Time-dependent Nonlinear Perturbations of Analytic Semigroups

Robert H. Martin (), Toshitaka Matsumoto (), Shinnosuke Oharu () and Naoki Tanaka ()
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Robert H. Martin: North Carolina State University, Department of Mathematics
Toshitaka Matsumoto: Hiroshima University, Department of Mathematical and Life Sciences, Graduate School of Science
Shinnosuke Oharu: Chuo University, Department of Mathematics, Faculty of Science and Engineering
Naoki Tanaka: Shizuoka University, Department of Mathematics, Faculty of Science

A chapter in Functional Analysis and Evolution Equations, 2007, pp 457-502 from Springer

Abstract: 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.

Keywords: Nonlinear perturbation; analytic semigroup; evolution operator; subtangential type condition; generation theorem; characterization of evolution operators; local multiple Laplace transform; mathematical model of bone remodeling (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-7643-7794-6_29

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DOI: 10.1007/978-3-7643-7794-6_29

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