Many physical problems are meaningfully formulated in a cylindrical domain. When the size of the cylinder goes to infinity, the solutions, under certain symmetry conditions, are expected to be identical in every cross-section of the domain. The proof of this, however, is sometimes difficult and almost never given in the literature. The present book partially fills this gap by providing proofs of the asymptotic behaviour of solutions to various important cases of linear and nonlinear problems in the theory of elliptic and parabolic partial differential equations. The book is a valuable resource for graduates and researchers in applied mathematics and for engineers. Many results presented here are original and have not been published elsewhere. They will motivate and enable the reader to apply the theory to other problems in partial differential equations.