On the Well-Posedness of the Kirchhoff String

Abstract

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 ) , \begin{align*} u_{tt} - m \left (\int _{_{\partial {\Omega }}} |\nabla _{x}u|^{2} \, dx \right ) \delta _{x}u= f(x,t) &&& (x\in \, \Omega , t > 0),\\ u(\cdot ,t)_{|\partial \Omega } =0 &&& (t\geq 0), \end{align*} where Ω \Omega is an open subset of R n \mathbb {R}^{n} and m m is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.