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This paper is concerned with the small smooth data problem for the 3-Dnonlinear 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$g_{ij}(u, \nablau)=c_{ij}+d_{ij}u+\dsize\sum_{k=0}^3e_{ij}^k\partial_ku+O(|u|^2+|\nabla u|^2)$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 thenull condition. For the 3-D nonlinear wave equations $\partial_t^2u-\left(1+u\right)\Delta u=0$ and $\partial_t^2u-\left (1+\partial_t u\right)\Deltau=0$, 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 finitetime. The explicit expression of the asymptotic lifespan $T_{\varepsilon}$ as$\varepsilon\to0^+$ is also given.

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