The concept of asymptotic stability is studied for a class of semi-linear distributed parameter dissipative systems with nonlinearity defined on a convex subset of the state space. This is achieved by using infinite-dimensional Banach state space description Stability criteria are established. which are based on a weaker technical condition of the m-dissipativity. These theoretical results are applied to a class of transport-reaction processes. Different types of nonlineanties are studied by adapting the criteria given in the early portions of the paper. (C) 2010 Elsevier Ltd. All rights reserved.