TY - JOUR

T1 - Final value problems for parabolic differential equations and their well-posedness

AU - Christensen, Ann-Eva

AU - Johnsen, Jon

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Compatibility condition

KW - Final value

KW - Hyponormal

KW - Non-selfadjoint

KW - Parabolic boundary problem

KW - Well posed

UR - http://www.scopus.com/inward/record.url?scp=85046699145&partnerID=8YFLogxK

U2 - 10.3390/axioms7020031

DO - 10.3390/axioms7020031

M3 - Journal article

VL - 7

SP - 1

EP - 36

JO - Axioms

JF - Axioms

SN - 2075-1680

IS - 2

M1 - 31

ER -