Global weak solutions to the weakly dissipative Degasperis-Procesi equation
The authors consider the Degasperis-Procesi equation with a dissipative term, \[ (1 - \partial_x^2) u_t + 4 u u_x - 3 u_x u_{xx} - u u_{xxx} + \lambda (1 - \partial_x^2) u = 0, \] where \(\lambda > 0\) is the dissipation constant. Wave breaking and decay of the solutions were analyzed for this equation by \textit{S. Wu} and \textit{Z. Yin} [SIAM J. Math. Anal. 40, No. 2, 475--490 (2008; Zbl 1216.35126)]. Local well-posedness, blow-up scenario, global existence of strong solutions were investigated by \textit{Z. Guo} [J. Differ. Equations 246, No. 11, 4332--4344 (2009; Zbl 1170.35083)]. In the present article, the authors consider the global weak solutions to the dissipative Degasperis-Procesi equation in the energy space \(H^1\). The technique is inspired by similar works of \textit{A. Constantin} and \textit{L. Molinet} [Commun. Math. Phys. 211, No. 1, 45--61 (2000; Zbl 1002.35101)] and \textit{J. Escher, Y. Liu} and \textit{Z. Yin} [J. Funct. Anal. 241, No. 2, 457--485 (2006; Zbl 1126.35053)] in the context of the Camassa-Holm and conservative Degasperis-Procesi equations, respectively. After details of analysis, the authors also compare their results with the outcomes of the recent work of \textit{Z. Yin, S. Lai} and \textit{Y. Guo} [Comput. Math. Appl. 60, No. 9, 2645--2652 (2010; Zbl 1205.76049)] for a model shallow-water equation.
scientific article; zbMATH DE number 5925791
scientific article