Time-dependent Nonlinear Perturbations of Analytic Semigroups

Abstract

This paper is concerned with time-dependent relatively continuous perturbations of analytic semigroups and applications to convective reaction-diffusion systems. A general class of time-dependent semilinear evolution equations of the form u _t = (A + B(t))u(t), t ∈ (s, τ); u(s) = v ∈ D(s) is introduced in a general Banach space X. Here A is the generator of an analytic semigroup in X and B(t) is a possibly nonlinear operator from a subset of the domain of a fractional power (−A)^α into X and D(t) = D(B(t)) ⊂ D((−A)^α). This type of semilinear evolution equations admit only local and mild solutions in general. In order to restrict the growth of mild solutions and formulate a Lipschitz conditions in a local sense for B(t), a lower semicontinuous functional ϕ: D((−A)^α) → [0,+∞] is introduced and the growth condition of u(·) is formulated in terms of the nonnegative function ϕ(u(·)) and the nonlinear operator B(t) is assumed to be Lipschitz continuous on D _ρ (t) ≡ {v ∈ D(t): ϕ(v) ≤ ρ for ρ > 0. The main objective is to establish a generation theorem for a nonlinear evolution operator which provides mild solutions to the semilinear evolution equation under the assumption that a consistent discrete scheme exists under a growth condition with respect to ϕ as well as closedness condition for the noncylindrical domain ∪({t}×D _ρ(t)). Moreover, a characterization theorem for the existence of such evolution operator is established in terms of the existence of ϕ-bounded discrete schemes. Our generation theorem can be applied to a variety of semilinear convective reaction-diffusion systems. We here make an attempt to apply our result to a mathematical model which describes a complex physiological phenomena of bone remodeling.

The third author is partially supported by a Grant-in-Aid for Scientific Research (B)(1) No.16340042 from JSPS.