Semiflows generated by Lipschitz perturbations of non-densely defined operators. (English) Zbl 0734.34059

Summary: A variety of problems in differential equations ((abstract) functional differential equations, age-dependent population models (with and without delay), evolution equations with boundary conditions, e.g.) can be written as semilinear Cauchy problems with a Lipschitz perturbation of a closed linear operator which is not densely defined but satisfies the resolvent estimates of the Hille and Yosida theorem. A natural generalized notion of solution is provided by the integral solutions in the sense of Da Prato and Sinestrari. We derive a variation of constants formula which allows us to transform the integral solutions of the evolution equation to solutions of an abstract semilinear Volterra integral equation. The latter can be used to find integral solutions to the Cauchy problem; moreover one finds sufficient and necessary conditions for the (forward) invariance of closed convex sets under the solution flow. The solution flow can be shown to form a dynamical system. Conditions for the regularity of the flow in time and initial state are derived. The steady states of the flow are characterized and sufficient conditions for local stability and instability are found. Finally the problems mentioned at the beginning are fitted into the general framework.


34G20 Nonlinear differential equations in abstract spaces
37-XX Dynamical systems and ergodic theory
34K30 Functional-differential equations in abstract spaces
35R10 Partial functional-differential equations
47H20 Semigroups of nonlinear operators
92D25 Population dynamics (general)