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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Well-posedness of higher order abstract Cauchy problems


Author: Frank Neubrander
Journal: Trans. Amer. Math. Soc. 295 (1986), 257-290
MSC: Primary 34G10; Secondary 47D05
MathSciNet review: 831199
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Abstract: The paper is concerned with differential equations of the type

$\displaystyle {u^{(n + 1)}}(t) - A{u^{(n)}}(t) - {B_1}{u^{(n - 1)}}(t) - \cdots - {B_n}u(t) = 0$ ($ \ast$)

in a Banach space $ E$ where $ A$ is a linear operator with dense domain $ D(A)$ and $ {B_1}, \ldots ,{B_n}$ are closed linear operators with $ D(A) \subset D({B_k})$ for $ 1 \leq k \leq n$. The main result is the equivalence of the following two statements: (a) $ A$ has nonempty resolvent set and for every initial value $ ({x_0}, \ldots ,{x_n}) \in {(D(A))^{n + 1}}$ the equation $ ( \ast )$ has a unique solution in $ {C^{n + 1}}({{\mathbf{R}}^ + },E) \cap {C^n}({{\mathbf{R}}^n},[D(A)])([D(A)]$ denotes the Banach space $ D(A)$ endowed with the graph norm); (b) $ A$ is the generator of a strongly continuous semigroup. Under additional assumptions on the operators $ {B_k}$, which are frequently fulfilled in applications, we obtain continuous dependence of the solutions on the initial data; i.e., well-posedness of $ ( \ast )$. Using Laplace transform methods, we give explicit expressions for the solutions in terms of the operators $ A$, $ {B_k}$. The results are then used to discuss strongly damped semilinear second order equations.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0831199-8
PII: S 0002-9947(1986)0831199-8
Keywords: Semigroups of operators, abstract Cauchy problem
Article copyright: © Copyright 1986 American Mathematical Society