Physica D 139 (2000) 28–47 Evolution equations, invariant surface conditions and functional separation of variables Edvige Pucci a , Giuseppe Saccomandi b a Dipartimento di Ingegneria Industriale, Università degli Studi di Perugia, 06125 Perugia, Italy b Dipartimento di Ingegneria dell’Innovazione, Università degli Studi di Lecce, 73100 Lecce, Italy Received 27 January 1999; received in revised form 24 August 1999; accepted 29 October 1999 Communicated by F.H. Busse Abstract This paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables. The relationship of these methods with nonclassical and weak point symmetries is stressed. Applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Evolution equation; Invariance surface condition; Nonclassical symmetry; Direct method 1. Introduction The classical theory of symmetries of differential equations due to Sophus Lie [3,29,34] is the main inspiring source for various generalizations aiming to find new methods to obtain exact solutions for partial differential equations. One of the more interesting extensions for the Lie theory has been first considered by Bluman and Cole [2] and named the nonclassical method. Let us consider a partial differential equation (PDE) in 1 + 1 independent variables Δ(x,t,u,u (k) ) = 0 (1) with u (k) denoting the derivatives of the unknown function u with respect to the x and t up to the order k. On the space R 2 × R, we introduce the vector field: v = ξ(x,t,u) ∂ ∂x + τ(x,t,u) ∂ ∂t + η(x,t,u) ∂ ∂u . (2) The graph of a solution of the differential equation (1) u = f(x,t), E-mail address: giuseppe@ibm.isten.ing.unipg.it (G. Saccomandi). 0167-2789/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII:S0167-2789(99)00224-9