Existence of Global Solutions to Supercritical Semilinear Wave Equations

Abstract

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).