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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Translation semigroups and their linearizations on spaces of integrable functions

Author: Annette Grabosch
Journal: Trans. Amer. Math. Soc. 311 (1989), 357-390
MSC: Primary 47H20; Secondary 34G20, 47B38, 47D05, 92A15
MathSciNet review: 974781
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Abstract: Of concern is the unbounded operator $ {A_\Phi }f = f'$ with nonlinear domain $ D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\} $ which is considered on the Banach space $ E$ of Bochner integrable functions on an interval with values in a Banach space $ F$. Under the assumption that $ \Phi$ is a Lipschitz continuous operator from $ E$ to $ F$, it is shown that $ {A_{\Phi}}$ generates a strongly continuous translation semigroup $ {({T_\Phi }(t))_{t \geq 0}}$. For linear operators $ \Phi$ several properties such as essential-compactness, positivity, and irreducibility of the semigroup $ {({T_\Phi }(t))_{t \geq 0}}$ depending on the operator $ \Phi$ are studied. It is shown that if $ F$ is a Banach lattice with order continuous norm, then $ {({T_\Phi }(t))_{t \geq 0}}$ is the modulus semigroup of $ {({T_\Phi }(t))_{t \geq 0}}$. Finally spectral properties of $ {A_\Phi}$ are studied and the spectral bound $ s({A_\Phi })$ is determined. This leads to a result on the global asymptotic behavior in the case where $ \Phi$ is linear and to a local stability result in the case where $ \Phi$ is Fréchet differentiable.

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Keywords: Linear and nonlinear strongly continuous semigroups, translation semigroups, first derivatives, compactness, positivity, irreducibility, domination, spectral properties, asymptotics, linearization, stability of equilibrium solutions, functional equations, Volterra equations
Article copyright: © Copyright 1989 American Mathematical Society

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