Weak solutions of impulsive pseudoparabolic equations with an infinitesimal transition layer. (March 2023) - Kuznetsov, Ivan; Sazhenkov, Sergey
Abstract: We study the multi-dimensional initial–boundary value problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term, which models a non-instantaneous impulsive impact. 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, in turn, models an instantaneous impulsive impact. We prove that the infinitesimal transition layer, associated with the Dirac delta function, is formed as ɛ → 0, and that the family of weak solutions of the original problem converges to the weak solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial, boundary, 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 equation inherits the full information about the profile of the original non-instantaneous impulsive impact.