ON PERIODIC SOLUTION FOR THE BOUSSINESQ SYSTEM ON REAL HYPERBOLIC MANIFOLDS

The paper proves existence of bounded T-periodic mild solutions to the Boussinesq system on real hyperbolic space H^d (d>2) for p>d via semigroup L^p-L^q estimates due to Pierfelice, a fixed-point on a periodic ball, and an unconditional uniqueness statement for large periodic solutions that assumes small limsup of h near t=0 (Theorems 3.3, 4.2, 4.4). The candidate solution obtains the same main existence result using closely related heat-kernel/smoothing bounds and the L^p-bounded Hodge projector, but formulates a past-history variation-of-constants map and proves uniqueness by a Volterra-type inequality with kernel (t−s)^{−α}, α=1/2+d/(2p)<1, which in fact yields the same uniqueness without the paper’s auxiliary smallness-on-h-near-0 assumption. Thus both arguments are sound; the model’s uniqueness route is slightly sharper but consistent with the paper’s claims. Key elements match: the same exponent α=1/2+d/(2p), an exponentially decaying semigroup on H^d, bilinear estimates for B, and the Poincaré/periodization mechanism for the linear problem (Poincaré map in the paper, (I−e^{−TA})^{-1} in the model). See the paper’s setup and semigroup estimates, matrix formulation and Poincaré map, bilinear bound and contraction, and the uniqueness section for details .