Finite-time convergence of solutions of Hamilton-Jacobi equations

The paper proves two main results: (i) finite-time convergence to a prescribed stationary viscosity solution u for suitable initial data ϕ in Au (Theorem 1.1), and (ii) a uniform finite-time reachability from a fixed neighbourhood of the unique forward weak KAM solution u+ (Theorem 1.2). The statements and their proofs are coherent and rely on the implicit action function representation, monotonicity, and a blow-up property for data in A+; see the formal statements and the structure of the proofs in Theorem 1.1 and Theorem 1.2, respectively . Key technical steps include confining the minimizers to a neighbourhood Oε of Iu (Steps 1–2) and then identifying T−tϕε with u for large t (Step 3) . The uniform inclusion A∞ ⊂ T−t(Bε(u+)) follows from an exponential separation estimate for the implicit action functions (Lemma 3.1 and Corollary 3.2) and a boundedness lemma (Lemma 3.4) specialized at t=1 to produce the constant C1 used in Theorem 1.2 . By contrast, the model’s outline contains several substantive issues: (a) it asserts an “exponential damping of vertical constant shifts” for T−t that contradicts the paper’s rigorous lower bound showing e^{K2 t}-growth of such shifts (Lemma 3.1, Cor. 3.2) ; (b) it constructs ϕε via a bump supported outside U but then incorrectly claims ϕε|U = u+, whereas the paper correctly requires ϕε = u on an open neighbourhood Oε of Iu to force equality in Step 3 ; (c) it relies on an unproven “entrance time of calibrated curves into U,” while the paper avoids this by a direct minimizer localization argument using the A+ blow-up property and monotonicity of implicit action functions . The model’s part (2) also sketches a different route than the paper’s clean argument A∞ ⊂ T−tA1 ⊂ T−t(Bε(u+)), leaving key steps unsupported. In sum, the paper’s results and proof strategy check out, but the model’s solution has sign errors, a flawed data modification, and missing justification of critical steps.