Existence of a polynomial attractor for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity

The uploaded paper establishes a contractive-function criterion and applies it to the same wave equation with nonlocal weak damping and critical nonlinearity, proving existence of a polynomial attractor and the explicit attraction rate in Theorem 3.2; the final bound matches the statement the candidate addresses, including the coefficient pk Cp/(2p+2) and the entering-time shift t*(B) + 1 . The paper’s proof is complete: it derives a key quasi-stability inequality via a concavity argument and a monotonicity estimate (cf. (3.12)–(3.15)), verifies precompact pseudometrics using Simon’s compactness framework, and then invokes its abstract Theorem 2.6 to obtain α-decay and the polynomial attractor (see (2.14)–(2.20) and (3.23)) . The candidate’s solution outlines the same main ingredients (absorbing set, quasi-stability, contractive-function framework, α-decay ⇒ φ-attractor), and gives the same final estimate. It differs technically (e.g., uses a one-step discrete “nonlinear contraction” q(s) and a slightly different set of pseudometrics), and cites the companion preprint for the abstract machinery. While some intermediate steps in the candidate’s sketch are asserted more tersely than in the paper (and one reference number differs), the approach is consistent with the paper’s framework and aims at the same conclusion. Hence, both are correct, with proofs that are conceptually related but not identical.