The asymptotic behaviour of perturbed evolution families. (Q1847676)

This paper is devoted to the study of bounded linear operators on a complex Banach space \(X\). It is considered a \({\mathcal C}_0\)-semigroup \({\mathcal T}:=(T(t))_{t\geq 0}\), perturbed by a family \({\mathcal B}\) of closed linear operators \((B(t),D(B(t)))_{t\geq 0}\) on \(X\), fulfilling a suitable condition, and it is proved that the perturbed evolution family \({\mathcal U}_{{\mathcal B}}:= (U_{B}(t,s))_{t\geq 0}\), related to \({\mathcal T}\) by a variation of constants formula, inherits, without any additional assumptions, the asymptotic behavior of \({\mathcal T}\). The results obtained are extended to a genetic evolution family \({\mathcal U}\) perturbed by a family \({\mathcal B}:=(B(t))_{t\in {\mathbb R}}\) of bounded linear operators on \(X\). Applications are given to non-autonomous partial differential equations with delay of the form \[ u^{\prime}(t)=Au(t)+L(t)u_t\quad (t>0), \] \[ u(0)=x,\quad u_0=f, \] where \((A,D(A))\) is the infinitesimal generator of a \({\mathcal C}_0\)-semigroup \((T(t))_{t\geq 0}\) on \(X\), \(x\in X\), \(f\in L^p([-1,0],X)\) for some \(1\leq p<\infty\) and \((L(t))_{t\geq 0}\) is a family of bounded linear operators from \(L^p([-1,0],X)\) to \(X\).