Nonlinear dispersive wave equations in two space dimensions. (English) Zbl 1331.35317

The paper is concerned with the Cauchy problem \[ \begin{cases} \left(\partial_t^2+\frac{1}{\rho^2}(-\Delta)^\rho\right)u = F(\partial_tu),\quad (t,x)\in\mathbb{R}\times\mathbb{R}^2,\\ u(0,x)=u_0(x),\;\partial_tu(0,x)=u_1(x),\quad x\in\mathbb{R}^2, \end{cases} \] where \((-\Delta)^\rho={\mathcal F}^{-1}|\xi|^{2\rho}{\mathcal F}\), \({\mathcal F}\) the Fourier transform, \(F(\partial_tu)=\lambda|\partial_tu|^{p-1}\partial_tu\) or \(F(\partial_tu)=\lambda|\partial_tu|^p\), \(\lambda\in\mathbb{C}\), and either \(0<\rho<1\), \(p>2\), or \(\rho=1\), \(p>3\), or \(1<\rho<2\), \(p>1+\rho\). In these cases the authors prove global existence of solutions and obtain time-decay estimates. This is a sequel of the authors’ previous paper [Differ. Integral Equ. 25, No. 7–8, 685–698 (2012; Zbl 1265.35331)] in one space dimension.


35Q55 NLS equations (nonlinear Schrödinger equations)
35R11 Fractional partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
35P25 Scattering theory for PDEs


Zbl 1265.35331
Full Text: DOI


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