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