Abstract
In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, D x D u),x∈R n,t>0; u=u 0 (x), u t =u 1 (x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions.
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Gao, J. A theorem of global existence of solutions to nonlinear wave equations in four space dimensions. Monatshefte für Mathematik 109, 123–134 (1990). https://doi.org/10.1007/BF01302932
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DOI: https://doi.org/10.1007/BF01302932