Weighted Sobolev spaces applied to nonlinear Klein-Gordon equation

In this work we study the decay properties of the semilinear Klein-Gordon equation with nonlinearity of fractional order. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the upper branch of the unit hyperboloid: (formula presented) In these spaces of fractional order we obtain a weighted Sobolev inequality and a nonlinear estimate. Using these estimates we study the decay property of the solution for large t provided the power of nonlinearity is greater than a critical value.