Translation semigroups and their linearizations on spaces of integrable functions

Abstract

Of concern is the unbounded operatorAΦf=f′{A_\Phi }f = f’with nonlinear domainD(AΦ)={f∈W1,1:f(0)=Φ(f)}D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\}which is considered on the Banach spaceEEof Bochner integrable functions on an interval with values in a Banach spaceFF. Under the assumption thatΦ\Phiis a Lipschitz continuous operator fromEEtoFF, it is shown thatAΦ{A_{\Phi }}generates a strongly continuous translation semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. For linear operatorsΦ\Phiseveral properties such as essential-compactness, positivity, and irreducibility of the semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}depending on the operatorΦ\Phiare studied. It is shown that ifFFis a Banach lattice with order continuous norm, then(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}is the modulus semigroup of(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. Finally spectral properties ofAΦ{A_\Phi }are studied and the spectral bounds(AΦ)s({A_\Phi })is determined. This leads to a result on the global asymptotic behavior in the case whereΦ\Phiis linear and to a local stability result in the case whereΦ\Phiis Fréchet differentiable.