Optimal regularity and stability analysis in the \(\alpha-\)norm for a class of partial functional differential equations with infinite delay
The paper deals with the partial functional differential equation with infinite delay \[ \begin{cases}\frac{dx}{dt}(t)=-Ax(t)+F(x_t),&\text{for }t\geq 0,\\ x_0=\varphi\in {\mathcal B},&{}\end{cases}\tag{P} \] where \(-A\) generates an analytic semigroup \((T(t))_{t\geq 0}\) on a Banach space \(X\), and the phase space \(\mathcal B\) is the Banach space of functions mapping \((-\infty,0]\) to \(X\), which satisfies some fundamental axioms. The function \(F:{\mathcal B}_{\alpha}\to X\) is continuously differentiable with \(F'\) locally Lipschitz, where \({\mathcal B}_{\alpha}\) is the subspace of \(\mathcal B\) defined by \[ {\mathcal B}_{\alpha}=\{\varphi\in {\mathcal B};\;\varphi(\theta)\in D(A^{\alpha})\text{ for }\theta\leq 0\text{ and }A^{\alpha}\varphi\in{\mathcal B}\}, \] for \(0<\alpha<1\), \(A^{\alpha}\) denotes the fractional power of \(A\), the function \(A^{\alpha}\varphi\) is defined by \((A^{\alpha}\varphi)(\theta):=A^{\alpha}(\varphi(\theta))\) for \(\theta\leq 0\), and, for every \(t\geq 0\), the history function \(x_t\in {\mathcal B}_{\alpha}\) is defined by \(x_t(\theta):=x(t+\theta)\) for \(\theta\leq 0\). The authors establish necessary and sufficient conditions for the existence of strict solutions for problem (P); then they give a new characterisation of the infinitesimal generator of the solution semigroup and study the regularity of mild solutions. Spectral properties of the infinitesimal generator of the semigroup solution and the asymptotic behavior for the solution semigroup are also investigated. An example of a partial differential equation with infinite delay which illustrates the obtained results is finally presented.
partial functional differential equation
scientific article; zbMATH DE number 5888332