On the Well-Posedness of the Kirchhoff String

Abstract:

Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation \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 $\Omega$ is an open subset of $\mathbb {R}^{n}$ and $m$ 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.