October  2009, 23(4): 1253-1275. doi: 10.3934/dcds.2009.23.1253

Well-posedness for regularized nonlinear dispersive wave equations

1. 

Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago , 851 S. Morgan Street MC 249, Chicago, Illinois 60607-7045

2. 

Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, and Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, Illinois

Received  August 2007 Revised  December 2007 Published  November 2008

In this essay, we study the initial-value problem

$ u_t+u_x+g(u)_x+Lu_t=0, \qquad x\in\mathbb R,\quad t>0,$
$u(x,0)=u_0(x), \qquad x\in\mathbb R, (0.1)$

where $u=u(x,t)$ is a real-valued function, $L$ is a Fourier multiplier operator with real symbol $\alpha(\xi),$ say, and $g$ is a smooth, real-valued function of a real variable. Equations of this form arise as models of wave propagation in a variety of physical contexts. Here, fundamental issues of local and global well-posedness are established for $L_p$, $H^s$ and bore-like or kink-like initial data. In the special case where $\alpha(\xi)=|\xi|^{r}$ wherein $r>1$ and $g(u)=1/2u^2,$ (0.1) is globally well-posed in time if $s$ and $r$ satisfy a simple algebraic relation.

Citation: Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253
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