Asymptotic behaviour of contraction non-autonomous semi-flows in a Banach space: Application to first-order hyperbolic PDEs

Abstract

The asymptotic behaviour is studied for a class of non-autonomous infinite-dimensional non-linear dissipative systems. This is achieved by using the concept of contraction semi-flow, which is a generalization of contraction non-linear semigroup. Conditions are presented under which the solution of the abstract differential equation converges to the omega limit set (the equilibrium profile, respectively). The general development is applied to semi-linear systems with time-varying non-linearity. Asymptotic behaviour and stability criteria are established on the basis of the conditions given in the early portion of the paper. The theoretical results are applied to a general class of first-order hyperbolic time-varying semi-linear PDEs.