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Compactness theorems for abstract evolution problems. (English) Zbl 1008.47057
Consider the evolution problem defined by a family of unbounded, multivalued, dissipative, nonlinear operators in a Banach space. The author gives conditions under which, for a given starting point, the problem has either no bounded solution or a bounded solution with precompact trajectory. He extends and supplements results by A. Pazy [Isr. J. Math. 20, 23-36 (1975; Zbl 0305.47022)], C. M. Dafermos and M. Slemrod [J. Funct. Anal. 13, 97-106 (1973; Zbl 0267.34062)], and G. F. Webb [Proc. Roy. Soc. Edinb., Sect. A 84, 19-33 (1979; Zbl 0414.34042)].

MSC:
47H20 Semigroups of nonlinear operators
35K90 Abstract parabolic equations
34G25 Evolution inclusions
35B37 PDE in connection with control problems (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
47H06 Nonlinear accretive operators, dissipative operators, etc.
93C25 Control/observation systems in abstract spaces
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