Unique Continuation Properties for solutions to the Camassa-Holm\n equation and other non-local equations

2019/02/08 by Felipe Linares, Gustavo Ponce, Linares, Felipe +1
Mathematics · Physics and Astronomy · #35Q51 #37K10 #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1902.03279

openalex publication_date 2019/02/08 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

It is shown that if ,u(x,t) , is a solution of the initial value problem\nfor the Camassa-Holm equation which vanishes in an open set ,\Ω\⊂\n mathbb R\× [0,T], then ,u(x,t)=0, ,(x,t)\∈ mathbb R\× [0,T]. This\nresult also applies to solutions of the initial periodic boundary value\nproblems associated to the Camassa-Holm equation. The argument of proof can be\nplaced in a general setting to extend the above results to a class of\nnon-linear non-local 1-dimensional models which includes the Degasperis-Procesi\nequation.\n

Citations

• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Wave breaking for nonlinear nonlocal shallow water equations

Related

• On special properties of solutions to Camassa-Holm equation and related models
• Persistence Properties and Unique Continuation of solutions of the Camassa-Holm equation
• Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited
• Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics
• Local well-posedness of the Camassa-Holm equation on the real line