On the behaviors of solutions of a nonlinear diffusion system with a source and nonlinear boundary conditions

We study the global solvability and unsolvability of a nonlinear diffusion system with nonlinear boundary conditions in the case of slow diffusion. We obtain the critical exponent of the Fujita type and the critical global existence exponent, which plays a significant part in analyzing the qualitative characteristics of nonlinear models of reaction-diffusion, heat transfer, filtration, and other physical, chemical, and biological processes. In the global solvability case, the key components of the asymptotic solutions are obtained. Iterative methods, which quickly converge to the exact solution while maintaining the qualitative characteristics of the nonlinear processes under study, are known to require the presence of an appropriate initial approximation. This presents a significant challenge for the numerical solution of nonlinear problems. A successful selection of initial approximations allows for the resolution of this challenge, which depends on the value of the numerical parameters of the equation, which are primarily in the computations recommended using an asymptotic formula. Using the asymptotics of self-similar solutions as the initial approximation for the iterative process, numerical calculations and analysis of the results are carried out. The outcomes of numerical experiments demonstrate that the results are in excellent accord with the physics of the process under consideration of the nonlinear diffusion system.