Blowup of classical solutions for a class of 3-D quasilinear wave equations with small initial data

Abstract

This paper is concerned with the small smooth data problem for the 3-D nonlinear wave equation $\partial_t^2u-\left (1+u+\p_t u\right)Δu=0$. This equation is prototypical of the more general equation $\dsize\sum_{i,j=0}^3g_{ij}(u, \nabla u)\partial_{ij}u=0$, where $x_0=t$ and $g_{ij}(u, \nabla u)=c_{ij}+d_{ij}u+\dsize\sum_{k=0}^3e_{ij}^k\partial_ku+O(|u|^2+|\nabla u|^2)$ are smooth functions of their arguments, with $c_{ij},d_{ij}$ and $e_{ij}^k$ being constants, and $d_{ij}\ne 0$ for some $(i,j)$; moreover, $\dsize\sum_{i,j,k=0}^3e_{ij}^k(\partial_ku)\p_{ij} u$ does not fulfill the null condition. For the 3-D nonlinear wave equations ${\partial }_t^{2}u-\left(1+u\right)\Delta u=0$ and ${\partial }_t^{2}u-\left(1+{\partial }_{t}u\right)\Delta u=0$, H. Lindblad, S. Alinhac, and F. John proved and disproved, respectively, the global existence of small smooth data solutions. For radial initial data, we show that the small smooth data solution of ${\partial }_t^{2}u-\left(1+u+{\partial }_{t}u\right)\Delta u=0$ blows up in finite time. The explicit expression of the asymptotic lifespan $T_{\epsilon }$ as $\epsilon \to 0^{+}$ is also given.

20 pages, 2 figures

References in corpus (1)

• On the lifespan of and the blowup mechanism for smooth solutions to a class of 2-D nonlinear wave equations with small initial data