{"id":"61d9b3d5-ad4c-4b77-846c-e85e19413b6a","arxiv_id":"2501.04998","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Small perturbations of a contact Hamilton-Jacobi equation preserve the existence and stability of viscosity solutions near a Lyapunov stable solution.","lead":"This paper proves that if a contact Hamilton-Jacobi equation has a stable solution, then small perturbations of the equation still have a nearby solution that is also stable. The result gives mathematicians a tool to analyze how small changes affect dissipative dynamical systems described by these equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's stability proof invokes the unperturbed semigroup T^-_t instead of the perturbed T^ε_t, so the Lyapunov-stability/uniqueness conclusion is not proved as written.","rationale":"The reader's weakest_assumption points to Proposition 2.6 (viscosity solutions of (HJε) coincide with fixed points of T^ε_t). I do not regard that as the most load-bearing concern: the equivalence is cited from [24, Proposition 2.7], and the paper explicitly extends Proposition 2.1-2.8 to H^ε, so unless the external theorem is faulty the proof step is legitimate. The genuinely internal soft spot is the wrong semigroup in the final proof of Theorem 1.2: the manuscript states convergence of T^-_t φ to u^ε_-, which cannot establish stability for (HJε). This is concrete and located, and it affects the second main theorem. The fix is readily available by applying Lemma 4.1 to the perturbed system, using Lemma 4.2. The other gaps noted by the reader (boundary handling in Theorem 1.1, brevity of the uniqueness argument) are minor and do not change the verdict. Thus I keep the reader's CONDITIONAL verdict unchanged.","tokens_in":17255,"tokens_out":20431,"duration_ms":191310,"concrete_test":"Re-examine the final paragraph of Section 4: replace 'T^-_t φ' by 'T^ε_t φ' and confirm that Lemma 4.1 (cited from [27, Lemma 2.3]) is applicable to the perturbed Hamiltonian H^ε and the solution u^ε_- whenever ∂H^ε/∂u>0 on Λ_{u^ε_-}. If Lemma 4.2 indeed supplies this condition, the corrected line proves both local asymptotic stability and uniqueness, and the current error is typographical; if Lemma 4.1 requires additional hypotheses that H^ε does not inherit, then Theorem 1.2's stability and uniqueness conclusions are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.2 (Section 4, final paragraph), the authors write 'lim_{t→+∞} T^-_t φ = u^ε_-' for φ near u^ε_-, and use this to conclude that u^ε_- is locally asymptotically stable and unique. The theorem is about the perturbed equation (HJε), whose evolution semigroup is T^ε_t. Convergence under the unperturbed semigroup T^-_t is a statement about the original equation (HJs) and does not imply stability for (HJε). The intended step is to apply Lemma 4.1 to the pair (H^ε,u^ε_-), using the conclusion of Lemma 4.2 that ∂H^ε/∂u>0 on Λ_{u^ε_-}; this would give lim_{t→∞}T^ε_t φ = u^ε_-, which is the needed statement. As written, however, the only bridge from the ∂H/∂u>0 condition to Lyapunov stability for the perturbed problem is broken. Because this is also the step that yields uniqueness among nearby stationary solutions, it is load-bearing for Theorem 1.2. The gap appears to be a correctable typo, not a flaw in the overall strategy, but it should be fixed before the theorem is accepted as proved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies contact Hamilton-Jacobi equations of the form H(x,Du,u)=0 on a closed manifold and their small perturbations H(x,Du,u)+εP(x,Du,u)=0. The main results are Theorem 1.1, which asserts that a locally Lyapunov asymptotically stable viscosity solution u_- of the unperturbed equation persists as a viscosity solution u^ε_- of the perturbed equation with ||u^ε_- - u_-|| ≤ δ for any prescribed δ>0 once ε is sufficiently small, and Theorem 1.2, which adds that if ∂H/∂u>0 on the set Λ_{u_-}, then the nearby perturbed solution is unique and locally asymptotically stable. The proofs rely on weak KAM theory for contact Hamiltonians, the identification of viscosity solutions with fixed points of the backward semigroup, and a new comparison estimate between the perturbed and unperturbed semigroups.","tokens_in":17499,"tokens_out":9196,"duration_ms":80608,"significance":"If the results are correct, they provide a natural robustness theorem for weakly stable solutions of contact Hamilton-Jacobi equations under perturbations, a question that has not been systematically treated before. The main novel technical ingredient is Proposition 2.10, the explicit estimate |T^ε_t φ - T^-_t φ| ≤ (ε/λ)(e^{λt}-1), which is clearly useful beyond this paper. The authors also give a careful compactness argument in Lemma 4.2 to transfer positivity of ∂H/∂u from Λ_{u_-} to Λ_{u^ε_-}. However, the paper as written contains two gaps in the proofs of the main theorems, one in the handling of the stability basin in Theorem 1.1 and one in the use of the unperturbed semigroup in Theorem 1.2; both are local and appear correctable, but they currently leave the central claims not fully proved.","major_comments":[{"comment":"The final paragraph asserts that Lemma 4.2 provides a δ′0 > 0 such that for every φ near u^ε_- one has lim_{t→∞} T^-_t φ = u^ε_-, and then concludes that u^ε_- is locally asymptotically stable and unique for (HJε). This is not correct as written. First, Lemma 4.2 does not state a basin-of-attraction property for u^ε_-; it only states positivity of ∂H^ε/∂u on Λ_{u^ε_-}. Second, the stability notion for a solution of (HJε) uses the perturbed semigroup T^ε_t, not T^-_t, and convergence under T^-_t does not imply convergence under T^ε_t. The intended step is to apply the perturbed analogue of Lemma 4.1 to the pair (H^ε, u^ε_-), using the positivity from Lemma 4.2; this would give lim_{t→∞} T^ε_t φ = u^ε_- and then uniqueness among nearby fixed points of T^ε_t. This load-bearing step is missing, so Theorem 1.2 is not proved as written.","section":"Section 4, proof of Theorem 1.2, final paragraph"}],"minor_comments":[{"comment":"The text says 'By Proposition 2.8, we can find (u^{ε_n}_-, L, 0)-calibrated curve' but this should be '(u^{ε_n}_-, L^{ε_n}, 0)-calibrated curve', since the relevant Lagrangian for the perturbed equation is L^{ε_n}.","section":"Section 4, proof of Lemma 4.2, Step 2"},{"comment":"The step 'u^ε_-(x) = lim_{t→∞} inf_{s≥t} T^ε_s u_-(x) uniformly' is stated as easy to prove; the authors should spell out that this follows from the equi-Lipschitz estimate established just above together with Dini's theorem, since the infimum over s≥t is monotone in t.","section":"Section 3, proof of Theorem 1.1"},{"comment":"The authors state that there is a θ such that H^ε satisfies (H1)-(H3) for ε∈[0,θ]; this should be justified by noting that positive definiteness in p is uniform on the compact support of P, rather than merely pointwise.","section":"Section 2.2, after definition of H^ε"},{"comment":"There are several English and typographical errors, for example 'pertu rbed' in the title and 'does exist viscosity solution' in the abstract; the manuscript should be carefully proofread.","section":"Title and Abstract"}],"recommendation":"major_revision","confidential_remarks":"The two gaps identified above are load-bearing but appear to be correctable within the scope of the manuscript. I recommend major revision rather than rejection, because the main strategy is sound and the required fixes are local: adjust the stability radius in Theorem 1.1 and apply the perturbed analogue of Lemma 4.1 in Theorem 1.2. I do not see a novelty or citation problem; the reliance on [24] and [27] is explicit and appropriate for this line of work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves that a locally Lyapunov asymptotically stable viscosity solution of a contact Hamilton–Jacobi equation persists under small perturbations; the perturbed equation has a nearby viscosity solution, and under a strict monotonicity condition that solution is locally unique and asymptotically stable. That is a legitimate, useful extension of the Wang–Yan–Xu–Zhao framework to parametric perturbations, and it is not in the cited literature. The main existence argument is sound in strategy: compare the perturbed and unperturbed semigroups, control the difference explicitly (Proposition 2.10 is a clean estimate), construct a fixed point by an inf–limiting procedure, and identify it as a viscosity solution. No circularity; the paper relies on external theorems from [21], [24], and [27], not on its own claims.\n\nThe soft spots are real but mostly cosmetic. In Theorem 1.1, the sub/supersolutions are chosen at distance min{δ,δ0} from u−; when δ ≥ δ0 that puts them on the boundary of the Lyapunov basin, not strictly inside it, so the claimed convergence of T^-_t uδ is not exactly justified. That is a minor boundary issue, fixed by taking min{δ,δ0/2}. More serious: the final paragraph of Theorem 1.2 uses the unperturbed semigroup T^-_t to conclude convergence to u^ε_−, the perturbed solution. That is the wrong semigroup. The correct step is to apply Lemma 4.1 to the perturbed Hamiltonian H^ε, using Lemma 4.2’s conclusion that ∂H^ε/∂u > 0 on Λ_{u^ε_−}; then T^ε_t φ → u^ε_−, which is exactly what proves both local asymptotic stability and uniqueness among nearby stationary solutions. As written, the stability and uniqueness conclusions of Theorem 1.2 are not proved. I agree with the stress-test note that this looks like a correctable typo rather than a broken strategy, but it must be fixed before the theorem is accepted.