Sergio Spagnolo

On the convergence of solutions of hyperbolic equations

(1978). On the convergence of solutions of hyperbolic equations. Communications in Partial Differential Equations: Vol. 3, No. 1, pp. 77-103.

On an Abstract Weakly Hyperbolic Equation Modelling the Nonlinear Vibrating String

Let V be a complex Banach space continuously embedded into its dual V′, and denote by the (sesquilinear) pairing on V′ x V. Then we can consider the Hilbert space H, completion of V with respect to the product (u, v)H = , and the increasing triple of spaces (V, H, V′) thus obtained is usually called a Hilbert triple.

On the absolute continuity of the roots of some algebraic equations

SuntoProviamo che per ogni polinomio di gradom≤3, i cui coefficienti dipendano con sufficiente regolarità da un parametro realet, si possono selezionarem radici (indipendenti) che sono assolutamente continue int.AbstractWe prove that for every polynomial of degreem≤3, with coefficients depending smoothly on a real parametert, it is possible to select am-tuple of roots absolutely continuous int.

On the 2 by 2 weakly hyperbolic systems

We study the wellposedness in the Gevrey classes s and in C1 of the Cauchy problem for 2 by 2 weakly hyperbolic systems. In this paper we shall give some conditions to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points.

Local and Semi-Global Solvability for Systems of Non-Principal Type

We consider the N X N system where A(t,ξ)is a matrix valued,homogeneous symbol of order 1,and B(t,ξ) a symbol of order≤0.Denoting byµ the maximum multiplicity of the eigevalues {tj(t,ξ)} of A?(t,ξ),and assuming some regularity of these,we prove that the system is locally solvable in the Gevrey class γs,for 1≤8

Analytic Propagation for Nonlinear Weakly Hyperbolic Systems

le;µ/(µ−1),as soon as,for each ξ,the imaginary parts of the Tj(t,ξ)s do not change sign for varying t and j.For 1<8<N/(N-1) the system is semi-globally solvable in γs,in particular for all f(t,x)analytic on Rn+1,and all U⊂⊂Rn+1,there is a C∞solution u(t,x) on U.

Hyperbolic equations and classes of infinitely differentiable functions

The propagation of analyticity for sufficiently smooth solutions to either strictly hyperbolic, or smoothly symmetrizable nonlinear systems, dates back to Lax [14] and Alinhac and Métivier [2]. 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.

Propagation of analyticity for a class of nonlinear hyperbolic equations

SummaryWe consider the solvability in the Mandelbrojt classes ɛ{Mh} (for the definition see (6), (7), (8), (9) below) of the Cauchy problem for hyperbolic equations of the type utt - - a(t) uxx=0, where a(t) is a strictly positive continuous function. More precisely, we give an example of a function a(t) for which the Cauchy problem is not well-posed in any class ɛ{Mh} containing a non-trivial function with compact support.

ERRATUM: "HOMOGENEOUS HYPERBOLIC EQUATIONS WITH COEFFICIENTS DEPENDING ON ONE SPACE VARIABLE"

The propagation of analyticity for a solution u(t,x) to a nonlinear weakly hyperbolic equation of order m, means that if u, and its time derivatives up to the order m-1, are analytic in the space variables x at the initial time, then they remain analytic for any time. Here we prove that such a property holds for the solutions bounded in C-infinity of a special class of homogeneous equations in one space variable, with time dependent coefficient.

Local in Space Energy Estimates for Second-order Hyperbolic Equations

We investigate the Cauchy problem for homogeneous equations of order m in the (t,x)-plane, with coefficients depending only on x. Assuming that the characteristic roots satisfy the condition we succeed in constructing a smooth symmetrizer which behaves like a diagonal matrix: this allows us to prove the well-posedness in .