The propagation of analyticity for sufficiently smooth solutions to either strictly hyperbolic, or smoothly symmetrizable nonlinear systems, dates back to Lax (Comm. on Pure Appl. Math. 1953) and Alinhac and Métivier (Invent. Math. 1984). Here we consider the general case of a system with real, possibly multiple, characteristics, and we ask which regularity should be a priori required of a given solution in order that it enjoys the propagation of analyticity. By using the technique of the quasi-symmetrizer of a hyperbolic matrix, we prove, in the one-dimensional case, the propagation of analyticity for those solutions which are Gevrey functions of order s for some s < m/(m − 1), m being the maximum multiplicity of the characteristics.

Let h be a system with characteristics of constant multiplicity. We prove that if there exists an operator A′ such that h∘A′ has diagonal principal part and admits a good decomposition, then h must satisfy the Levi conditions.

This paper concerns the Cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients. We give a sufficient condition for the C-infinity-well-posedness which is also necessary if the space dimension is equal to one. The main point of the paper consists in expressing our condition only in terms of the coefficients of the operator, without needing to know the behavior of the characteristic roots. This is made possible by using the so-called standard symmetrizer of a companion hyperbolic matrix.

We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed.

We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed.

We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class gs0s0 and the Cauchy data belong to gs1s1, then the Cauchy problem has a solution in gs0([0,T*];gs1(\mathbbR))s0([0T];s1(R)) for some T *>0, provided 1≤s 1≤2−1/s 0. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤s 1≤s 0.

In questo lavoro si propone una caratterizzazione della risolubilità di un problema differenziale associato all'equazione di~Black-Scholes-Merton che utilizza alcune tecniche della teoria dei semigruppi. A tal riguardo, una piccola sezione preliminare fornisce dei richiami utili ad illustrare un'interconnessione fra la teoria degli operatori e quella delle equazioni alle derivate parziali.