Abstract
<abstract><p>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.</p></abstract>
Highlights
Well-posedness of final value problems for a large class of parabolic differential equations was recently obtained in a joint work of the author and given an ample description for a broad audience in [5], after the announcement in [4]
The present paper substantiates the indications made in the concise review [21], namely, that the abstract parts in [5] extend from V -elliptic Lax–Milgram operators A to those that are merely V -coercive—despite that such A may be non-injective
The basic analysis is made for a Lax–Milgram operator A defined in H from a V -coercive sesquilinear form a(·, ·) in a Gelfand triple, i.e., three separable, densely injected Hilbert spaces V ֒→ H ֒→ V ∗ having the norms ·, | · | and · ∗, respectively
Summary
Well-posedness of final value problems for a large class of parabolic differential equations was recently obtained in a joint work of the author and given an ample description for a broad audience in [5], after the announcement in [4]. Returning to the final value problem (2) it would be natural to seek solutions u in the same space X This turns out to be possible only when the data ( f , uT ) are subjected to substantial further conditions. Y f is a priori a vector in V ∗, but y f lies in H as Proposition 2 shows it equals the final state of a solution in C([0, T ], H) of a Cauchy problem having u0 = 0 These remarks on y f make it clear that in the following main result of the paper—which relaxes the assumption of V -ellipticity in [4, 5] to V -coercivity—the difference in (9) is a member of H: Theorem 1. There is a strictly descending chain of domains as in (14)
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