  We consider linear and non-linear Cauchy equations in the context of Sobolev
spaces. In particular, we show the global existence of solutions to the
Kirchhoff equation with initial data in the Sobolev spaces, a problem that has
been open for more than eighty years. Our proof is based on a new uniform
estimate for solutions to the linear equation with time-dependent coefficient
and a fixed point argument. As an immediate consequence of our result, the
global solvability of the Kirchhoff equation with initial data in the Gevrey
spaces is also obtained.
