Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations

Abstract

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity (see the results of T. T. Li and Y. M. Chen in 1992). For this purpose, the authors consider the following Cauchy problem: $\left\{ \begin{gathered} \square u = \left( {u_t } \right)^3 , n = 2, \hfill \\ t = 0: u = 0, u_t = \varepsilon g\left( x \right), x \in \mathbb{R}^2 , \hfill \\\end{gathered} \right.$ where $\square = \partial _t^2 - \sum\limits_{i = 1}^n {\partial _{x_i }^2 }$ is the wave operator, g(x) ≢ 0 is a smooth non-negative function on ℝ² with compact support, and ɛ > 0 is a small parameter. It is shown that the solution blows up in a finite time, and the lifespan T(ɛ) of solutions has an upper bound T(ɛ) ≤ exp(Aɛ ⁻²) with a positive constant A independent of ɛ, which belongs to the same kind of the lower bound of the lifespan.

Keywords

Nonlinear wave equation / Cauchy problem / Lifespan

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