Abstract
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.
Highlights
In this article, we establish well-posedness of final value problems for a large class of parabolic differential equations
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
Taking the heat equation as a first example, we address the problem of characterising the functions u(t, x ) that, in a C ∞ -smooth bounded open set Ω ⊂ Rn with boundary ∂Ω, fulfil the equations, where
Summary
We establish well-posedness of final value problems for a large class of parabolic differential equations. This clarifies a longstanding gap in the comprehension of such problems. We provide here a theoretical analysis of such problems and prove that they are well-posed, that is, they have existence, uniqueness and stability of solutions u ∈ X for given data ( f , g, u T ) ∈ Y, in certain normed spaces X, Y to be specified below. Our method is to provide a useful structure on the reachable set for a general class of parabolic differential equations
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