TY - JOUR

T1 - Strong solutions of a semilinear impulsive pseudoparabolic equation with an infinitesimal initial layer

AU - Antontsev, Stanislav

AU - Kuznetsov, Ivan

AU - Sazhenkov, Sergey

AU - Shmarev, Sergey

N1 - The first, second, and third authors are supported by the Ministry of Science and Higher Education of the Russian Federation under project no. FWGG-2021-0010 , Russian Federation. The fourth author acknowledges the support of the Research Grant MCI-21-PID2020-116287GB-I00, Spain.

PY - 2024/2/1

Y1 - 2024/2/1

N2 - We study the multi-dimensional initial-boundary value problem for the semilinear pseudoparabolic equation with a regular nonlinear minor term, which, in general, may be superlinear. This term models a non-instantaneous but a very rapid absorption with q(x)-growth. The minor term depends on a positive integer parameter n and, as n→+∞, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as n→+∞, and that the family of regular weak solutions to 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, boundary, and matching conditions, so that the ‘outer’ macroscopic solution beyond the initial layer is governed by the linear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the initial 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 absorption. In general, the research is devoted to pseudoparabolic equations with measure data depending on an unknown solution.

AB - We study the multi-dimensional initial-boundary value problem for the semilinear pseudoparabolic equation with a regular nonlinear minor term, which, in general, may be superlinear. This term models a non-instantaneous but a very rapid absorption with q(x)-growth. The minor term depends on a positive integer parameter n and, as n→+∞, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as n→+∞, and that the family of regular weak solutions to 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, boundary, and matching conditions, so that the ‘outer’ macroscopic solution beyond the initial layer is governed by the linear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the initial 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 absorption. In general, the research is devoted to pseudoparabolic equations with measure data depending on an unknown solution.

KW - Impulsive partial differential equation

KW - Initial layer

KW - Measure data

KW - Pseudoparabolic equation

KW - Semilinear

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85171464062&origin=inward&txGid=eaa5db697f1a3ff538b2579d7019fc4f

UR - https://www.mendeley.com/catalogue/202eb13a-63f6-3283-a44d-7680cb8f5679/

U2 - 10.1016/j.jmaa.2023.127751

DO - 10.1016/j.jmaa.2023.127751

M3 - Article

VL - 530

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

M1 - 127751

ER -