Periodic Solutions of Quasi-Monotone Semilinear Multidimensional Hyperbolic Systems

This paper deals with the Cauchy problem for a class of first-order semilinear hyperbolic equations of the form partial derivative(t)f(i) +Sigma(d)(j=1) lambda(ij)partial derivative(xj)f(i) =Q(i)(f). where f(i)=f(i)(x, t)(i = 1,.. . , n) and x = (x(1), center dot center dot center dot , x(d)) is an element of R-d (n >= 2, d >= 1). Under assumption of the existence of a conserved quantity Sigma(i) alpha(i)f(i) for some alpha(1), ... , alpha(n) > 0, of (strong) quasimonotonicity and an additional assumption on the speed vectors Lambda(i) = (lambda(i1), center dot center dot center dot , lambda(id)) is an element of R-d-namely, span {Lambda(j) - Lambda(k) : j = 1, ... , n} = R-d for any k-it is proved that the set of constant steady state { f is an element of R-n : Q(f) = 0} is asymptotically stable with respect to periodic perturbations, i.e., any initial data given by an periodic L-1-perturbations of a constant steady state (f) over bar leads to a solution converging to another constant steady state (g) over bar (uniquely determined by the initial condition) as t -> +infinity.