\n\nThe paper is for researchers in weak KAM theory and contact Hamiltonian dynamics. It deserves a serious referee: the problem is meaningful, the main existence theorem is plausible and carefully argued, and the gap in Theorem 1.2 is localized and repairable. I would send it to review and ask for a revised version that corrects the semigroup in Theorem 1.2 and tightens the δ0 boundary point in Theorem 1.1.","headline":"Theorem 1.1 is basically right; Theorem 1.2 has a wrong-semigroup typo that is load-bearing but fixable.","tokens_in":18034,"tokens_out":2162,"would_cite":true,"duration_ms":21586,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35F21","35D40","37J51"],"pacs":[],"model":"deepseek-v4-flash","headline":"Small perturbations of contact Hamilton–Jacobi equations preserve stable viscosity solutions.","keywords":["contact Hamilton-Jacobi equations","viscosity solutions","Lyapunov stability","perturbation theory","backward semigroup","asymptotic stability","Hamilton-Jacobi equations"],"falsifier":"Fix a simple contact Hamiltonian on the circle with a known locally Lyapunov asymptotically stable viscosity solution u_- and a compactly supported P with |P|≤1; if direct computation shows the perturbed equation H+εP=0 has no viscosity solution within δ of u_- for arbitrarily small ε, Theorem 1.1 fails, and if two distinct such solutions coexist for arbitrarily small ε while ∂H/∂u>0 on Λ_{u_-}, Theorem 1.2 fails.","tokens_in":1972,"feed_emoji":"📐","tokens_out":2038,"duration_ms":92789,"temperature":0.7,"pith_summary":"The paper proves that a Lyapunov asymptotically stable viscosity solution of a contact Hamilton–Jacobi equation persists under small smooth perturbations. If u_- solves H(x,D_xu,u)=0 and is locally Lyapunov asymptotically stable, then for every tolerance δ>0 there is an ε_δ>0 such that for all ε∈[0,ε_δ] the perturbed equation H(x,D_xu,u)+εP(x,D_xu,u)=0 has a viscosity solution u^ε_- within δ of u_-. The proof identifies viscosity solutions with fixed points of a backward evolution semigroup and shows the perturbed semigroup has a fixed point near u_-. It then shows that when ∂H/∂u>0 along the lifted solution set, this nearby solution is unique and itself locally Lyapunov asymptotically stable. The significance is that dissipative contact equations, which are not covered by classical conservative Hamilton-Jacobi theory, are shown to be structurally stable around stable stationary states.","feed_headline":"Stable Hamilton-Jacobi solutions survive perturbations","feed_subtitle":"For every small ε the perturbed equation has a nearby solution; strict u-monotonicity makes it unique.","key_machinery":"The central mechanism is the backward solution semigroup {T^ε_t}_{t≥0} acting on C(M,ℝ), whose value T^ε_t φ(x) is the unique viscosity solution of the evolutionary equation ∂_t u + H(x,D_xu,u)+εP(x,D_xu,u)=0 with initial data φ. A theorem from the variational theory of contact Hamiltonians says that a function is a stationary viscosity solution of the time-independent equation exactly when it is a fixed point of every T^ε_t. The paper compares the unperturbed and perturbed semigroups through the bound ‖T^ε_t φ - T^-_t φ‖_∞ ≤ (ε/λ)($e^{{λt}}$-1), obtained from the Legendre transforms and the Lipschitz-in-u property. Lyapunov asymptotic stability of u_- is then used to trap the iterates T^ε_t u_- in a shrinking interval, and a liminf construction produces a fixed point, hence a viscosity solution, uniformly close to u_-. The strict sign condition ∂H/∂u>0 on Λ_{u_-} transfers to the perturbed solution by a compactness and Arzelà-Ascoli argument, yielding uniqueness and asymptotic stability.","core_discovery":"On a closed Riemannian manifold, for a $C^{3}$ contact Hamiltonian H(x,p,u) satisfying positive definiteness, superlinearity, and uniform Lipschitz dependence on u, the authors establish the following. If u_-∈S^- is a viscosity solution of H(x,D_xu,u)=0 that is locally Lyapunov asymptotically stable with respect to the backward semigroup {T^-_t}, then for any δ>0 there exists ε_δ>0 such that for every ε∈[0,ε_δ] the perturbed equation admits a viscosity solution u^ε_- with ‖u^ε_- - u_-‖_∞≤δ. If in addition ∂H/∂u>0 on Λ_{u_-}, the closure of the set of points (x,D_xu_-(x),u_-(x)) at differentiability points of u_-, then for δ,ε small the solution is unique in the δ-neighbourhood of u_- and is itself locally Lyapunov asymptotically stable.","pith_inferences":["A natural testable extension, not pursued in the paper, is to allow unbounded C^3 perturbations P with bounded u-derivative; the comparison estimate behind Proposition 2.10 may still hold with modified constants, so the persistence theorem could extend beyond compact support.","Because