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

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

2010-07-05

arxiv.org/abs/1007.0733v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

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 $H^s\times H^{s-1}$ with certain angular regularity. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space $L^2_t L_{|x|}^\infty L^2_\theta ([0,T]\times \R^2)$. In the last section, we also consider the general semilinear wave equations with the spatial dimension $n\ge 2$ and the order of nonlinearity $p\ge 3$.

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