Global existence of solutions to nonlinear dispersive wave equations. (English) Zbl 1265.35331

The authors study the Cauchy problem to the equation \[ \partial _t^2u+\frac {1}{\rho ^2}| \partial _x| ^{2\rho }u=\lambda | \partial _tu| ^{p-1}\partial _tu,\; \; t>0,\; x\in \mathbb {R}^1, \] where \(0<\rho \leq 2\), \(\rho \neq 1\), \(p>3\), \(\lambda \in \mathbb {C}\) and \(| \partial _x| ^{2\rho }=\mathcal {F}^{-1}| \zeta | ^{2\rho }\mathcal {F}\), \(\mathcal {F}\) being the Fourier transform. They prove that the problem has a unique global solution \(u\) such that \(| \partial _x| ^{\rho }u, \partial _tu\in C([0,\infty );L^2(\mathbb {R}^1)\) with time decay estimate \[ \| | \partial _x| ^{\rho }u(t)\| _{L^{\infty }(\mathbb {R}^1)}+\| \partial _tu(t)\| _{L^{\infty }(\mathbb {R}^1)}\leq C(1+t)^{-1/2} \] for sufficiently small and smooth initial data.


35Q55 NLS equations (nonlinear Schrödinger equations)
35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs