Almost global existence for some semilinear wave equations with almost critical regularity

For any subcritical index of regularity $s>3\mathrm{/}2$, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space $H^s\times H^{s-1}$ with certain angular regularity. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space $L_t^2{L}_{\mathrm{\mid }x\mathrm{\mid }}^{\mathrm{\infty }}{L}_{\theta }^2([0,T]\times {\mathbb{R}}^2)$. In the last section, we also consider the general semilinear wave equations with the spatial dimension $n\ge 2$ and the order of nonlinearity $p\ge 3$.