First-Order Hyperbolic Systems. Energy Estimates and Exponential

The paper is devoted to studying asymptotic properties of solutions for first-order hyperbolic systems with nearly constant coefficients. It is shown that if the matrix symbol of the system in question has no eigenvalues in an open strip containing the real line, then the inhomogeneous problem is uniquely solvable in the space of time-bounded functions. As a consequence, we establish the important property of exponential dichotomy and prove the existence of periodic and almost periodic solutions. In the particular case where the above-mentioned strip coincides with a half-plane, we show that the zero solution of the homogeneous problems exponentially asymptotically stable. The proofs of all these properties are based on some energy estimates that follow from early-established results on hyperbolic matrices depending on several parameters.