Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators

Jon Johnsen; Jon Johnsen

doi:10.3934/era.2019008

Electronic Research Archive

2019, Volume 27, 20-36. doi: 10.3934/era.2019008

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Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators

Jon Johnsen

Department of Mathematics, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmark

Received: 01 August 2019 Revised: 01 September 2019
Primary: 35A01; Secondary: 47D06

This paper treats parabolic final value problems generated by coercive Lax–Milgram operators, and well-posedness is proved for this large class. The result is obtained by means of an isomorphism between Hilbert spaces containing the data and solutions. Like for elliptic generators, the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, and the resulting compatibility condition extends to the coercive context. Lax–Milgram operators in vector distribution spaces is the main framework, but the crucial tool that analytic semigroups always are invertible in the class of closed operators is extended to unbounded semigroups, and this is shown to yield a Duhamel formula for the Cauchy problems in the set-up. The final value heat conduction problem with the homogeneous Neumann boundary condition on a smooth open set is also proved to be well posed in the sense of Hadamard.
- Duhamel formula,
- coercive operator,
- final value data,
- Neumann heat problem,
- well-posed
Citation: Jon Johnsen. Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators[J]. Electronic Research Archive, 2019, 27: 20-36. doi: 10.3934/era.2019008

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Abstract

This paper treats parabolic final value problems generated by coercive Lax–Milgram operators, and well-posedness is proved for this large class. The result is obtained by means of an isomorphism between Hilbert spaces containing the data and solutions. Like for elliptic generators, the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, and the resulting compatibility condition extends to the coercive context. Lax–Milgram operators in vector distribution spaces is the main framework, but the crucial tool that analytic semigroups always are invertible in the class of closed operators is extended to unbounded semigroups, and this is shown to yield a Duhamel formula for the Cauchy problems in the set-up. The final value heat conduction problem with the homogeneous Neumann boundary condition on a smooth open set is also proved to be well posed in the sense of Hadamard.

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