Global well-posedness for the radial defocusing cubic wave equation on \(\mathbb R^3\) and for rough data. (English) Zbl 1133.35095
Summary: We prove global well-posedness for the radial defocusing cubic wave equation
\[ \begin{aligned}\partial_{tt} u - \Delta u &= -u^{3},\\ u(0,x) &= u_{0}(x),\\ \partial_{t} u(0,x) &= u_{1}(x), \end{aligned} \]
with data \((u_0, u_1) \in H^{s} \times H^{s-1}\), \(1 > s >7/10\). The proof relies upon a Morawetz-Strauss-type inequality that allows us to control the growth of an almost conserved quantity.
\[ \begin{aligned}\partial_{tt} u - \Delta u &= -u^{3},\\ u(0,x) &= u_{0}(x),\\ \partial_{t} u(0,x) &= u_{1}(x), \end{aligned} \]
with data \((u_0, u_1) \in H^{s} \times H^{s-1}\), \(1 > s >7/10\). The proof relies upon a Morawetz-Strauss-type inequality that allows us to control the growth of an almost conserved quantity.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
