Сибирский федеральный университет (СФУ) входит в топ-15 вузов России. Ведущие научные школы Красноярска. Лучший в России кампус. Мировые индустриальные партнёры

In this paper, we study the conditions of global solvability and unsolvability in time of solutions to the system of multidimensional diffusion equations with nonlinear boundary conditions. We constructed various self-similar solutions of the system of multidimensional diffusion equations in the slow diffusion case. We established critical exponents of the Fujita type and critical exponents for the global existence of the solution.

We study the qualitative behaviour of weak solutions to a doubly nonlinear cross-diffusion system in an inhomogeneous medium with nonlinear boundary flux. The main novelty of the paper lies in the analysis of a cross-diffusive p-Laplacian system weighted by a spatially varying density in the form $\rho \left(x\right)={(1+|x\left|\right)}^n$, combined with nonlinear boundary interactions. By constructing self-similar weak solutions of Barenblatt type and employing a nonlinear separation of variables, the system is reduced to a coupled family of ordinary differential equations that characterise admissible similarity profiles. This approach allows us to identify Fujita-type critical conditions separating global solutions from finite-time blow-up and to derive explicit estimates for the solution profiles. Consequently, we establish critical thresholds driven by the interaction between cross-diffusion, degeneracy, boundary nonlinearity, and medium inhomogeneity.

This article studies the asymptotic behavior of self-similar solutions within a nonlinear cross-diffusion system characterized by nonlocal boundary conditions. The principal term of the asymptotic behavior of self-similar solutions has been established. A method for selecting the initial approximation for the iterative process in the numerical study of the problem is proposed. Numerical computations and result analysis were conducted utilizing asymptotic formulas as a preliminary approximation for the iterative procedure.

В работе исследуется асимптотика автомодельных решений нелинейной кросс диффузии системы, с нелокальными граничными условиями и плотносю. Построены различные автомодельные решения задачи для случая медленной диффузии, являющиеся асимптотикой решений рассматриваемой задачи. Получен главный член асимптотики автомодельных решений. Для численного исследования рассматриваемой задачи предложен способ выбора оптимального начального приближения для итерационного процесса. Используя асимптотические формулы в качестве начального приближения для итерационного процесса, произведены численные расчеты. Результаты расчета визуализированы по времени и проведены анализ результатов.

This article investigates the asymptotic behavior of self-similar solutions of a nonlinear cross-diffusion system with nonlocal boundary conditions. The main term of the asymptotics of self-similar solutions has been determined. For the numerical study of the problem under consideration, a method for selecting the initial approximation corresponding to the iterative process is proposed. Numerical calculations and analysis of the results were carried out using asymptotic formulas as an initial approximation for the iteration process.