Semilinear formulation of a hyperbolic system of partial differential equations

Abstract In this paper, we solve the Cauchy problem for a hyperbolic system of first-order PDEs defined on certain Banach space X . The has special semilinear structure because, one hand, evolution law can be expressed as sum linear unbounded operator and nonlinear Lipschitz function but, other perturbation takes values not in but larger Y which is related to order deal with situation use theory dual semigroups. Stability results around steady states are also given when Fréchet differentiable. These based two propositions: relating local dynamics semiflow linearised semigroup equilibrium, second dynamical properties spectral its generator. later proven by showing that Spectral Mapping Theorem always applies semigroups obtains linearised. Some epidemiological applications involving gut bacteria commented