Conditions for parabolicity of second order abstract differential equations (Q1176212)

The authors study conditions ensuring the differential equation \(u''+Bu'+Au=0\), with closed operators \(A\) and \(B\) in a Banach space, to be parabolic. For \(B\) subordinated to \(A\) of order \(\alpha<0.5\), they state that parabolicity is possible only if \(A\) and \(B\) are bounded. When \(B=cA^ \alpha\) \((0.5\leq\alpha<1)\) they establish necessary and sufficient conditions of parabolicity on the asymptotic spectral behaviour of \(A\). When \(B\) is stronger than \(A^{0.5}\), the authors prove some sufficient conditions of parabolicity which are of different nature from those from \textit{S. Chen} and \textit{R. Triggiani}'s article [Pac. J. Math. 136, No. 1, 15-55 (1989; Zbl 0696.47041)], whose results for the equation in a Hilbert space are extended in the first part of the paper. There are examples of applications to initial boundary value problems for partial differential equations.