### Abstract

We consider the critical semilinear wave equation (Equation Presented) set in ℝ^{d}, d ≥ 3, with 2* = 2d/d-2. Shatah and Struwe [22] proved that, for finite energy initial data (ie if (u_{0}, u _{1}) ∈ Ḣ^{1} × L^{2}), there exists a global solution such that (u, ∂_{t}u) ∈ C(ℝ, Ḣ^{1} × L^{2}). Planchon [17] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u_{0}, u_{1} ) in Ḃ_{2,∞} ^{1} × Ḃ_{2,∞}^{0} is small enough. In this article, we build up global solutions of (NLW)_{2*-1} for arbitrarily big initial data of infinite energy, by using two methods which enable to interpolate between finite and infinite energy initial data: the method of Calderón, and the method of Bourgain. These two methods give complementary results.

Original language | English (US) |
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Pages (from-to) | 463-497 |

Number of pages | 35 |

Journal | Revista Matematica Iberoamericana |

Volume | 24 |

Issue number | 2 |

DOIs | |

State | Published - 2008 |

### Keywords

- Besov spaces
- Critical wave equation
- Global solutions
- Infinite energy

### ASJC Scopus subject areas

- Mathematics(all)