Sull'equazione della corda vibrante

Abstract: In the Cauchy-Dirichlet problem related to the non-linear equation of a vibrating string partial derivative/partial derivative x([1 + (partial derivative u/partial derivative x)2]partial derivative u/partial derivative x) - partial derivative2 u/partial derivative t2 = f(x, t) in OMEGA x [0, T] (OMEGA being a bounded interval of the real line) local existence and uniqueness of a solution is established. To this aim we use a generalized version of Gronwall's inequality: if u(t) is a solution in [0, T] of the inequality u(t) less-than-or-equal-to u0 + integral-t/0 {f(s) + g(s)u(beta)(s)} ds (beta > 1), then there exists T less-than-or-equal-to T such that in [0, T]: \u(t)\ less-than-or-equal-to c(T,beta,f,g,u0)