442 research outputs found
Characterisation of Log-Convex Decay in Non-Selfadjoint Dynamics
The short-time and global behaviour are studied for an autonomous linear
evolution equation, which is defined by a generator inducing a uniformly
bounded holomorphic semigroup in a Hilbert space. A general necessary and
sufficient condition is introduced under which the norm of the solution is
shown to be a log-convex and strictly decreasing function of time, and
differentiable also at the initial time with a derivative controlled by the
lower bound of the generator, which moreover is shown to be positively
accretive. Injectivity of holomorphic semigroups is the main technical tool.Comment: 11 pages. Version to appear in Electronic Research Announcements in
Mathematical Sciences (a precision in Lemma 3.2, plus minor improvements
On parabolic final value problems and well-posedness
We prove that a large class of parabolic final value problems is well
posed.This results via explicit Hilbert spaces that characterise the data
yielding existence, uniqueness and stability of solutions. This data space is
the graph normed domain of an unbounded operator, which represents a new
compatibility condition pertinent for final value problems. The framework is
evolution equations for Lax--Milgram operators in vector distribution spaces.
The final value heat equation on a smooth open set is also covered, and for
non-zero Dirichlet data a non-trivial extension of the compatibility condition
is obtained by addition of an improper Bochner integral.Comment: 6 pages. Accepted version; a short announcement of results from our
full paper on final value problems. Appeared in Comptes Rendu Mathematique
Final value problems for parabolic differential equations and their well-posedness
This article concerns the basic understanding of parabolic final value
problems, and a large class of such problems is proved to be well posed. The
clarification is obtained via explicit Hilbert spaces that characterise the
possible data, giving existence, uniqueness and stability of the corresponding
solutions. The data space is given as the graph normed domain of an unbounded
operator occurring naturally in the theory. It induces a new compatibility
condition, which relies on the fact, shown here, that analytic semigroups
always are invertible in the class of closed operators. The general set-up is
evolution equations for Lax--Milgram operators in spaces of vector
distributions. As a main example, the final value problem of the heat equation
on a smooth open set is treated, and non-zero Dirichlet data are shown to
require a non-trivial extension of the compatibility condition by addition of
an improper Bochner integral.Comment: 39 pages. Revised version, with minor improvements. Essentially
identical to the accepted version, which appeared in Axioms on 9 May 201
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