TY - JOUR

T1 - Strong solutions of impulsive pseudoparabolic equations

AU - Kuznetsov, Ivan

AU - Sazhenkov, Sergey

N1 - Funding Information: The authors would like to thank the anonymous referees for their valuable remarks and recommendations that helped improve the earlier version of the paper. Publisher Copyright: © 2022 Elsevier Ltd

PY - 2022/6

Y1 - 2022/6

N2 - 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.

AB - 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.

KW - Impulsive equations

KW - Pseudoparabolic equations

KW - Strong solutions

KW - Transition layer

UR - http://www.scopus.com/inward/record.url?scp=85123307899&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/69d22013-be05-349f-8362-4fb5cc493449/

U2 - 10.1016/j.nonrwa.2022.103509

DO - 10.1016/j.nonrwa.2022.103509

M3 - Article

AN - SCOPUS:85123307899

VL - 65

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

SN - 1468-1218

M1 - 103509

ER -