Some generalizations of the Kadomtsev-Petviashvili equations
The article is devoted to the initial value problem \[ (u_t+ u^p u_x- D^\alpha_x u_x)_x+\varepsilon u_{yy}= 0,\quad u|_{t= 0}= u_0, \] where \(u= u(x,y,t)\) is a real-valued function of variables \((x,y)\in \mathbb{R}^2\) and \(t\geq 0\), \(D_x= (-\partial^2_x)^{1/2}= H\partial_x\) (the Hilbert transform \(H\)), \(\alpha\geq 1\), \(\varepsilon= \pm 1\). The existence of the local solution \[ u\in C(0, T; H^s(\mathbb{R}^2))\cap C^1(0, T; H^{s-(\alpha+ 1)}(\mathbb{R}^2)) \] with \(\partial^{-1}_x u_y\in C(0, T; H^{s- 1}(\mathbb{R}^2))\) is proved for the initial data satisfying \[ u_0\in H^1(\mathbb{R}^2),\quad\partial^{-1}_x u_0\in H^1(\mathbb{R}^2)\qquad (s\geq \alpha+ 1) \] such that \(\partial^2_y u_0= \partial^2_x\varphi\) for appropriate \(\varphi\in L^2(\mathbb{R}^2)\). Then the investigation of certain conserved quantities yields a global estimate in the favourable case \(\varepsilon=-1\), \(\alpha\geq 2\), \(p< {4\alpha\over 4+\alpha}\) and even the existence of global solutions \((T=\infty)\) for all \(\alpha\) large enough, e.g., for \(\alpha\geq 10\).
scientific article; zbMATH DE number 1429695
scientific article