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

We investigate the evolution problem where H is a Hilbert space, A is a self-adjoint nonnegative operator on H with domain D(A), δ>0 is a parameter, and m(r) is a nonnegative function such that m(0)=0 and m is nonnecessarily Lipschitz continuous in a neighborhood of 0. We prove that this problem has a unique global solution for positive times, provided that the initial data (u0,u1) D(A)×D(A1/2) satisfy a suitable smallness assumption and the nondegeneracy condition . Moreover, we study the decay of the solution as t→+∞. These results apply to degenerate hyperbolic PDEs with nonlocal nonlinearities.