Geodesic
Existence of Global Solutions to Supercritical Semilinear Wave Equations
Serdica Mathematical Journal, Tome 22 (1996) no. 2, pp. 125-164.

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In this work we study the existence of global solution to the semilinear wave equation (1.1) (∂2t − ∆)u = F(u), where F(u) = O(|u|^λ) near |u| = 0 and λ > 1. Here and below ∆ denotes the Laplace operator on R^n. The existence of solutions with small initial data, for the case of space dimensions n = 3 was studied by F. John in [13], where he established that for 1 λ 1+√2 the solution of (1.1) blows-up in finite time, while for λ > 1 + √2 the solution exists globally in time. Therefore, the value λ0 = 1 + √2 is critical for the semilinear wave equation (1.1).
Keywords: Semilinear Wave Equation, Strichartz Estimate 
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     author = {Georgiev, V.},
     title = {Existence of {Global} {Solutions} to {Supercritical} {Semilinear} {Wave} {Equations}},
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Georgiev, V. Existence of Global Solutions to Supercritical Semilinear Wave Equations. Serdica Mathematical Journal, Tome 22 (1996) no. 2, pp. 125-164. http://geodesic.mathdoc.fr/item/SMJ2_1996_22_2_a5/