{"id":"69579e0e-0ef8-45ae-82fb-42dce3f2f0d4","arxiv_id":"2502.08315","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Morse-Smale semigroups in Hilbert spaces satisfy Lipschitz shadowing on their global attractor and Hölder shadowing in any positively invariant bounded neighborhood, without requiring an inertial manifold.","lead":"This paper proves that Morse-Smale semigroups in Hilbert spaces have a shadowing property: every approximate trajectory is close to a true trajectory, both on the global attractor and in a bounded neighborhood. This extends a classical finite-dimensional result to infinite-dimensional systems, including damped wave equations, where previous methods requiring inertial manifolds do not apply.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted proof of Lemma 3.19 is load-bearing: lower semicontinuity of S is used in Lemma 3.23 to prove induction hypothesis (3.14) for Theorem 3.21, and the stated justification is effectively circular.","rationale":"The reader's weakest assumption coincides with what I regard as the most load-bearing gap. The fixed-point application in Theorem 3.32 is sketched and would also benefit from verification, but the subbundle construction is the structurally novel part and it is where the omitted lemma sits. The authors explicitly acknowledge not proving Lemma 3.19, and the remark that the proof is similar to Theorem 3.21 creates a circular dependency because Theorem 3.21 uses Lemma 3.23, which uses Lemma 3.19. The surrounding architecture is coherent: the Holder-shadowing part is largely independent via Lemma 4.1, the exponential-attraction statement is standard, and the applications are plausible conditional on the main theorem. Thus the gap is fixable rather than fatal, and the appropriate recommendation is unchanged from the reader's CONDITIONAL verdict.","tokens_in":44349,"tokens_out":9743,"duration_ms":99436,"concrete_test":"Write out the proof of Lemma 3.19 for the family S constructed in Proposition 3.8 using only (H1), (H2), and the local C^1 graph representation of W^s_loc near each equilibrium. If the proof requires the S_i construction from Theorem 3.21, re-derive Theorem 3.21 in a way that avoids Lemma 3.19, for example by proving (3.31) directly from the explicit structure of J_j and the continuity of U_j. A second useful check: in the linearization of the damped wave equation around a hyperbolic equilibrium, compute the extended stable subspaces along a heteroclinic connection and verify the lower-semicontinuity condition of Definition 3.18 at a point where the first-hitting time t_0(x) is discontinuous.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.21 constructs the compatible subbundles S_i and U_i. The construction of U_i proceeds by induction and assumes condition (3.14): S(x)+U_j(x)=X for x in gamma^+(O_j). This condition is not automatic when D_xT(t) is only an isomorphism onto its range, so Lemma 3.23 is the dedicated proof. The proof of Lemma 3.23 passes from the compact set J_j, where S(x)+U_j(x)=X, to the neighborhood B_A(J_j,epsilon_2) by invoking Lemma 3.19, the lower semicontinuity of the extended stable subbundle S. But Lemma 3.19 is stated without proof; the text says the strategy is 'similar to the construction of the subbundles S_i in Theorem 3.21'. Since Theorem 3.21 itself depends on Lemma 3.23, and hence on Lemma 3.19, to justify (3.14), there is no independent proof of the lower semicontinuity of S. Lower semicontinuity is not a formal consequence of continuity of projection-valued maps in infinite dimensions, and the extension of S in Proposition 3.8 involves preimages under non-surjective derivatives, so this is a real hypothesis. If Lemma 3.19 fails, (3.31) is unsupported, the induction hypothesis (3.14) cannot be propagated, and the compatible subbundles S_i, U_i -- and therefore the Lipschitz shadowing conclusion of Theorem 3.32 -- do not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims an infinite-dimensional generalization of the Lipschitz and Hölder shadowing theorems for Morse-Smale semigroups. Specifically, Theorem 1.1 asserts that for a C^1 Morse-Smale semigroup on a Hilbert space with global attractor A, satisfying growth and continuity conditions (H1) and (H2) on the Fréchet derivative, the time-one map T(1) restricted to A has the Lipschitz shadowing property, and its restriction to any positively invariant bounded neighborhood U of A has the α-Hölder shadowing