On a nonlinear equation of the vibrating string (Q1344564)

The equation \[ u_{tt} - \left( \varphi \bigl( l \sqrt {1 + u^ 2_ x} - l_ 0 \bigr) \cdot {u_ x \over \sqrt {1 + u^ 2_ x}} \right)_ x = f(t,x), \quad u(t,0) = u(t,l) = 0 \tag{*} \] was proposed some years ago by L. Amerio and G. Prouse as a model for the nonlinear vibrating string, where \(x \in I \equiv [0,l]\), \(l \geq l_ 0\), \(\varphi (r) \sim r^ \gamma (0 < \gamma \leq 1)\) and \(0 < \varphi' (r) \leq M\). In this short note, the authors study this equation by approximating with the addition of a penalising term \(\varepsilon \cdot u_{xxxx}\), and state (without proof) the following existence result: For any initial data \((u(0,x), u_ t(0,x))\) in \(H^ 1(I) \times L^ 2 (I)\) and any \(f \in L^ 2 ((0,T) \times I)\), the Cauchy problem for \((*)\) has a weak solution \(u \in L_ \infty (0,T; W_ 0^{1,1 + \gamma} (I))\), with \(u_ t \in L_ \infty (0,T; L^ 2(I))\). A kind of result of uniqueness is also stated. The paper concludes with a comparison, with the aid of numerical experiments, of the above model with the linear model (D'Alembert equation) and the nonlocal model proposed by G. Kirchhoff.