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**On the global solvability of the Cauchy problem for damped Kirchhoff equations.**
*(English)*
Zbl 1213.35311

The author deals with an \(n\)-dimensional generalisation of Kirchhoff equation that was proposed initially to describe better the motion of a stretched string in a three dimensional Euclidean space. A damping term which is proportional to the velocity is also added. As such the field equation is a nonlinear second order hyperbolic integro-differential equation to determine \(n+1\) variable displacement function. The author considers the Cauchy problem of this system with prescribed initial conditions for the displacement and the velocity fields. These functions are chosen from appropriate Sobolev spaces equipped with an inner product. The problem is studied in the phase space and the author proves two theorems concerning the global existence of the solution and the decay estimate of the solution as time goes to infinity by ingeniously employing techniques of functional analysis. A very detailed investigation is made relative to types of Sobolev spaces that are taken into account.

Reviewer: Erdoǧan S. Şuhubi (İstanbul)