Lamé's system in large size domains: Existence, asymptotic behaviour and explicit solution

Abstract

This work is devoted to study the parabolic Lamé system defined in noncylindrical domains. We focus on the asymptotic behaviour of the solution as t and the state variable domain become very large. Different rates of convergence are established according to the growth of the domain. The “parabolic-elliptic regularization” method is used to treat the existence of the solution of such problems in time growing domains with multi-initial conditions. The steady state problem is a Lamé system defined on unbounded domains. In the cases of polynomial data and cylindrical symmetries, the limit solution will be given explicitly and some examples are provided.