Periodic Solutions of Quasi-Monotone Semilinear Multidimensional Hyperbolic Systems

Abstract

This paper deals with the Cauchy problem for a class of first-order semilinear hyperbolic equations of the form ∂tfi+∑j=1dλij∂xjfi=Qi(f). where fi=fi(x,t) (i=1,⋯,n) and x=(x1,⋯,xd)∈IRd (n≥2,d≥1). Under assumption of the existence of a conserved quantity ∑iαifi for some α1,⋯,αn>0, of (strong) quasimonotonicity and an additional assumption on the speed vectors Λi=(λi1,⋯,λid)∈IRd—namely, span{Λj−Λk:j=1,⋯,n}=IRd for any k—it is proved that the set of constant steady state {f¯∈IRn:Q(f¯)=0} is asymptotically stable with respect to periodic perturbations, i.e., any initial data given by an periodic L1–perturbations of a constant steady state f¯ leads to a solution converging to another constant steady state g¯ (uniquely determined by the initial condition) as t→+∞.