On the Well-Posedness of the Kirchhoff String

Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation u t t − m ( ∫ ∂ Ω | ∇ x u | 2 d x ) δ x u = f ( x , t ) a m p ; a m p ; a m p ; ( x ∈ Ω , t > 0 ) , u ( ⋅ , t ) | ∂ Ω = 0 a m p ; a m p ; a m p ; ( t ≥ 0