Global attractor for Camassa-Holm type equations with dissipative term
The global existence of classical solution and global attractor for Camassa-Holm type equations with dissipative term are established by using fixed-point theorem and a priori estimates. The authors consider the Camassa-Holm type equations with dissipative term \[ u_t-u_{xxt}+\delta u_{xxxx}+f(u)_x= \varepsilon ({2u_x u_{xx}+uu_{xxx} })+g(x),\quad x \in [0,1],\;t > 0.\tag{1} \] Under nonlinear boundary condition \[ u({0,t})= u_x ({1,t})= u_{xx} ({0,t})= 0,\quad t > 0,\tag{2} \] \[ \delta u_{xxx} ({1,t})-\varepsilon u({1,t})u_{xx} ({1,t})= \varphi ({u({1,t})}),\quad t > 0, \tag{3} \] and the initial value \[ u({x,0})= u_0 (x),\quad x \in [0,1], \tag{4} \] where \(\delta > 0,\) and \(\varepsilon \) are real numbers. When \(\delta > 0, \varepsilon= 1,\quad f(u)= \frac{3}{2}u^2,\) and \(g(x)= 0,\) equation (1) becomes the Camassa-Holm equation which was proposed as a model for one directional long waves of small-amplitude in shallow water. In this paper, the existence of global classical solution and global attractor for problem (1)--(4) is studied. In Section 2, the local existence of classical solution is proved. In Section 3, the authors establish the global existence of smooth solution and in Section 4, they prove the existence of global attractor.
scientific article; zbMATH DE number 5008209