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

Abstract

This paper mainly concerns with the small data solution problem for the 3-D nonlinear wave equation: $\partial_t^2u-(1+u+\partial_t u)\Delta u=0$. This equation is prototypical of the more general equation $$ \sum_{i,j=0}^3g_{ij}(u, \nabla u)\partial_{ij}^2u =0 , $$ where $x_0=t$, $\nabla=(\partial_0, \partial_1, ..., \partial_3)$, and $$ g_{ij}(u, \nabla u)=c_{ij}+d_{ij}u+ \sum_{k=0}^3e_{ij}^k\partial_ku+O(|u|^2+|\nabla u|^2) $$ are smooth functions of their arguments with $c_{ij}, d_{ij}$ and $e_{ij}^k$ being constants, and $d_{ij}\not=0$ for some $(i,j)$; moreover, $ \sum_{i,j,k=0}^3e_{ij}^k\partial_ku\partial_{ij}^2u$ does not fulfill the null condition. For the 3-D nonlinear wave equations $\partial_t^2u-(1+u)\Delta u=0$ and $\partial_t^2u-(1+\partial_t u)\Delta u=0$, H. Lindblad, S. Alinhac, and F. John proved and disproved, respectively, the global existence of small data solutions. For radial initial data, we show that the small data solution of $\partial_t^2u-(1+u+\partial_t u)\Delta u=0$ blows up in finite time. The explicit asymptotic expression of the lifespan $T_{\varepsilon}$ as $\varepsilon\to 0+$ is also given.