Recently, there has been a surge in the analysis and modeling of mathematical models of the reaction-diffusion type in the presence of the so-called cross diffusion. Mathematical models of nonlinear cross diffusion are described by a system of nonlinear partial parabolic equations associated with nonlinear boundary conditions. In this paper, based on a self-similar analysis and the method of standard equations, the properties of a nonlinear cross-diffusion system coupled via nonlocal boundary conditions are studied. It is proved that for certain values of the numerical
parameters of the nonlinear cross-diffusion system coupled via nonlinear boundary conditions, they may not have global solutions in time. Based on a self-similar analysis and the principle of comparing solutions, a critical exponent of the Fujita type and a critical value of global solvability are established. Using the comparison theorem, upper bounds for global solutions and lower bounds for blow-up solutions are obtained.