ABSTRACT

The BBM or regularized long-wave equation was originally proposed as an alternative to the Korteweg-de Vries equation. It was shown in the paper of Benjamin et al. (1972) to be globally well-posed in H 1(ℝ), the class of square-integrable-functions whose derivative is also square-integrable. Recently, Bona and Tzvetkov (2002) have shown that the initial-value problem () u t + u x + u u x − u x x t = 0 , x ∈ ℝ ,   t > 0 , u ( x ,   0 ) = u 0 ( x )   , x ∈ ℝ , } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187599/b1a4101e-e019-448a-8402-e9c5cf3b70e4/content/eq422.tif"/>

is globally well posed in Hs (ℝ) for any s ≥ 0.

It is our purpose here to extend this well-posedness theory in weak spaces to some members of a more general class of evolution equations of the form () u t + u x + g ( u ) x + L u t = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187599/b1a4101e-e019-448a-8402-e9c5cf3b70e4/content/eq423.tif"/>

where L is a Fourier-multiplier related to the linearized dispersion relation and g is a smooth, real-valued function of a real variable. Results are established analogous to those for equation (0.1), for (0.2) posed on the entire real axis. In addition, local 36and global well-posedness theory is established for bore-like or kink-like initial data, wherein u 0 has different limits as x tends to ±∞.