the proof constructs u^ε_- as a liminf of the semigroup trajectory, one could ask whether the limsup construction gives a different fixed point when uniqueness fails, and whether the interval of viscosity solutions shrinks to a point as ε→0.","The result suggests a selection principle: among several unperturbed stable solutions, a small perturbation keeps a nearby solution near each one, leaving open the global question of which stable branch is selected as ε varies.","In applications to dissipative contact Hamiltonians, such as thermodynamic models, the theorem means small parameter changes do not destroy stationary regimes that are already asymptotically stable; this is a structural-stability statement in the C^0 topology of solutions."],"forward_implications":["For every small ε, the perturbed Hamiltonian H+εP is admissible at zero: equation (HJε) admits a viscosity solution, not merely for specially chosen perturbations P.","The solution u^ε_- converges uniformly to u_- as ε→0, so stable stationary states of the unperturbed dissipative system have nearby stationary states after perturbation.","Under the strict monotonicity condition ∂H/∂u>0, the nearby stationary state is the unique viscosity solution within the specified neighbourhood, so the perturbation does not create alternative solutions there.","The perturbed stationary state inherits local Lyapunov asymptotic stability, so the qualitative dynamics near u_- is preserved for small perturbations.","The explicit choice ε_δ := λ min{δ,δ_0}/(2(e^{λ t_δ}-1)) makes the allowed perturbation size depend on the unperturbed contraction time t_δ and the Lipschitz constant λ."],"supporting_citations":[{"why":"Supplies the fixed-point characterization: backward weak KAM solutions coincide with viscosity solutions, and a function is a viscosity solution exactly when it is a fixed point of the backward semigroup.","marker":"[24, Proposition 2.7]"},{"why":"Provides the representation formula for the backward solution semigroup and its monotonicity, Lipschitz, and semigroup properties that the paper transfers to the perturbed semigroup.","marker":"[23, Theorem 1.1]"},{"why":"Supplies the implicit variational principle and implicit action function, the foundation for the semigroup representation and the calibrated-curve arguments used in the proof of Theorem 1.2.","marker":"[22, Theorem A]"},{"why":"Gives the criterion that ∂H/∂u>0 on Λ_{u_-} implies local asymptotic stability, which the paper uses to transfer stability to the perturbed solution.","marker":"[27, Lemma 2.3]"},{"why":"Defines the ergodic interval of constants for which contact Hamilton-Jacobi equations admit viscosity solutions; the admissibility assumption 0∈C is the premise that the unperturbed equation has a viscosity solution.","marker":"[25]"}],"fun_headline_variants":["Perturbed contact HJ equations retain stable viscosity solutions","Stable viscosity solutions persist under small perturbations","Nearby viscosity solution exists for each small epsilon","Uniform convergence to stable solution under perturbation","Stable viscosity solutions for perturbed contact HJ"],"cache_read_input_tokens":20224,"weakest_assumption_plain":"The load-bearing premise is that viscosity solutions of the contact Hamilton-Jacobi equation are exactly the fixed points of the backward solution semigroup {T^-_t}; if that equivalence fails for contact Hamiltonians satisfying (H1)-(H3), the construction of a fixed point of {T^ε_t} would not produce a viscosity solution of the perturbed equation.","fun_headline_variants_meta":{"raw":{"variants":["Perturbed contact HJ equations retain stable viscosity solutions","Stable viscosity solutions persist under small perturbations","Nearby viscosity solution exists for each small epsilon","Uniform convergence to stable solution under perturbation","Stable viscosity solutions for perturbed contact HJ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000583,"raw_usage":{"total_tokens":2725,"prompt_tokens":910,"completion_tokens":1815,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":1746}},"tokens_in":526,"tokens_out":1815,"duration_ms":11535,"temperature":1.0,"reasoning_tokens":1746,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:20:35.702696+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a simple contact Hamiltonian on the circle with a known locally Lyapunov asymptotically stable viscosity solution u_- and a compactly supported P with |P|≤1; if direct computation shows the perturbed equation H+εP=0 has no viscosity solution within δ of u_- for arbitrarily small ε, Theorem 1.1 fails, and if two distinct such solutions coexist for arbitrarily small ε while ∂H/∂u>0 on Λ_{u_-}, Theorem 1.2 fails.","supporting_citations":[],"review_version":1}