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

We investigate the evolution problem{u″+δu′+m(|A1/2u|H2)Au=0,u(0)=u0,u′(0)=u1, where H is a Hilbert space, A is a self-adjoint nonnegative operator on H   with domain D(A)D(A), δ>0δ>0 is a parameter, and m(r)m(r) is a nonnegative function such that m(0)=0m(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)(u0,u1)∈D(A)×D(A1/2) satisfy a suitable smallness assumption and the nondegeneracy condition m(|A1/2u0|H2)>0. Moreover, we study the decay of the solution as t→+∞t→+∞.These results apply to degenerate hyperbolic PDEs with nonlocal nonlinearities.