property for some α∈(0,1). The proof adapts Pilyugin's finite-dimensional compatible-subbundle construction, with substantial new work to handle the non-surjectivity of derivatives in infinite dimensions. Applications to structural stability and to rates of convergence of global attractors are also given.","tokens_in":44648,"tokens_out":4009,"duration_ms":42243,"significance":"If the proof is completed, this is a significant result: it removes the inertial-manifold assumption that was previously needed to obtain shadowing in a neighborhood of the attractor, and it applies to equations such as the damped wave equation. The paper also provides a new route to Hölder estimates between attractors and to orbit-tracking statements for perturbed Morse-Smale semigroups. The claimed theorems are concrete and falsifiable, and the overall strategy is well motivated by the finite-dimensional theory. However, the central proof currently rests on a lemma that the authors explicitly decline to prove, and that lemma is used in a load-bearing way, so the manuscript is not yet complete.","major_comments":[{"comment":"Lemma 3.19 asserts that the extended stable subbundle S(x) is lower semicontinuous on the global attractor, but the proof is omitted; the text says the strategy is 'similar to the construction of the subbundles S_i in Theorem 3.21' and that the authors 'choose to prove Theorem 3.21 rigorously instead.' This is circular: Theorem 3.21 uses Lemma 3.19 (via Lemma 3.23) to establish the induction hypothesis (3.14), and Lemma 3.19 is not independently proved. Lower semicontinuity of S is not a formal consequence of the continuity of projection-valued maps in infinite dimensions, especially because the extension in Proposition 3.8 uses preimages under non-surjective derivatives. The main Lipschitz shadowing conclusion of Theorem 3.32 therefore depends on an unproved statement. A complete proof of Lemma 3.19, or an alternative argument that does not rely on Theorem 3.21, must be supplied.","section":"Section 3.1, Lemma 3.19"},{"comment":"Equation (3.31) is the key step that moves the transversality condition S(y)+U_j(y)=X from the compact set J_j to a neighborhood, and the proof of (3.31) invokes Lemma 3.19. Without Lemma 3.19, the induction hypothesis (3.14) cannot be propagated from j+1,...,p to i, and the construction of the compatible subbundles U_i and S_i collapses. This is not a presentation issue but a load-bearing gap in the proof of the paper's central theorem.","section":"Section 3.2, Lemma 3.23 and Theorem 3.21"},{"comment":"The displayed definition of S'_i(x) as R(P_u(x)) is apparently a typo: it should be R(P_s(x)). As written, both S'_i and U'_i are defined by the same projection, which would make (3.4) and (3.5) inconsistent. The intended construction is clear from the preceding sentence, but the typo should be corrected because Lemma 3.9 is used to produce the local subbundles in Theorem 3.21.","section":"Section 3.1, Lemma 3.9"},{"comment":"The proof states that items (4) and (5) follow from hyperbolicity and continuity of the subbundles, but continuity of S_i is only asserted on O_i, while the estimates (3.12) and (3.13) are claimed for all x in O_i along forward/backward times in [0,1]. The passage from the local subbundle continuity to these uniform exponential estimates is only sketched, and the non-surjectivity of D_xT(t) makes this step nontrivial. A few lines of justification are needed here, especially for the backward estimate involving U_i.","section":"Section 3.2, Theorem 3.21, item (5)"}],"minor_comments":[{"comment":"There are several typographical errors: 'eloborated' should be 'elaborated', 'stablish' should be 'establish', 'Bihrkhoff' should be 'Birkhoff', and 'It holds' should be lowercase in Theorem 4.7.","section":"Throughout"},{"comment":"In the statement of Lemma 3.27, the hypotheses introduce subbundles P(x), Q(x), Z(x), but the body of the lemma then refers to S(x), U(x), V(x). These notations should be aligned for readability.","section":"Section 3.3, Lemma 3.27"},{"comment":"In the compactness argument, the use of Proposition 6.4 with σ_k=k and ξ_k(s)=ψ_k(s+k) is confusing because ψ_k is defined as ξ_k(·−k), making the substitution tautological. The intended limiting argument is clear, but the exposition should be rephrased.","section":"Section 3.2, Lemma 3.22"},{"comment":"The constants C_0 and C_1 introduced in (3.42) collide with the Lipschitz