Parabolicity of a class of higher order abstract differential equations

Abstract:

Let $E$ be a complex Banach space, ${c_i} \in \mathbb {C}\;(1 \leqslant i \leqslant n - 1)$, and $A$ be a nonnegative operator in $E$. We discuss the parabolicity of the higher-order abstract differential equations \begin{equation} \tag {$\ast $} {u^{(n)}}(t) + \sum \limits _{i = 1}^{n - 1} {{c_i}{A^{{k_i}}}{u^{(n - i)}}(t) + Au(t) = 0} \end{equation} and some perturbation cases of ($\ast$). A sufficient and necessary condition for ($\ast$) to be parabolic is obtained, provided ${k_1} > {k_2} - {k_1} > \cdots > 1 - {k_{n - 1}} > 0,\;{c_i} \ne 0\;(1 \leqslant i \leqslant n - 1)$. For $A$ strictly nonnegative (Definition 1.3), $n = 3,{c_1},{c_2} \geqslant 0$, a sharp criterion is given.