Erschienen in:

1991 | OriginalPaper | Buchkapitel

A Decay Estimate for the Three-Dimensional Inhomogeneous Klein-Gordon Equation and Global Existence for Nonlinear Equations

verfasst von : Thomas C. Sideris

Erschienen in: Microlocal Analysis and Nonlinear Waves

Verlag: Springer New York

Enthalten in: Professional Book Archive

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We will describe a pointwise decay estimate for solutions u(t, x) of the inhomogeneous Klein-Gordon equation (IKG)$$ \partial_t^2u - \Delta u + u = F(t,x) $$ on R+ × R3, with zero initial data $$ u(0,x) = {\partial_t}u(0,x) = 0 $$. The desired bound takes the form (1)$$ \left| {u(t,x)} \right| = \frac{C}{{{{(1 + t + \left| x \right|)}^{{ \frac{\hbox{$\scriptstyle 3$}}{\hbox{$\scriptstyle 2$}} }}}}}\left| {\left\| F \right\|} \right| $$, with the norm ||| • ||| to be made precise later on.

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