Compactness theorems for abstract evolution problems. (English) Zbl 1008.47057

Couchouron, J.-F.

J. Evol. Equ. 2, No. 2, 151-175 (2002).

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)].

Reviewer: H.Hering (Göttingen)

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

Keywords:

precompact trajectory; precompactness property; compact resolvent; mild solution; integral solution; control theory; stabilization