2009, 24(4): 1307-1323. doi: 10.3934/dcds.2009.24.1307

Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$

1. 

UCLA Mathematics Department, Box 951555 Los Angeles, CA 90095-1555, United States

Received  February 2008 Revised  March 2009 Published  May 2009

We prove global well-posedness for the defocusing cubic wave equation

tt $u - \Delta u = -u^{3} $
$u(0,x) = u_{0}(x) $
$\partial_{t} u(0,x) = u_{1}(x)$

with data $( u_{0}, u_{1} ) \in H^{s} \times H^{s-1}$, $1 > s > \frac{13}{18} $≈ 0.722. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We divide it into subintervals. On each of these subintervals we write the solution as the sum of its linear part adapted to the subinterval and its corresponding nonlinear part. Some terms resulting from this decomposition have a controlled global variation and other terms have a slow local variation.

Citation: Tristan Roy. Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1307-1323. doi: 10.3934/dcds.2009.24.1307
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