Global Existence and Blow–up of Solutions for Parabolic Systems Involving Cross–Diffusions and Nonlinear Boundary Conditions

In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ {\begin{array}{*{20}l} {{u_{t} - a(u,v)\Delta u = g(u,v),} \hfill} & {{v_{t} - b(u,v)\Delta v = h(u,v),} \hfill} \\ {{\frac{{\partial u}} {{\partial \eta }} = d(u,v),} \hfill} & {{\frac{{\partial u}} {{\partial \eta }} = f(u,v).} \hfill} \\ \end{array} } \right. $$\end{document} Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.