Asymptotically Almost Periodic Solutions to Parabolic Equations on the Real Hyperbolic Manifold

The paper proves existence/uniqueness of an asymptotically almost periodic (AAP) mild solution and exponential stability for the semilinear problem under the abstract semigroup bounds (2.5)–(2.6) and a smallness/Lipschitz regime (Theorem 3.6), using a fixed point on the AAP space and a cone-inequality argument for decay, with explicit constants M and an admissible range for γ (Eq. (3.11)) . The model reproduces the weighted-Volterra estimate and the same explicit admissible γ (up to typographical issues in the paper, e.g., σ^2 appearing where σ/2 is meant), but it does not establish AAP under the general hypotheses of Theorem 3.6: it only guarantees AAP either in the unforced case G(0,·)≡0 or when t↦G(0,t) is already AAP, and it even warns AAP “may fail” if G(0,t) is merely bounded. In contrast, the paper’s contraction on the AAP ball BAAP_ρ yields AAP without assuming G(0,·) is AAP (only bounded and small) . Moreover, the model invokes a shift-commutation ℒS_T = S_Tℒ on R+, which is false due to the causal lower limit 0; the paper correctly circumvents this via a Massera-type decomposition on the whole line and a careful C0/ AP splitting (Theorem 3.5) . Finally, while the paper has minor typos in the γ-range (σ^2 vs σ/2; and denominators (σ−αL) vs (σ−2αL)), the logical route to stability is sound and matches the model’s quantitative mechanism .