Local Lipschitz continuity of a nonlinear bounded operator induced by a generalized Benjamin-Ono-Burgers equation
The author is concerned with the long-time behaviour of the solution to the following Cauchy problem for the Benjamin-Ono-Burgers equation: \[ u_t+u_{xxx}+ Hu_{xx}- \alpha u_{xx}+ u^2u_x=0, \quad u(x,0)= u_0 (x),\;x\in \mathbb{R}, \] where \(H\) stands for the Hilbert transform operator \[ Hu(x,t)={1\over \pi} P.V.\int^\infty_{-\infty} {u(y,t)\over y-x} dy. \] After stating a result of global existence and uniqueness in the space \(L^\infty (0,T;H^2) \cap\break W^{1,2} (0,T;L^2)\), for an arbitrary \(T>0\), the author obtains various estimates for the solution, with respect to various norms related to the above mentioned function spaces. For instance, when \(u_0\in H^2\), there results \(\sup \|u(t)\|^2 \leq\|u_0\|^2\) on \(\mathbb{R}_+\); when \(u_0\in L^1\cap L^2\), then there exists a constant \(K>0\), such that \[ \bigl\|u(t) \bigr\|^2\leq K\bigl(\|u_0\|^2_{L^1} +\|u_0\|^6 \bigr)\bigl[ \ln(1+t) \bigr]^{-3/2}. \] As a byproduct of various estimates obtained in the paper, the author derives the local Lipschitz continuity (in various norms) of the operator \(u_0 \to u(x,t)\).
scientific article; zbMATH DE number 1389534