constant C_1 used in Lemma 3.24; this notational overlap makes the text harder to follow.","section":"Section 3.3, proof of Lemma 3.31"},{"comment":"The statement says 'S_0 is a Morse-Smale semigroup (autonomous)' while the family {S_ϵ} consists of evolution processes; this is understandable but should be phrased more carefully to avoid ambiguity.","section":"Section 5, Theorem 5.2"}],"recommendation":"major_revision","confidential_remarks":"The core idea is promising and the applications are relevant, but the omitted proof of Lemma 3.19 is a genuine gap in the main theorem, not a cosmetic issue. The authors' own text acknowledges the omission, so it cannot be treated as an artifact. I would be willing to review a revised version that supplies a complete proof of Lemma 3.19 or replaces the circular justification with an independent argument. The paper otherwise has solid context and appears to follow the standard roadmap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one for the main theorem, not for the proof as written. Arrieta, Carvalho and Takaessu claim the first Lipschitz shadowing result on the global attractor of an infinite-dimensional Morse-Smale semigroup that avoids an inertial manifold, plus a Holder shadowing statement in a neighborhood. If correct, that is a genuine advance: the damped wave equation satisfies their hypotheses while failing the spectral gap condition needed for inertial manifolds, so the result covers a natural class that previous shadowing results did not. The applications to structural stability and attractor continuity are plausible and well motivated.\n\nThe paper does real work. Section 3 adapts Pilyugin and Robinson's compatible subbundles to the non-invertible infinite-dimensional setting, and the technical machinery (Lemmas 3.12, 3.15, 3.17, Proposition 3.28) is nontrivial. The Holder neighborhood part in Section 4 is cleaner and mostly independent; Lemma 4.1's distance-to-attractor estimate is a useful tool in its own right.\n\nThe soft spot is exactly where the stress-test lands. Lemma 3.19 says S and U are lower semicontinuous, and for S it is stated with no proof, with the text pointing to Theorem 3.21. But Lemma 3.23 uses Lemma 3.19 to prove the induction hypothesis (3.14) that Theorem 3.21 needs. So the crucial lower semicontinuity of the stable subbundle is not established independently. This is load-bearing, not cosmetic. The proof of Theorem 3.21 as written is therefore conditional on an unproved lemma that is justified by the very construction it supports. The same section also has local typos: in Lemma 3.9 the definition of S'_i uses R(Pu) where it should be R(Ps). These are fixable, but the lower semicontinuity gap is substantive and should be resolved before the claim is relied on.\n\nMy guess is the theorem is true and the gap is repairable. The strategy is coherent and the finite-dimensional roadmap is credible. But this is not a paper where a referee can rubber-stamp the proof. It is for researchers in infinite-dimensional dynamics who want the latest on shadowing for dissipative PDEs, and it deserves serious refereeing. I would not cite it as a finished result until the gap in Lemma 3.19 is closed.","headline":"First shadowing result for infinite-dimensional Morse-Smale systems without inertial manifolds, but the proof has a load-bearing gap: lower semicontinuity of S is unproved and justified circularly.","tokens_in":45171,"tokens_out":2250,"would_cite":false,"duration_ms":22986,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37L05","35R15","37D05","37L45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Morse-Smale semigroups in Hilbert spaces have Lipschitz shadowing on their global attractors and Hölder shadowing in neighborhoods, without requiring an inertial manifold.","keywords":["Shadowing","Morse-Smale semigroups","Global attractor","Hilbert space","Hölder shadowing","Lipschitz shadowing","Structural stability","Continuity of attractors"],"falsifier":"Check whether the stable subbundle S(x) constructed in Proposition 3.8 is lower semicontinuous at every point of the global attractor for a concrete Morse-Smale damped wave equation on a bounded three-dimensional domain; if some point admits no continuous family of subspaces below S(x0), then Lemma 3.19 is false as stated and the induction in Theorem 3.21 loses its justification.","tokens_in":44153,"feed_emoji":"🔄","tokens_out":6283,"duration_ms":65151,"temperature":0.7,"pith_summary":"The