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

Abstract

We investigate the evolution problem { u ″ + δ u ′ + m ( | A 1 / 2 u | H 2 ) A u = 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , 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 ( u 0 , u 1 ) ∈ D ( A ) × D ( A 1 / 2 ) satisfy a suitable smallness assumption and the nondegeneracy condition m ( | A 1 / 2 u 0 | H 2 ) > 0 . Moreover, we study the decay of the solution as t → + ∞ . These results apply to degenerate hyperbolic PDEs with nonlocal nonlinearities.