Smooth dynamical solution for the damped second-order equation (Q1273253)

The authors study the second-order differential equation \[ u''+ Bu'+Au=f(t)\tag{1} \] in Banach spaces with closed, densely defined linear operators as coefficients. It is defined a main propagator family, or simply, a smooth dynamical solution to (1) as a very natural extension of the classical \(C_0\)-semigroup theory. A smooth dynamical solution to (1) is a family of bounded operators \(\{D(t)\}\) defined on a Banach space \(X\) for all nonnegative \(t\), where \(D(t)_x\) is continuously differentiable for every \(x\) in \(X\), \(D'(t)\) is bounded in \(X\) for all \(t\geq 0\), and the functions \(u(t)=D(t)x\) and \(u(t)= D_1(t)y\) \((D_1(t)=:D'(t)+D(t)B)\) are classical solutions to the homogeneous equation associated with (1): \[ {d^2u\over dt^2}(t)+ B{du\over dt} (t)+Au(t)=0, \tag{2} \] with the Cauchy data \(u(0)=0\), \(u'(0)=x\) and \(u(0)=y\), \(u'(0)=0\), respectively. \(D(t)\) is the main propagator, whereas \(D_1(t)\) is the static operator or the secondary propagator. The existence of a dynamical solution \(D(t)\) to (1) which satisfies the exponential estimate \[ \bigl\| D(t)x \bigr\|\leq Me^{\omega t}\| x\|,\quad \bigl\| D'(t)x\bigr\|\leq Me^{\omega t}\| x\|,\text{ for all }x\text{ in }X\;(\omega\geq 0,\;M>0)\tag{3} \] is equivalent with the fact that the operator \(\Delta (\lambda)= \lambda^2I+ \lambda B+A\) has an inverse for \(\text{Re} \lambda>\omega\), this inverse is analytic and satisfies the following condition \[ \left\| {d^n\over d\lambda^n} \lambda\Delta^{-1} (\lambda) \right \|\leq {Mn!\over (\text{Re} \lambda-\omega)^{n+1}},\;n=0,1,2,\dots \] Another main result states that if \(D(t)\) is a dynamical solution to (1) with the exponential estimate (3), then the mild solution to (1) defined by \[ u(t)= D(t)_x+ D_1(t)_y+ \int^t_0 D(t-s)f(s)ds \] is a classical one for \(x\in D_1:=D(A)\cap D(B)\), \(y\in D_2:=D(A^2)\cap D(B^2)\), \(f(t)\in D_1\). Afterwards, the authors present generalizations of the Miyadera-Phillips-Feller theorem, a Hille type theorem and the Trotter-Kato approximation type theorem.