Global existence and energy decay for mildly degenerate Kirchhoff’s equations on ℝ N

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We study the initial-boundary value problem for the following degenerate non-lineardissipative wave equations of Kirchhoff type: with initial conditions u(x, 0) = u 0(x) and ut (x, 0) = u 1(x) , in the case where N ≥ 3 , δ > 0 , γ ≥ 1 , f (u) is a nonlinear C 1 function and (ϕ(x))-1 = g(x) is a positive function lying in L N/2( ℝ N ) ∩ L ∞( ℝ N ) . It is proved that if the initial data {u 0, u 1} are small and ║∇u 0║ > 0 , then the unique solution exists globally in time and has certain decay properties.