Semilinear perturbations of Hille-Yosida operators

Horst R. Thieme, and Hauke Vosseler. "Semilinear perturbations of Hille-Yosida operators." Banach Center Publications 63.1 (2003): 87-122. <http://eudml.org/doc/281610>.

@article{HorstR2003,
abstract = {The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), $u(0) = x ∈ \overline\{D(A)\}$, with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, $B(λ - A)^\{-1\}x = λ ∫_0^∞ e^\{-λt\} V(s)xds$, where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0,∞) × \overline\{D(A)\}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤ Bv whenever x,y,v ∈ D(A), |x-y| ≤ v, with B being positive. We illustrate our results by applications to age-structured population models.},
author = {Horst R. Thieme, Hauke Vosseler},
journal = {Banach Center Publications},
keywords = {Hille-Yosida condition; integrated semigroups; F-solutions; nonlinear perturbations; -dissipativeness; Banach lattices; age structured population dynamics},
language = {eng},
number = {1},
pages = {87-122},
title = {Semilinear perturbations of Hille-Yosida operators},
url = {http://eudml.org/doc/281610},
volume = {63},
year = {2003},
}

TY - JOUR
AU - Horst R. Thieme
AU - Hauke Vosseler
TI - Semilinear perturbations of Hille-Yosida operators
JO - Banach Center Publications
PY - 2003
VL - 63
IS - 1
SP - 87
EP - 122
AB - The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), $u(0) = x ∈ \overline{D(A)}$, with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, $B(λ - A)^{-1}x = λ ∫_0^∞ e^{-λt} V(s)xds$, where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0,∞) × \overline{D(A)}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤ Bv whenever x,y,v ∈ D(A), |x-y| ≤ v, with B being positive. We illustrate our results by applications to age-structured population models.
LA - eng
KW - Hille-Yosida condition; integrated semigroups; F-solutions; nonlinear perturbations; -dissipativeness; Banach lattices; age structured population dynamics
UR - http://eudml.org/doc/281610
ER -