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On the nonlinear equation of a vibrating string. (Italian. English summary) Zbl 0780.35016
Il est question du problème de Cauchy-Dirichlet \[ {\partial \over {\partial x}} \bigl\{ \bigl[1+ \bigl( {{\partial u} \over {\partial x}}\bigr)^ 2\bigr] {{\partial u} \over {\partial x}}\bigr\} - {{\partial^ 2 u} \over {\partial t^ 2}}= f(x,t) \quad \text{en} \quad \Omega\times [0,T]; \] \(u(x,0)=u_ 0(x)\), \({{\partial u(x,0)} \over {\partial t}}= u_ 1(x)\) en \(\Omega\); \(u(x,t)=0\) en \(\partial\Omega\) pour presque tous les \(t\in [0,T]\), où \(\Omega\) est un intervalle borné de \(\mathbb{R}\). – Les auteurs démontrent l’éxistence locale et l’unicité de la solution et en donnent une majoration. La démonstrtion fait appel à plusieurs considerations, des quelles l’auteur remarque les deux suivantes: considération d’un problème pénalisé correspondant à l’équation \[ \varepsilon {{\partial^{2m}u} \over {\partial x^{2m}}}+ {\partial \over {\partial x}} \left\{\left[ 1+ \left( {{\partial u} \over {\partial x}}\right)^ 2 \right] {{\partial u} \over {\partial x}}\right\}- {{\partial^ 2u} \over {\partial t^ 2}}= f(x,t); \] nouvelle généralisation du lemme de Gronwall \(u(t)\leq u_ 0+ \int^ t_ 0 [f(s)+g(s) u^ \beta(s)]ds\) aux cas \(\beta\neq 1\) positif.
Reviewer: S.Cinquini (Pavia)

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
74H45 Vibrations in dynamical problems in solid mechanics
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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