LARGE-TIME ASYMPTOTICS FOR HYPERBOLIC SYSTEMS WITH NON-SYMMETRIC RELAXATION: AN ALGORITHMIC APPROACH

The paper proves Theorem 2.4 via hypocoercive Lyapunov functionals built from Bs(iξA+Ba)^k, obtaining frequency-differential inequalities that, together with the two-frequency bounds defining α and β (Lemma 2.2), yield exactly the stated high/low-frequency L2 decay rates (polynomial with loss governed by α at high frequencies; heat-like t^{-1/(4β)} at low frequencies), including exponential decay when α=0 or β=0. The candidate solution constructs an equivalent frequency-dependent hypocoercive energy E(ξ,t)=⟨P(ξ)Û,Û⟩ with cross-terms between S T^k and S T^{k+1}, derives dE/dt + c Σ w_k(ξ)|S T^kÛ|^2 ≤ 0, invokes the same two-frequency inequalities, and integrates in ξ to reach the same rates. This aligns with the paper’s pointwise-in-ξ version (Remark 3.4) and the dyadic proof in Section 3, differing mainly in presentation (matrix multiplier vs. scalar perturbed energies). No logical gap or contradiction was found. See the paper’s statements of Lemma 2.2 and Theorem 2.4 and the companion Lyapunov constructions for high/low frequencies .