Decay of all global CGL solutions on noncompact manifolds

Establish that every global solution of the complex Ginzburg–Landau equation on a noncompact three-dimensional Riemannian manifold satisfying Assumption A converges to zero in the H¹ norm for the full exponent range 1 < p ≤ 5, extending the proved range 7/3 ≤ p ≤ 5.

Background

Theorem 3.1 proves global existence and convergence to zero in H¹ for solutions on noncompact three-dimensional Riemannian manifolds satisfying Assumption A only when 7/3 ≤ p ≤ 5. The authors note that, in the Euclidean case, the full range 1 < p ≤ 5 can be treated using a truncated virial argument.

The same full-range decay statement is expected to hold on the broader class of noncompact manifolds covered by Assumption A, but the manuscript does not provide a proof. This is a concrete unresolved extension of the noncompact long-time theory.