Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation (Q630041)

This paper studies long time asymptotics of large time solutions of the generalized Benjamin-Ono-Burgers equation of the form \[ u_t+u_x+(P(u))_x-\nu u_{xx}-\mathbb H u_{xx}=0 \] where \(x \in\mathbb R\), \(t>0\), \(\nu>0\) is a viscosity coefficient, \(P\) is a nonlinear function and \(\mathbb H\) is the Hilbert transform. This is a damped wave equation, which is complemented with an initial condition \(u(x,0)=f(x)\). After a detailed study of the linearized equation, using Fourier transform techniques, the authors provide global well-posedness results for arbitrary data in \(H^s(\mathbb R)\), \(s \geq 1\) for a wide class of nonlinearities. Then detailed estimates for the decay rates of the solutions of the equation, depending on the behaviour of the initial conditions are provided, for instance, \[ \text{if }\;f\in H^1(\mathbb R) \cap L^1(\mathbb R),\quad\text{then}\quad \|u(\cdot, t)\|_{L^2(\mathbb R)}\leq C(1+t)^{-1/4}, \] where \(C\) is independent of \(t\), whereas if the initial condition has Fourier transform which behaves as \(|y|^\alpha\) as \(y \rightarrow 0\), then the decay rate of the corresponding solutions of the equations will increase by \(\alpha/2\) over what can be expected for generic data. Such results rely on Fourier transform techniques, which reduce the equation to a nonlinear integral equation, along with detailed estimates in terms of differential inequalities.