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

Author： Fang Daoyuan   Wang Chengbo  

Publisher： Taylor & Francis Ltd

ISSN： 0360-5302

Source： Communications in Partial Differential Equations, Vol.38, Iss.9, 2013-09, pp. : 1467-1491

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Abstract

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 × H s−1 with certain angular regularity. The lifespan is known to be sharp in general. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space . 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.

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