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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., Comment: 20 pages, 2 figures

This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and $(c_1'(0))^2+(c_2'(0))^2\neq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$ with $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$, and $\ve>0$ is small, we will show that the classical solution $u(t,x)$ stops to be smooth at some finite time $T_{\ve}$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $\na_{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time $T_{\ve}$., Comment: 22 pages