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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Parabolicity of a class of higher order abstract differential equations

Authors: Ti Jun Xio and Jin Liang
Journal: Proc. Amer. Math. Soc. 120 (1994), 173-181
MSC: Primary 34G10; Secondary 47D09
MathSciNet review: 1182708
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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

$\displaystyle {u^{(n)}}(t) + \sum\limits_{i = 1}^{n - 1} {{c_i}{A^{{k_i}}}{u^{(n - i)}}(t) + Au(t) = 0}$ ($ \ast$)

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.

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Keywords: Parabolicity, higher-order abstract differential equation, nonnegative operator, strictly nonnegative, perturbation
Article copyright: © Copyright 1994 American Mathematical Society