Translation Semigroups and Their Linearizations on Spaces of Integrable Functions

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.