Global and blow-up solutions for a non-local integrable equation with applications to geometry

We establish the global existence of higher-order Sobolev solutions for a non-local integrable evolution equation arising in the study of pseudospherical surfaces and non-linear wave propagation. Under a natural assumption on the initial momentum, we prove that the solution remains globally regular in arbitrary finite-order Sobolev spaces. The proof relies on an inductive energy method involving a hierarchy of functional estimates and applies to both the periodic and non-periodic settings. We determine a criterion for the existence of blow-up solutions. The consequences of these qualitative properties of the solutions on Riemannian surfaces determined by the solutions of the equation are investigated.