This work deals with the dissipative generalized Korteweg-de Vries (gKdV) equations of the formu t + u 2u x + u xxx-bu xx+ ru = f, t≥0, u(0,x) = u 0(x)∈V = H 2 2π,with periodic boundary conditions. It is proved tha...This work deals with the dissipative generalized Korteweg-de Vries (gKdV) equations of the formu t + u 2u x + u xxx-bu xx+ ru = f, t≥0, u(0,x) = u 0(x)∈V = H 2 2π,with periodic boundary conditions. It is proved that there exists an inertial manifold for the semiflow generated by this equation in space V. Since such a manifold is finite dimensional, positively invariant, and exponentially attracting of all the solution trajectories, the long-time dynamics of the dissipative gKdV equations are determined by a finite number of modes without the soliton phenomena.展开更多
文摘This work deals with the dissipative generalized Korteweg-de Vries (gKdV) equations of the formu t + u 2u x + u xxx-bu xx+ ru = f, t≥0, u(0,x) = u 0(x)∈V = H 2 2π,with periodic boundary conditions. It is proved that there exists an inertial manifold for the semiflow generated by this equation in space V. Since such a manifold is finite dimensional, positively invariant, and exponentially attracting of all the solution trajectories, the long-time dynamics of the dissipative gKdV equations are determined by a finite number of modes without the soliton phenomena.
