Critical conditions of nonlinearities in the John problem

Abstract

In this paper, we determine the sharp criteria for the nonlinearities in the John problem ${\partial }_t^{2}u-{\Delta }_{{\mathbb{R}}^3}u=F(u)$ so that the problem admits global solutions for small initial data with compact support. The criteria is of Dini type. For the situation in which the criteria is not satisfied, we show that the solution will blow up in finite time for some small initial data, together with an almost sharp estimates of the lifespan. For the proof, we mainly rely on the precise pointwise estimates of the solution, based on the fundamental solution, John's iteration argument as well as an adaption of the slicing method.

22 pages