Notes on the Propagators of Evolution Equations
© Yu Lin et al. 2010
Received: 8 January 2010
Accepted: 1 February 2010
Published: 16 February 2010
We consider the propagator of an evolution equation, which is a semigroup of linear operators. Questions related to its operator norm function and its behavior at the critical point for norm continuity or compactness or differentiability are studied.
As it is well known, each well-posed Cauchy problem for first-order evolution equation in Banach spaces
gives rise to a well-defined propagator, which is a semigroup of linear operators, and the theory of semigroups of linear operators on Banach spaces has developed quite rapidly since the discovery of the generation theorem by Hille and Yosida in 1948. By now, it is a rich theory with substantial applications to many fields (cf., e.g., [1–6]).
In this paper, we pay attention to some basic problems on the semigroups of linear operators and reveal some essential properties of theirs.
For a comprehensive theory of -semigroups we refer to .
We ask the following question
To show (III) and (II), we write
First we note from (2.3) that
which is a contradiction to (2.16).
Open Problem 1.
3. The Critical Point of Norm-Continuous (Compact, Differentiable) Semigroups
Therefore, if we write
and suppose ( , ), then ( , ) takes the form of for a nonnegative real number . In other words, if ( , ), then is norm continuous (compact, differentiable) on the interval but not at any point in . We call the critical point of the norm continuity (compactness, differentiability) of operator semigroup .
A natural question is the following
Suppose that is the critical point of the norm continuity compactness, differentiability of the operator semigroup . Is also norm continuous (compact, differentiable) at ? Of course, concerning norm continuity or differentiability at we only mean right continuity or right differentiability.
We show that the answer is "yes" in some cases and "no" for other cases.
The authors acknowledge the support from the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).
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