paper proves that Morse-Smale semigroups in Hilbert spaces possess a strong approximation property known as shadowing: any approximate orbit of the time-one map on the global attractor is within a bounded distance of a true orbit, and the error scales linearly with the approximation error. Outside the attractor, in any positively invariant bounded neighborhood, the same is true with a Hölder rather than linear error bound. This matters because the main examples, such as damped wave equations, cannot be reduced to a finite-dimensional inertial manifold, and the prior finite-dimensional shadowing theorems did not apply. The proof adapts the finite-dimensional compatible-subbundle construction, handling the fact that the derivative need not be surjective in infinite dimensions.","feed_headline":"Time-one maps shadow pseudo-orbits on infinite-dimensional attractors","feed_subtitle":"Morse-Smale systems shadow pseudo-orbits on the attractor and Hölder-shadow nearby, with no inertial manifold.","key_machinery":"The central object is the family of compatible subbundles: closed subspaces S(x) and U(x) splitting the Hilbert space at each point x of the attractor, positively invariant under the derivative and exponentially contracting in forward time on S(x) and in backward time on U(x). These subbundles let the proof transfer the finite-dimensional Lipschitz shadowing argument to infinite dimensions, despite the derivative not being surjective. The construction proceeds by induction over the equilibria, using continuity and lower semicontinuity of the subbundles, inclination estimates near unstable manifolds, and a final Banach fixed point argument that produces the shadowing orbit.","core_discovery":"The paper's central claim is that Morse-Smale semigroups in Hilbert spaces, even when the dynamics cannot be projected onto a finite-dimensional inertial manifold, still satisfy strong shadowing. Specifically, for a semigroup {T(t)} with global attractor A and non-wandering set consisting only of hyperbolic equilibria, and with Fréchet derivatives obeying the exponential-growth and continuity conditions (H1) and (H2), the time-one map T(1) restricted to A has Lipschitz shadowing: every δ-pseudo-orbit in A lies within Lδ of a true orbit. Moreover, on any positively invariant bounded neighborhood U of A, the map T(1)|_U has α-Hölder shadowing for some 0<α<1. This is the first such neighborhood shadowing result in infinite dimensions that does not pass through a finite-dimensional reduction.","pith_inferences":["If Lemma 3.19's lower semicontinuity claim fails for some admissible Morse-Smale semigroup, the induction proving S(x)+U_j(x)=X in Theorem 3.21 loses its justification; the theorem's conclusion could survive, but the proof would need a different transversality argument.","The proof does not identify the Hölder exponent α or the shadowing constants explicitly; reading the estimate in Lemma 4.1 as a recipe would give concrete exponents in terms of the Lipschitz constant of T(1)|_U and the exponential attraction rate, making the result quantitative.","Because the construction only needs the derivative to be an isomorphism onto its range along the attractor, the same compatible-subbundle method should apply to gradient semigroups whose time-one map is non-invertible outside the attractor, as long as the attractor dynamics remain invertible.","A direct stress test of the method would be to check whether the stable subbundle S(x) is lower semicontinuous for parabolic systems with critical nonlinearities, where unstable manifolds may have infinite codimension; a failure there would not automatically disprove the shadowing conclusion, but it would require replacing the current proof."],"forward_implications":["For the damped wave equation, T(1) has Hölder shadowing in any positively invariant bounded neighborhood of its attractor, even though no inertial manifold exists.","Small perturbations of a Morse-Smale semigroup have global orbits that stay close to global orbits of the unperturbed system, with the gap bounded by a Hölder power of the perturbation size.","The distance between global attractors can be controlled by the time-one map distance with any Hölder exponent less than one, under exponential attraction and subexponential Lipschitz growth.","The finite-dimensional Lipschitz shadowing theorem for Morse-Smale systems with only equilibrium non-wandering