Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation
The authors study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili equations \[ u_t+u_{xxx}+\sigma \partial_x^{-1} u_{yy}= -(u^p)_x, (t,x,y)\in \mathbb{R}\times \mathbb{R}^2, \] \[ u(0,x,y) =u_0(x,y), (x,y)\in \mathbb{R}^2, \] where \(\sigma=1\) or \(\sigma=-1\). If \(\rho\geq 3\) is an integer and the initial data are sufficiently small, then the solution \(u\) satisfies the following estimates: \[ \|u(t)\|_\infty \leq C(1+|t|)^{-1} (log(2+|t|))^k, \|u_x (t)\|_\infty \leq C(1+|t|)^{-1} \] for all \(t\in \mathbb{R}\), where \(k=1\) if \(\rho=3\) and \(k=0\) if \(\rho \geq 4\).
generalized Kadomtsev-Petviashvili equation
scientific article; zbMATH DE number 1334917