Asymptotic Stability of Monotone Increasing Kinked Solitary Wave Solution for the Generalized Boussinesq Equation with Dissipative Term

In this paper, the asymptotic stability of monotonically increasing kinked soliton solution to the generalized Boussinesq equation with dissipation term is studied by using the the antiderivative method. In order to overcome the difficulty of integral estimate about perturbed solution caused by higher-order derivatives in the equation, several important properties of the nonlinear part for the studied equation and the wave function are first given and proved in this paper. Furthermore, according to the correspondence between the solitary wave solution and the solution trajectory in the plane system, the second-order derivative expression concerning the wave function is given by considering the asymptotic behavior of the solution trajectory at infinity, so that we can successfully obtain the integral estimates of product terms about the second-order derivative of the wave function and the perturbed solution, and then obtain the uniform a priori estimate of studied solitary wave solution with respect to the perturbation, and prove the monotonically increasing kinked soliton solution for the generalized Boussinesq equation with dissipation term is asymptotically stable.