Global existence of weak solutions for parabolic triangular reaction

Abstract

Over the years, reaction-diffusion systems have attracted the attention of a great number of investigators and were successfully developed on the theoretical backgrounds. Not only it has been studied in biological and chemical fields, some investigations range as far as economics, semiconductor physics, and star formation. Recently particular interests have been on the impact of environmental changes, such as climate. This work is devoted to the existence of weak solutions for  m x m reaction-diffusion systems arises from an energy balance climate model. We consider a time evolution model for the climate obtained via energy balance. This type of climate model, independently introduced in 1987 by V. Jentsch, has a spatial global nature and involves a relatively long-time scale. Our study concerns the global existence of periodic solutions of the nonlinear parabolic problem. The originality of this study persists in the fact that the non-linearities of our system have critical growth with respect to the gradient of solutions. For this reason new techniques will used to show the global existence. This is our main goal in this article.