Backward uniqueness on a half-space

The Gaussian estimate and its propagation argument prove thm:bu.

theorem ESS.Main.backwardUniqueness (c₁ M : ℝ) (hc₁ : 0 < c₁) (hM : 0 < M) (w : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (Dw : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3) (D2w : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3) (Dtw : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (hcont : ContinuousOn w ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico 0 1)) (hinit : ∀ (x : CKN.Foundation.Parabolic.Vec3), 0 < x 2 → w (x, 0) = 0) (hderiv : CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) w Dw D2w Dtw) (hL2 : ∀ S ⊆ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1), Bornology.IsBounded S → ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in S, ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤) (hineq : ∀ᵐ (z : CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict (CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1)), (CKN.Foundation.Parabolic.vec3EuclideanNorm fun (i : Fin 3) => Dtw z i + ∑ j : Fin 3, D2w z i j j) ≤ c₁ * (√(CKN.spatialGradientSq w Dw z) + CKN.Foundation.Parabolic.vec3EuclideanNorm (w z))) (hgrowth : ∀ z ∈ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1), CKN.Foundation.Parabolic.vec3EuclideanNorm (w z) ≤ Real.exp (M * CKN.Foundation.Parabolic.vec3EuclideanNorm z.1 ^ 2)) (z : CKN.Foundation.Parabolic.ParabolicPoint) : z ∈ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) → w z = 0

Backward uniqueness on a half-space (thm:bu; ESS Theorem 5.1).