Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation
By Aynur Bulut. 2010
48 citations recorded by the source corpus.
In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ in the energy-supercritical regime, in dimensions $d\geq 6$, with no radial assumption on the initial data. We prove that if a solution satisfies an a priori bound in the critical homogeneous Sobolev space throughout its maximal interval of existence, that is, $u\in L_t^\infty(\dot{H}_x^{s_c}\times\dot{H}_x^{s_c-1})$, then the solution is global and it scatters. Our analysis is based on the methods of the recent works of Kenig-Merle \cite{KenigMerleSupercritical} and Killip-Visan \cite{KillipVisanSupercriticalNLS,KillipVisanSupercriticalNLW3D} treating the energy-supercritical nonlinear Schr\"odinger and wave equations.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d394e13eed5ed01ff2401105e52990ec448a5bdc
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In Applied Mathematics
In Control and Systems Engineering
In Mathematical Physics
Citations in the corpus
- Scattering for the radial 3D cubic wave equation (2014, methodology): “This technique was developed by Tao in [47] and utilized in the Kenig-Merle framework for nonlinear Schrödinger problems by Killip and Visan [27, 28, 30], and for semilinear wave equations in [29, 5, 6].”