Qualitative analysis of a system of non-homogeneous doubly nonlinear parabolic equations

We consider the qualitative properties of solutions to a coupled system of nonhomogeneous doubly nonlinear parabolic equations on the whole line with an exponentially varying density. The characteristic features of degeneracy at vanishing values and gradients are analyzed, and the need for weak solutions and reliable comparison estimates is identified and justified. Using a nonlinear splitting method, we construct explicit comparison functions and, on this basis, apply a comparison principle to obtain global existence of nonnegative solutions for sufficiently small initial data in the slow-diffusion regime. In addition, a self-similar reduction is performed via a nonlinear change of variables, which converts the problem into an auxiliary system for similarity profiles. An asymptotic representation of these self-similar solutions is derived, and the dependence of the solution behaviour on the governing parameters is clarified. It is shown how the parameters affect spatial localization and finite-speed propagation, and a Fujita-type criterion is obtained that provides conditions for the existence and nonexistence of global solutions. To support the analytical results, numerical simulations implemented in Python produce solution profiles and graphical illustrations of the nonlinear diffusion dynamics. The computations agree with the qualitative predictions and help visualize the transition between parameter regimes.

Hali tarjima qilinmagan