Dynamics of a state-dependent delay-differential equation

The paper formulates the scalar DDE with threshold state-dependent delay, defines the delay via the threshold integral, and proves that the delay functional δ is C^1 with an explicit derivative formula (Proposition 2.1). It then verifies the smoothness hypotheses needed to obtain a C^1 solution manifold X_G and a C^1 semiflow S_G (by checking property (2) in Walther's framework), derives uniform bounds a/v_U ≤ δ(φ) ≤ a/v_0 under v bounded above, and establishes global existence, absorption, and positivity (Propositions 3.1–3.2). Finally, it proves the existence of a compact global attractor A_G with strictly positive histories (Theorem 3.4). These steps and conclusions match the candidate solution’s outline almost point-for-point: the candidate derives the same δ-existence and derivative formula via the Implicit Function Theorem, obtains the same delay bounds, formulates the same right-hand side G, proves local well-posedness and a C^1 semiflow on X_G, derives a priori bounds leading to point-dissipativity, shows asymptotic compactness (via an x″ bound and Arzelà–Ascoli), and concludes the existence of a compact global attractor with strict positivity. The only minor difference is stylistic: the paper uses Walther’s smoothness condition (S) rather than asserting outright Fréchet C^1-ness of G on all of C^1, and establishes precompactness by a direct equicontinuity argument, whereas the model bounds x″ to invoke Arzelà–Ascoli in C^1. Substantively, however, they prove the same results with closely aligned arguments (paper definitions and proofs: model setup and steps). Key paper loci: model definition and threshold integral (equations (1.1)–(1.2)), δ defined by the threshold integral and C^1 with derivative (equation (2.3) and Proposition 2.1), solution manifold and semiflow (verification of property (2) and ensuing statement), bounds a/v_U ≤ δ ≤ a/v_0, global existence and absorption (Propositions 3.1–3.2), and the global attractor with positivity (Theorem 3.4) .