Comparison Results for Parabolic Differential Equations at Resonance

Abstract

**Please note that the full text is embargoed** ABSTRACT: It is very well known that comparison principles for initial and boundary value problems for nonresonance cases have been very much used in the existence of solutions and the development of the monotone method [1,2,3,4,5,9]. These comparison techniques do not cover the resonance cases. Hence it is of practical interest to look at such results for resonance cases. With this view, different comparison results were recently developed for first and second order periodic boundary value problems [10]. Some special cases of these have been used in [6,7] and in developing the monotone method for first order periodic systems in [11]. In this paper we develop systematically general comparison techniques for semilinear parabolic equations with periodic and homogeneous Neumann boundary conditions, since special cases of such equations occur in many physical situations as reaction diffusion equations [5,8].