Summary
Motivated by the ideas of integrated or regularized operator families, we in this chapter select to treat several types of higher order abstract Cauchy problems which are no longer wellposed or strongly wellposed in the sense of Section 2.1, but wellposed in some generalized sense. Our main purpose is to establish some concise and useful criteria for these (ACP n ) to be wellposed in some sense.
In Section 3.1, fixing r ≥ 0 we give a set of conditions on λ k−r−1 A k R λ (1 ≤ k ≤ n) to lead to existence and uniqueness, as well as continuous dependence on initial data (in some sense) of solutions for (ACP n ). One of the conditions is that each of λ k−r−1 A k R λ is a Laplace transform. When n = 1, λ k−r−1 A k R λ reduces to λ −r R(λ; −A 0), and the condition reduces to the characterization for −A 0 to generate an r-times integrated semigroup.
Recall that Corollary 2.4.8 gave an interesting result about the equation whose ‘principal’ coefficient operator is the generator of a strongly continuous semigroup. There arises the problem of what if the generators of integrated semigroups in place of strongly continuous semigroups act as the ‘principal’ coefficient operators. Sections 3.2 and 3.3 are devoted to such problems. Section 3.4 deals with the case when the ‘principal’ coefficient operators are the generators of integrated cosine functions.
Section 3.5 defines C-wellposedness of (ACP n ), which is a reflection of the idea of C-regularized semigroups in higher order Cauchy problems. In particular, we consider complete (ACP 2) with differential operators as coefficient operators and explore the conditions under which such Cauchy problems are C-wellposed.
Finally in Section 3.6, we consider the Cauchy problem for u (n)(t) = Au(t). A classical result indicates that for n ≥ 3, its wellposedness implies the boundedness of A. We will give an extension of this result, Theorem 6.5. In fact, the main aim of this section is to show conditions on A, which are valid for many unbounded operators, such that the underlying Cauchy problem is C-wellposed (in some sense).
Throughout this chapter, E denotes a Banach space.
Keywords
- Unique Solution
- Linear Operator
- Cauchy Problem
- Bounded Linear Operator
- Continuous Semigroup
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). Generalized wellposedness. 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_3
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DOI: https://doi.org/10.1007/978-3-540-49479-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65238-0
Online ISBN: 978-3-540-49479-9
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