On The Initial Periodic Boundary Value Problems For A Family Of Generalized Camassa-Holm Equations

Camassa-Holm equation is an important model for the shallow water equation and also a model for the propagation of axially symmetric waves in hyperelastic rods,which has a wide range of physical background and has many interesting properties,for instance,blow-up phenomena.The thesis mainly studies blow-up phenomena on the initial periodic boundary value problems for a family of generalized Camassa-Holm equations.Firstly,we study blow-up phenomena by geodesic flow,locally well-posedness,blow-up criterion and blow-up scenario of the initial periodic boundary value problems for the generalized Camassa-Holm equations.We give four blow-up phenomena by applying three methods to construct differential equations in connection with ?u³_xxdx,u_xx(t,q(t,x₀))and (?) u_xx(t,x),which takes advantage of the generalized Camassa-Holm equations.Secondly,we study blow-up phenomena on the initial periodic boundary value problems for two-component Camassa-Holm equations by applying three methods and constructing ordinary differential inequalities in connection with ?u³xdx,u_x(t,q(t,x₀))and (?)u_x(t,x) by using two-component Camassa-Holm equations,and we give 4 theorems about blow-up phenomena for two-component Camassa-Holm equations.Finally,we study blow-up phenomena on the initial periodic boundary value problems for modified two-component Camassa-Holm equations.We get some differential inequations in connection with ?u³_xdx,u_x(t,q(t,x₀)) and (?) u_x(t,x) by making use of similar methods,and we give four blow-up phenomena.