### 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) |
---|---|

Pages (from-to) | 463-497 |

Number of pages | 35 |

Journal | Revista Matematica Iberoamericana |

Volume | 24 |

Issue number | 2 |

State | Published - 2008 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Global infinite energy solutions of the critical semilinear wave equation.** / Germain, Pierre.

Research output: Contribution to journal › Article

*Revista Matematica Iberoamericana*, vol. 24, no. 2, pp. 463-497.

}

TY - JOUR

T1 - Global infinite energy solutions of the critical semilinear wave equation

AU - Germain, Pierre

PY - 2008

Y1 - 2008

N2 - We consider the critical semilinear wave equation (Equation Presented) set in ℝd, d ≥ 3, with 2* = 2d/d-2. Shatah and Struwe [22] proved that, for finite energy initial data (ie if (u0, u 1) ∈ Ḣ1 × L2), there exists a global solution such that (u, ∂tu) ∈ C(ℝ, Ḣ1 × L2). Planchon [17] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u0, u1 ) in Ḃ2,∞1 × Ḃ2,∞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.

AB - We consider the critical semilinear wave equation (Equation Presented) set in ℝd, d ≥ 3, with 2* = 2d/d-2. Shatah and Struwe [22] proved that, for finite energy initial data (ie if (u0, u 1) ∈ Ḣ1 × L2), there exists a global solution such that (u, ∂tu) ∈ C(ℝ, Ḣ1 × L2). Planchon [17] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u0, u1 ) in Ḃ2,∞1 × Ḃ2,∞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.

KW - Besov spaces

KW - Critical wave equation

KW - Global solutions

KW - Infinite energy

UR - http://www.scopus.com/inward/record.url?scp=54049149760&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=54049149760&partnerID=8YFLogxK

M3 - Article

VL - 24

SP - 463

EP - 497

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 2

ER -