Abstract
Theory of parabolic differential inequalities, flow-invariance of solutions and comparison theorems are discussed relative to a cone. In this paper we investigate the theory of parabolic differential inequalities in arbitrary cones. After discussing the fundamental results concerning parabolic inequalities in cones, we prove a result on flow-invariance which is then used to obtain a comparison theorem. This comparison result is useful in deriving upper and lower bounds on solutions of parabolic differential equations in terms of the solutions of ordinary differential equations. We treat the Dirichlet problem in this paper since its theory follows the general pattern of ordinary differential equations and requires less restrictive assumptions. The treatment of Neumann problem, on the other hand, demands stronger smoothness assumptions and depends heavily on strong maximum principle. The study of the corresponding results relative to Neumann problem is discussed elsewhere.
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