Abstract: We study the multi-dimensional Cauchy--Dirichlet problem for the p(x,t)-parabolic equation with a regular nonlinear minor term, which models a non-instantaneous but very rapid absorption with the q(x,t)-growth. The minor term depends on a positive integer parameter n and, as n tends to infinity, converges weakly* to the expression incorporating the Dirac delta-function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta-function, is formed as n tends to infinity, and that the family of regular weak solutions of the original problem converges to the solution of a limit two-scale microscopic-macroscopic model. Furthermore, the equation of the microstructure can be integrated explicitly, which leads to upscaling of the limit model.

Cite: Sazhenkov S.
Impulsive p(x,t)-parabolic equations with an infinitesimal initial layer
The 15th Congress of the International Society for Analysis, its Applications, and Computation (ISAAC) 21-25 Jul 2025