Central manifold theory for the generalized equation of Kirchhoff strings on [formula omitted

Abstract

We consider the generalized quasilinear dissipative Kirchhoff's String problem u tt = - ∥ A 1 / 2 u ∥ H 2 Au - δ Au t + f ( u ) , x ∈ R N , t ⩾ 0 with the initial conditions u ( x , 0 ) = u 0 ( x ) and u t ( x , 0 ) = u 1 ( x ) , in the case where N ⩾ 3 , δ > 0 . The purpose of our work is to study the stability of the initial solution u = 0 for this equation using central manifold theory.