CAUCHY AND GOURSAT PROBLEM FOR SOME NONLINEAR PARTIAL-DIFFERENTIAL EQUATIONS

We generalize a paper of A. Friedman [1] giving Cauchy-Kowalewski and Goursat Theorems for equations and systems of type D(t)p(i)u(i)=f(i)(x, t, integral-b/a \D(x)q(isigma)(j)D(t)r(isigma)(j)u(sigma)\2dx), with functions f(i) being of class Gevrey in x and analytic with respect to other variables. Then we show existence of global solution of class C2 in t and analytic in x for Cauchy problem with analytic data associated to more particular system, which is weakly hyperbolic, D(t)u=phi(integral-b/a \D(x)u\2dx)D(x)2u+psi(integral-b/a\D(x)u\2dx)u (phi(r) greater-than-or-equal-to 0) using a method of S. Spagnolo [2], when phi and psi are continuous and bounded on R+ (with scalar phi and matricial r x r psi).