Strong solutions of impulsive pseudoparabolic equations

Abstract

We study the two-dimensional Cauchy problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term endowed with periodic initial data and periodicity conditions. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which models an instantaneous impulsive impact. We establish that the transition (shock) layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of strong solutions of the original problem converges to the strong solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter is derived based on the microstructure of the transition layer profile.