TY - JOUR

T1 - Global existence and regularity for a pseudo-parabolic equation with p(x,t)-Laplacian

AU - Antontsev, Stanislav

AU - Kuznetsov, Ivan

AU - Shmarev, Sergey

N1 - Acknowledgements: The authors would like to thank the anonymous referees for their valuable remarks and recommendations that helped improve the earlier version of the paper. The first and second authors are supported by the Ministry of Science and Higher Education of the Russian Federation under project no. FWGG-2021-0010-2.3.1.2.11, Russian Federation. The third author acknowledges the support of the Research Grant MCI-21-PID2020-116287GB-I00, Spain.

PY - 2023/10/1

Y1 - 2023/10/1

N2 - We study the Dirichlet problem for the pseudo-parabolic equation ut=div(|∇u|p(x,t)−2∇u)+Δut+f(x,t,u,∇u) in the cylinder (x,t)∈QT=Ω×(0,T), Ω⊂Rd, d≥2. It is shown that under appropriate conditions on the regularity of the data and the growth of the source f with respect to the second and third arguments, the problem has a global in time solution with the properties u∈L∞(0,T;H02(Ω)),ut,|∇ut|∈L2(QT),|∇u|∈L∞(0,T;Lp(⋅)(Ω))∩Lp(⋅,⋅)+δ(QT) with some δ>0. For special choices of the source f, sufficient conditions of uniqueness are derived, stability of solutions with respect to perturbations of the nonlinear structure of the equation is proven, and the rate of vanishing of ‖u‖W1,2(Ω) is found.

AB - We study the Dirichlet problem for the pseudo-parabolic equation ut=div(|∇u|p(x,t)−2∇u)+Δut+f(x,t,u,∇u) in the cylinder (x,t)∈QT=Ω×(0,T), Ω⊂Rd, d≥2. It is shown that under appropriate conditions on the regularity of the data and the growth of the source f with respect to the second and third arguments, the problem has a global in time solution with the properties u∈L∞(0,T;H02(Ω)),ut,|∇ut|∈L2(QT),|∇u|∈L∞(0,T;Lp(⋅)(Ω))∩Lp(⋅,⋅)+δ(QT) with some δ>0. For special choices of the source f, sufficient conditions of uniqueness are derived, stability of solutions with respect to perturbations of the nonlinear structure of the equation is proven, and the rate of vanishing of ‖u‖W1,2(Ω) is found.

KW - Higher regularity

KW - Singular pseudo-parabolic equation

KW - Variable nonlinearity

KW - Weak solutions

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85150334927&origin=inward&txGid=db41240b9282616f8940f3aa5b38819f

UR - https://www.mendeley.com/catalogue/4797fb20-a094-3b6c-b65f-29095ca63ee7/

U2 - 10.1016/j.jmaa.2023.127202

DO - 10.1016/j.jmaa.2023.127202

M3 - Article

VL - 526

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

M1 - 127202

ER -