Isomorphic Well-Posedness of the Final Value Problem for the Heat Equation with the Homogeneous Neumann Condition
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Integral Equations and Operator Theory
Isomorphic Well-Posedness of the Final Value Problem for the Heat Equation with the Homogeneous Neumann Condition Jon Johnsen Abstract. This paper concerns the final value problem for the heat equation subjected to the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that there exists a linear homeomorphism between suitably chosen Hilbert spaces containing the solutions and the data, respectively. This improves a recent work of the author, in which the same problem was proven well-posed in the original sense of Hadamard under an additional assumption of H¨ older continuity of the source term. Like for its predecessor, the point of departure is an abstract analysis in spaces of vector distributions of final value problems generated by coercive Lax–Milgram operators, now yielding isomorphic well-posedness for such problems. Hereby the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, resulting in a non-local compatibility condition on the data. As a novelty, a stronger version of the compatibility condition is introduced with the purpose of characterising the data that yield solutions having the regularity property of being square integrable in the generator’s graph norm (instead of in the form domain norm). This result allows a direct application to the class 2 boundary condition in the considered inverse Neumann heat problem. Mathematics Subject Classification. 35A01, 47D06. Keywords. Compatibility condition, Final value data, Inverse Neumann heat problem, Isomorphically well-posed.
1. Introduction The purpose of the present paper is to show rigorously that the heat conduction final value problem with the homogeneous Neumann condition is isomorphically well-posed, in the sense that there exists an isomorphism between suitably chosen spaces for the data and the corresponding solutions.
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This result is obtained below by improving the recent results in [27] by application of classical regularity properties in spaces of low regularity. The central theme below is to characterise the u(t, x) that in a fixed bounded open set Ω ⊂ Rn (n ≥ 1) with C ∞ -smooth boundary Γ = ∂Ω fulfil the following equations, whereby Δ = ∂x21 + · · · + ∂x2n denotes the Laplace operator and ν(x) stands for the exterior unit normal vector field at Γ: ⎫ for t ∈ ]0, T [ , x ∈ Ω,⎪ ∂t u(t, x) − Δu(t, x) = f (t, x) ⎬ (ν · grad)u(t, x) = 0 for t ∈ ]0, T [ , x ∈ Γ, (1.1) ⎪ ⎭ u(T, x) = uT (x) for t = T, x ∈ Ω. In view of the final value condition at t = T , this problem is also called the inverse Neumann heat equation. One area of interest of this could be a nuclear power plant hit by power failure at t = 0: when at t = T > 0 power is regained and the reactor temperatures uT (x) are measured, a backwards calculation could possibly settle whether at some t < T the temperatures u(t, x) were high enough to cau
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