This paper is concerned with the small smooth data problem for the 3-D nonlinear wave equation $\partial_t^2u-\left (1+u+\p_t u\right)\Delta u=0$. 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 $d_{ ij}\neq0$ 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 $\partial_t^2u-\left (1+u\right)\Delta u=0$ and $\partial_t^2u-\left (1+\partial_t u\right)\Delta u=0$, 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 $\partial_t^2u-\left(1+u+\partial_t u\right)\Delta u=0$ blows up in finite time. The explicit expression of the asymptotic lifespan $T_{ \varepsilon}$ as $\varepsilon\to0^+$ is also given.
Categories: math.AP