On a Problem of Multidimensional Cross-Diffusion With Nonlinear Boundary Conditions

Recently, the cross-diffusion problem has gained importance in solving problems of mathematical modelling of complex processes such as population dynamics, diffusion in multiphase media and reaction-diffusion systems and therefore has attracted considerable attention. Studies have shown that cross-diffusion elements in models can significantly change the qualitative and quantitative properties of solutions. However, many aspects of solutions, especially nonlinear boundary value problems, have not been sufficiently studied, which requires further and more in-depth study of these issues and creates the need for a more in-depth theoretical analysis. Based on the above considerations, the objective of this study is to formulate and analyse the scientific problem associated with the dynamic behaviour of the cross-diffusion problem based on the conditions of existence and non-existence of a global solution and the influence of boundary and parametric conditions on solutions based on self-similar analysis. The research methodology is based on the use of a self-similar approach, which allows simplifying the system of differential equations by introducing variables that are invariant with respect to changes in measurements. The cross-diffusion problem with nonlinear boundary conditions is considered, resulting in a qualitative analysis based on the conditions for the existence and non-existence of solutions. In addition, the methods of the theory of ordinary and partial differential equations, analytical and numerical methods are used to study the behaviour of solutions over large time intervals. The analysis shows that, depending on the choice of system parameters and boundary conditions, both the existence of global solutions and their non-existence are possible. The conditions under which the solution of the system retains its regularity over a finite time interval are established, as well as the one under which the emergence or failure of the solution in a finite time is observed. In addition, the self-similar approach allows us to determine the key parameters responsible for the system's critical behavior. The main conclusion of the work is that the use of self-similar variables not only significantly simplifies the study of complex cross-diffusion models but also allows us to obtain important information about the stability limits of solutions. The results obtained can be useful for research related to nonlinear diffusion problems, as well as for developing more accurate mathematical models in practical problems.