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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.

MSC:

35L71 Second-order semilinear hyperbolic equations
74K05 Strings
35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35R09 Integro-partial differential equations
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Full Text: Euclid