Nonlinear semigroup of a class of abstract semilinear functional-differential equations with a non-dense domains
Let \(A\) be a Hille-Yosida operator with non-dense domain in a Banach space \(X\). Here, the abstract semilinear autonomous functional-differential equation \[ \overset{\cdot}{u}(t)=Au(t)+F(u_{t}), \quad t>0,\;u_{0}=\varphi, \] is considered. Let \(\{ U( t) :t\geq0\} \) be the family of nonlinear operators defined on \[ Y=\{ \varphi\in C([ -r,0] ,X) :\varphi(0) \in\overline{D(A)}\} \] by \[ U( t) =u_{t} (\cdot,\varphi) \] for \(t\geq0\) and \(\varphi\in Y\). In case \(F\) is globally Lipschitzian, \(\{ U( t) :t\geq0\} \) defines a nonlinear strongly continuous semigroup of continuous operators on \(Y\). Using the Crandall-Liggett approach, the authors study the semigroup \(\{ U( t) :t\geq0\} \).
semilinear functional-differential equation
scientific article; zbMATH DE number 2174206
scientific article