Solutions of impulsive p(x,t)-parabolic equations with an infinitesimal initial layer

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 $n\to +\infty$, converges weakly${}^{\star }$ 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 $n\to +\infty$, and that the family of regular weak solutions of the original problem converges to the so-called ‘strong-weak’ solution of a two-scale microscopic–macroscopic model. Furthermore, the equation of the microstructure can be integrated explicitly, which leads in a number of cases to the purely macroscopic formulation for the $p(x,t)$-parabolic equation provided with the corrected initial data.