Fredholmness and smooth dependence for linear time-periodic hyperbolic systems

Abstract

This paper concerns n × n linear one-dimensional hyperbolic systems of the type ∂ t u j + a j ( x ) ∂ x u j + ∑ k = 1 n b j k ( x ) u k = f j ( x , t ) , j = 1 , … , n , with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data a j and b j k and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right-hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur. We show that perturbations of the coefficients a j lead to essentially different results than perturbations of the coefficients b j k , in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients a j and b j k , but they are new even in the case of strict hyperbolicity and of smooth coefficients.