Asymptotic behavior of a generalized Burgers' equation solutions on a finite interval

Physics and Mathematics of Nonlinear Phenomena 2013

Samokhin, Alexey^1

Department of Math., Moscow State Technical University of Civil Aviation, 20 Kronshtadtsky blvd., Moscow, 125493, Russia^1

关键词: Asymptotic behavior of solutions;    Asymptotic behaviors;    Asymptotic limits;    Boundary problems;    Burgers equations;    Finite intervals;    Invariant solutions;    Time-independent solutions;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/482/1/012039/pdf
DOI  :  10.1088/1742-6596/482/1/012039

来源: IOP

【 摘 要 】

The article is concerned with the study of asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value - boundary problem on a finite interval, with constant boundary conditions. Since these equations take a dissipation into account, it is naturally to presuppose that any initial profile will evolve to an invariant time-independent solution with the same boundary values. Yet the answer happens to be slightly more complex. There are three possibilities: the initial profile may regularly decay to an invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or, exotically, an asymptotic limit is a 'frozen multi-oscillation' piecewise-differentiable solution, composed of different smooth invariant solutions.