A Critical Value for Global Nonexistence of Solution of a Wave Equation

Abstract

Consider the Cauchy problem for a wave equation on R²: u_{tt} - Δu = |u|^{p-1}u. In 1981 Glassey gave a guess to a critical value p(2) = \frac{1}{2}(3 + \sqrt{17}): when p > p(2) there may exist a global solution and when 1 < p < p(2) the solution may blow up. By our main result in this paper a counter example to the guess is given that the solution may also blow up in finite time even if p(2) < p < 5.