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

2025/05/18 by Mutlubas, Nilay Duruk, Freire, Igor Leite
#35B45 #37K10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.12232

Abstract

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.

Citations

• A look on equations describing pseudospherical surfaces
• Global analytic solutions of a pseudospherical Novikov equation
• Correspondence theorems for hierarchies of equations of pseudo-spherical type
• Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations
• Wave breaking for nonlinear nonlocal shallow water equations
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation

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