In this talk, we are concerned with the existence and stability of time periodic solutions to a forced weakly dissipative Camassa-Holm equation in the real line, which models the shallow water wave propagation with combined dissipative effects. For such a quasilinear equation, the lack of spatial compactness and parabolic smoothing and the nonlocal structure of equation give rise to difficulties in constructing time periodic solutions. First, we transform the original equation into nonlinearly coupled equations by Fourier decomposition and the coefficient-freezing argument. Second, by combining the Poincare map and Duhamel's principle, we reformulate the time periodic problem for transformed equations as the solvability of integral equations. Third, we deduce the estimates of solution operators involved in the integral form and establish the a-prior estimates of solutions by the energy method. Fourth, we prove the unique existence of time periodic solution by the iteration argument and the time periodic extension. The key ingredient of proof lies in the coefficient-freezing argument and the regularizing effect of the inverse Helmholtz operator, which compensates for the regularity loss caused by the lack of smoothing effect. Finally, we introduce the Lyapunov stability of time periodic solutions by the continuity argument. Moreover, we justify the exponential decay of perturbation by constructing the time-weighted functional.
李延硕,河南省科学院数学研究所,助理研究员,研究方向为微分方程与动力系统。主要从事无穷维动力系统时间周期问题的研究,在JNVA,JMP等学术期刊上发表论文6篇,主持河南省级项目1项。