On Global Solvability for Some Degenerate Dissipative Nonlinear Kirchhoff Strings

Consider the initial boundary value problem for degenerate dissipative wave equations of Kirchhoff type with attractive force terms.We are interested in the case of 0 < γ < 1 for the degeneracy of nonlinear term Φ(r) = rγ. We prove the global solvability problem, provided that the initial data belong to the potential well and satisfy a suitable smallness condition. Moreover, we derive optimal decay estimates of the solutions.