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The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation. (English) Zbl 0974.35109

Summary: We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers equation \[ u_t+ u_{xxx}+ u^pu_x+ \varepsilon v_y- \nu u_{xx}= 0,\quad v_x= u_y,\quad u(0)= \varphi \] in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts. After showing local existence by the contraction principle for initial data \(\varphi\in H^s(\mathbb{R}^2)\) such that \({\mathcal F}^{-1}({k_2\over k_1}\widehat\varphi)\in H^r(\mathbb{R}^2)\), \(0\leq r\leq s-1\), we extend the solutions for all positive times. Whereas for \(\varepsilon= -1\) and \(1\leq p< 4/3\) this is done without any assumption on the initial data, we require a smallness condition on the initial data otherwise. In a last part, we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in \(L^2(\mathbb{R}^2)\) when \(\varepsilon= -1\) and \(1\leq p< 4/3\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
35B60 Continuation and prolongation of solutions to PDEs
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