaise, S., Some realizations of interaction problems, (), 15-28 · Zbl 0816.35160
[18] Ali Mehmeti, F.; Nicaise, S., Compact imbeddings and interaction problems, (), 143-152
[19] Ali Mehmeti, F.; Ali Mehmeti, F., Reflection and refraction of singularities for wave equations with interface conditions given by Fourier integral operators, (), Teubner-texte zur math., 131, 6-19, (1992) · Zbl 0828.35073
[20] \scNicaise S., Exact controllability of a pluridimensional coupled problem, Revista Matemàtica de la Universidad Complutense de Madrid (to appear). · Zbl 0760.35012
[21] Haraux, A., Nonlinear evolution equations—global behavior of solutions, () · Zbl 0583.35007
[22] Yosida, K., ()
[23] Duvaut, G.; Lions, J.-L., LES inéquations en mécanique et en physique, (1972), Dunod Paris · Zbl 0298.73001
[24] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[25] Ciarlet, P.G., The finite element method for elliptic problems, () · Zbl 0285.65072
[26] Ciarlet, P.G.; Le Dret, H.; Nzengwa, R., Junctions between three-dimensional and two-dimensional linearly elastic structures, J. math. pures appl., 68, 261-295, (1989) · Zbl 0661.73013
[27] Grisvard, P., Elliptic problems in nonsmooth domains, () · Zbl 0695.35060
[28] Pavel, N.H., Nonlinear evolution operators and semigroups, () · Zbl 0481.47048
[29] Lions, J.-L., Sur quelques questions d’analyse, de mécanique et de contrôle optimal, (1976), Presses de l’Univ. de Montréal · Zbl 0339.49003
[30] Brezis, H., Problémes unilatéraux, J. math. pures appl., 51, 1-168, (1972) · Zbl 0237.35001
[31] Ali Mehmeti, F., Existence and regularity of solutions of Cauchy problems for inhomogeneous wave equations with interaction, Operator theory, advances and applications, Vol. 50, 23-34, (1991) · Zbl 0749.35019
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.