Almost Global Existence for Some Semilinear Wave Equations with Almost Critical Regularity

For any subcritical index of regularity s > 3/2, 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 Hs × Hs−1 with certain angular regularity. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space LtL ∞ |x|L 2 θ([0, T ] × R 2). In the last section, we also consider the general semilinear wave equations with the spatial dimension n ≥ 2 and the order of nonlinearity p ≥ 3.