In this paper a model for the recovery of human and economic activities in a region, which underwent a serious disaster, is proposed. The model treats the case that the disaster region has an industrial collaboration with a non-disaster region in the production system and, especially, depends upon each other in technological development. The economic growth model is based on the classical theory of R. M. Solow (1956), and the full model is described as a nonlinear system of ordinary differential equations.
Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize it as a parabolic inclusion by using the Lagrange multiplier and the nonlinear maximal monotone operator associated with the time differential under time-dependent double obstacles.