The global solvability of the Kirchhoff equation with Sobolev data

Abstract

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.

arXiv commentDue to an error in the proof of Theorem 2.2, it is unsure if any of the subsequent results are correct