Summary
When more than one coefficient operators are involved, the theory of (ACP n ) is considerably more complicated than that of (ACP 1) or incomplete (ACP 2) (i.e. the Cauchy problem for u″(t) + A 0 u(t) = 0). As made clear by Fattorini [7], the usual wellposedness of (ACP n ) can not ensure the exponential growth of its solutions. This motivates the introduction of the notion of strong wellposedness for general (ACP n ), which makes it possible to evolve a rich theory for the propagators of (ACP n ), just as that for the classical strongly continuous semigroups.
We start in Section 2.1 with the explicit definitions of wellposedness and strong wellposedness of (ACP n ). One see easily that, in the case of (ACP 1) or incomplete (ACP 2), strong wellposedness is equivalent to wellposedness. Moreover, some basic facts regarding a strongly wellposed (ACP n ) are also discussed.
Section 2.2 is devoted to the proof of the characterization (Theorem 2.2) for (ACP n ) to be strongly wellposed. This is a Hille-Yosida-Feller-Miyadera-Phillips type theorem.
Keywords
- Banach Space
- Linear Operator
- Cauchy Problem
- Continuous Semigroup
- Order Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© 1998 Springer-Verlag Berlin Heidelberg
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Xiao, TJ., Liang, J. (1998). Wellposedness and solvability. In: The Cauchy Problem for Higher Order Abstract Differential Equations. Lecture Notes in Mathematics, vol 1701. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-49479-9_2
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DOI: https://doi.org/10.1007/978-3-540-49479-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65238-0
Online ISBN: 978-3-540-49479-9
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