Kurzweil's impulsive inverse problem for the Kelvin--Voigt equations with $p$-Laplacian

In the present paper, we proceed with the research initiated in (S. Antontsev \emph{et al.}, Inverse Problems, 2021). We study the inverse problem for the Kelvin–Voigt equations with $p$-Laplacian and with a right-hand side in the integral over-determination condition, which approximates a discontinuous function at $t=0$. We rescale the problem in a neighbourhood $[0,1\mathrm{/}n]$ and pass to the limit as $n$ approaches infinity. This leads to the inverse problem on an infinitesimal initial layer, which transforms the initial velocity into a new one.