Periodic solutions of a nonlinear evolution problem from heterogeneous catalysis. (English) Zbl 1032.34061

Bothe, Dieter

Differ. Integral Equ. 14, No. 6, 641-670 (2001).

Author’s 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 in 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 \[ u'+Au\ni f(t,u) \quad\text{on }\mathbb{R}_{+}. \] Here, \(A\) is an \(m\)-accretive operator in a Banach space \(X\) and \(f: \mathbb{R}_{+}\times K\to K\) 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 the existence of \(T\)-periodic mild solutions to this evolution problem are provided and applied to the class of reaction-diffusion systems mentioned above.

Reviewer: Mouffak Benchohra (Sidi Bel Abbes)

MSC:

34G25	Evolution inclusions
34G20	Nonlinear differential equations in abstract spaces
35K55	Nonlinear parabolic equations
47H06	Nonlinear accretive operators, dissipative operators, etc.
47H20	Semigroups of nonlinear operators

Keywords:

nonconvex evolution inclusion; monotone operator; selection; accretive operator; periodic solution