TY - JOUR
T1 - Global infinite energy solutions of the critical semilinear wave equation
AU - Germain, Pierre
N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
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
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U2 - 10.4171/RMI/543
DO - 10.4171/RMI/543
M3 - Article
AN - SCOPUS:54049149760
VL - 24
SP - 463
EP - 497
JO - Revista Matematica Iberoamericana
JF - Revista Matematica Iberoamericana
SN - 0213-2230
IS - 2
ER -