Research on the time global solutions for the partial differential equations of Kirchhoff type

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1.We studied the initial value problem for the dissipative hyperbolic equation of Kirchhoff type associated to a non-negative self-adjoint operator A in a Hilbert space, with a dissipative term and a forcing term.We gave sufficient conditions on the forcing term, related with the spectral decompo-sition of the operator A, for the global unique solvability of this initial value problem with small inital value.2.We studied the unique global solvability of initial(boundary)value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three.The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case ; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. We gave sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation.