Periodic solutions of a nonlinear evolution problem from heterogeneous catalysis

Abstract

We consider a class of reaction-diffusion systems with macroscopic convection and nonlinear diffusion plus a nonstandard boundary condition which results as a model for heterogeneous catalysis in a stirred multiphase chemical reactor. Since the appearance of $T$-periodic feeds is a common feature in such applications, we study the problem of existence of a $T$-periodic solution. The model under consideration admits an abstract formulation in an appropriate $L^1$-setting, which leads to an evolution problem of the type \[ u' + Au \ni f(t,u) \ \mbox{ on } \ \mathbb R_+. \] Here $A$ is an $m$-accretive operator in a Banach space $X$ and $f:\mathbb R_+ \times K \to X$ is $T$-periodic and of Carathéodory type where $K$ is a closed, bounded, convex subset of $X$. Sufficient conditions on $A$, $f$ and $K$ to assure existence of $T$-periodic mild solutions for this evolution problem are provided and applied to the class of reaction-diffusion systems mentioned above.