A shock layer arising as the source term collapses in the $p(\boldsymbol{x})$-Laplacian equation

We study the Cauchy–Dirichlet problem for the $p(\boldsymbol{x})$-Laplacian equation with a regular finite nonlinear minor term. The minor term depends on a small parameter $\varepsilon>0$ and, as $\varepsilon\to 0$, converges weakly$^\star$ to the expression incorporating the Dirac delta function, which models a shock (impulsive) loading. We establish that the shock layer, associated with the Dirac delta function, is formed as $\varepsilon\to 0$, and that the family of weak solutions of the original problem converges to a solution of a two-scale microscopic-macroscopic model. This model consists of two equations and the set of initial and boundary conditions, so that the ‘outer’ macroscopic solution beyond the shock layer is governed by the usual homogeneous $p(\boldsymbol{x})$-Laplacian equation, while the shock layer solution is defined on the microscopic level and obeys the ordinary differential equation derived from the microstructure of the shock layer profile.