Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity
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MULTIVALUED DYNAMICS OF SOLUTIONS OF AN AUTONOMOUS DIFFERENTIAL-OPERATOR INCLUSION WITH PSEUDOMONOTONE NONLINEARITY UDC 517.9
P. O. Kasyanov
Abstract. This article considers a nonlinear autonomous differential-operator inclusion with a pseudomonotone dependence between determinative problem parameters. The dynamics of all weak solutions defined on the positive semi-axis of time is studied. The existence of trajectory and global attractors is proved and their structure is investigated. A class of high-order nonlinear parabolic equations is considered to be a possible application.
Keywords: differential-operator inclusion, global attractor, trajectory attractor, pseudomonotone mapping.
INTRODUCTION Qualitative investigations of nonlinear mathematical models of evolutionary processes and fields of different nature, in particular, problems of dynamics of solving nonstationary problems, are performed by many collectives of mathematicians, mechanicians, geophysicists (mainly theorists), and engineers. A list of relevant results that is far from complete is presented in [1–17]. The latest data on the study of multivalued (in the general case) dynamics of solutions of mathematical models with nonlinear nonsmooth discontinuous multivalued nonmonotone interaction functions are based on the theory of global and trajectory attractors for m-semiflows of solutions [1, 5–7]. In this case, to solve the evolutionary problem being considered, the properties connected with system dissipativity and closeness (in a sense) of the resolving operator [1, 5–8, 11, 13, 14] must be fulfilled. Note that such properties of solutions are individually checked for each inclusion on the basis of the linearity or monotonicity of the leading part of the differential operator appearing in the problem [1, 6, 11, 13, 14]. In most cases, quasilinear equations are considered. At the same time, energy extensions and Nemytskii operators for differential operators occurring in generalized statements of various problems of mathematical physics, problems on a manifold with boundary and without boundary, problems with delay, stochastic partial differential equations, and problems with degeneration, as a rule, possess (if the phase space is properly chosen) common properties connected by growth conditions (the growth often is no more than polynomial), sign conditions, and pseudo-monotonicity [2–4, 12, 15, 16]. Under such constraints imposed on key problem parameters, it is possible to prove in the general case only the existence of weak solutions of a differential-operator inclusion, but the proof is not always cons tructive [2–4, 12, 15, 16]. Thus, the problem of existence and investigation of the structure of trajectory and global attractors for weak solutions of evolutionary inclusions in infinite-dimensional spaces with multivalued interaction multifunctions of pseudomonotone type is an urgent problem.
National Technical University “Kyiv Polytechnic Institute,” National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, Ky
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