Global solutions for dissipative Kirchhoff strings with a non Lipschitz nonlinear term

We study the abstract Kirchhoff equation with a dissipative term depending on a small parameter. Here we assume that the nonlocal nonlinear term is a nonnegative, nonnecessarily Lipschitz continuous function in a neighborhood of the origin. We prove that this problem has a unique global solution for positive times, provided that the initial data satisfy a suitable smallness assumption and a nondegeneracy condition Moreover, we study the decay of the solution