Infinite Propagation Speed And Asymptotic Behavior For Two Generalized Camassa-Holm Equations

As a model for surface water waves in the shallow water regime,the Camassa-Holm equation attracts much attention and interest from scholars both at home and abroad.This paper is devoted to the infinite propagation speed and the asymptotic behavior for a generalized Camassa-Holm equation with a complex parameter ε and a fifth-order Camassa-Holm equation.Firstly,assuming initial datas m0 and u0 to the two equations have compact supports,we establish that the solution m(x,t)has the compact support by introducing the family {φ(·,t)}t∈[0,T)of diffeomorphisms and prove that the solution u(x,t)which keeps the property of having compact support is the trivial solution u≡0.Although the nontrivial solution u(x,t)is no longer compactly supported,we prove that the solution u(x,t)has an exponential decay as x goes to infinity.At last,we prove that the solution keeps the corresponding properties at infinity within its lifespan as the initial data decays algebraically.