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Global well-posedness for the defocusing, cubic, nonlinear wave equation in three dimensions for radial initial data in \(\dot{H}^s \times \dot{H}^{s - 1}\), \(s> \frac{1}{2}\). (English) Zbl 1428.35224

Summary: In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is \(\dot{H}^{1/2} \times \dot{H}^{-1/2}\). We show that if the initial data is radial and lies in \((\dot{H}^s \times \dot{H}^{s-1}) \cap (\dot{H}^{1/2} \times \dot{H}^{-1/2})\) for some \(s> \frac{1}{2}\), then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [B. Dodson, “Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when \(d = 2, 3\), \(u_0\) radial”, Preprint, arXiv:1405.0218].

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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