Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

Abstract

We construct small global classical solutions to boundary value problems for nonautonomous quasilinear first-order hyperbolic systems in a strip, assuming that the right hand side is small. We provide stable conditions on the coefficients of the hyperbolic system and on the boundary operator ensuring that the nonhomogeneous version of a linearized problem has a unique bounded classical solution for any smooth right-hand side. We also establish conditions under which this solution smoothly depends on the coefficients of the linearized hyperbolic system. In the case that the data of the problem are almost periodic (respectively, periodic), we prove that the bounded solution is also almost periodic (respectively, periodic).