points is recovered as a special case.","Non-autonomous perturbations of Morse-Smale semigroups also inherit orbit proximity, as shown in Theorem 5.2."],"supporting_citations":[{"why":"Supplies the finite-dimensional proof of Lipschitz shadowing for Morse-Smale systems, including the Banach fixed point theorem that the infinite-dimensional proof reuses.","marker":"[42]"},{"why":"Introduced the compatible subbundles whose construction the paper adapts to non-surjective derivatives.","marker":"[48]"},{"why":"Provides the Morse-Smale semigroup framework, stable and unstable manifolds, and structural stability results that the paper extends to orbit proximity.","marker":"[8]"},{"why":"Establishes well-posedness, differentiability, and conditions (H1) and (H2) for the damped wave equation, the main example without an inertial manifold.","marker":"[17]"},{"why":"Shows exponential attraction of global attractors for gradient semigroups, used to estimate pseudo-orbit distance to the attractor.","marker":"[14]"},{"why":"Provides the Lipschitz shadowing result in R^n and the fixed point lemma that the neighborhood argument leans on.","marker":"[53]"},{"why":"Supplies the inclination argument used to glue continuously extended subbundles near unstable manifolds.","marker":"[37]"}],"fun_headline_variants":["Lipschitz shadowing on infinite-dimensional attractors","Infinite-dimensional shadowing without inertial manifolds","Morse-Smale semigroups: shadowing in Hilbert space","No finite reduction: shadowing in infinite dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the stable subbundles S(x) to be lower semicontinuous across the whole attractor; the paper states this without a full proof, and the compatible-subbundle induction leans on it at each step.","fun_headline_variants_meta":{"raw":{"variants":["Lipschitz shadowing on infinite-dimensional attractors","Infinite-dimensional shadowing without inertial manifolds","Morse-Smale semigroups: shadowing in Hilbert space","No finite reduction: shadowing in infinite dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000766,"raw_usage":{"total_tokens":3369,"prompt_tokens":890,"completion_tokens":2479,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":2414}},"tokens_in":506,"tokens_out":2479,"duration_ms":19709,"temperature":1.0,"reasoning_tokens":2414,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T05:35:00.530302+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the stable subbundle S(x) constructed in Proposition 3.8 is lower semicontinuous at every point of the global attractor for a concrete Morse-Smale damped wave equation on a bounded three-dimensional domain; if some point admits no continuous family of subspaces below S(x0), then Lemma 3.19 is false as stated and the induction in Theorem 3.21 loses its justification.","supporting_citations":[{"cited_title":"Shadowing in Dynamical Systems.Springer-Verlag (1999)","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-dimensional proof of Lipschitz shadowing for Morse-Smale systems, including the Banach fixed point theorem that the infinite-dimensional proof reuses."},{"cited_title":"Structural stability of vector fields.Annals of Mathematics","cited_arxiv_id":null,"evidence_quote":"Introduced the compatible subbundles whose construction the paper adapts to non-surjective derivatives."},{"cited_title":"& Langa, J","cited_arxiv_id":null,"evidence_quote":"Provides the Morse-Smale semigroup framework, stable and unstable manifolds, and structural stability results that the paper extends to orbit proximity."},{"cited_title":"& Robinson, J","cited_arxiv_id":null,"evidence_quote":"Establishes well-posedness, differentiability, and conditions (H1) and (H2) for the damped wave equation, the main example without an inertial manifold."},{"cited_title":"& Cholewa, J","cited_arxiv_id":null,"evidence_quote":"Shows exponential attraction of global attractors for gradient semigroups, used to estimate pseudo-orbit distance to the attractor."},{"cited_title":"Distance of attractors of evolutionary equations.Ph","cited_arxiv_id":null,"evidence_quote":"Provides the Lipschitz shadowing result in R^n and the fixed point lemma that the neighborhood argument leans on."},{"cited_title":"On morse-smale dynamical systems.Topology","cited_arxiv_id":null,"evidence_quote":"Supplies the inclination argument used to glue continuously extended subbundles near unstable manifolds."}],"review_version":1}