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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.

MSC:
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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