TY - JOUR

T1 - Nonlocal evolution equations with p[u(x, t)]-Laplacian and lower-order terms

AU - Antontsev, Stanislav

AU - Shmarev, Sergey

PY - 2020/6/1

Y1 - 2020/6/1

N2 - We study the homogeneous Dirichlet problem for a class of nonlocal singular parabolic equations ut-div(|∇u|p[u]-2∇u)=f((x,t),u,l(u))inQT=Ω×(0,T),where Ω⊂ Rd, d≥ 2 , is a smooth domain, p[u] = p(l(u)) is a given function with values in the interval [p-,p+]⊂(2dd+2,2), and l(u)=∫Ω|u(x,t)|αdx, α∈ [1 , 2] , is a functional of the unknown solution. We prove the existence of a strong solution such that ut∈L2(QT),u∈L∞(0,T;W01,2(Ω)),|Dij2u|p[u]∈L1(QT).Conditions of uniqueness of strong solutions are obtained.

AB - We study the homogeneous Dirichlet problem for a class of nonlocal singular parabolic equations ut-div(|∇u|p[u]-2∇u)=f((x,t),u,l(u))inQT=Ω×(0,T),where Ω⊂ Rd, d≥ 2 , is a smooth domain, p[u] = p(l(u)) is a given function with values in the interval [p-,p+]⊂(2dd+2,2), and l(u)=∫Ω|u(x,t)|αdx, α∈ [1 , 2] , is a functional of the unknown solution. We prove the existence of a strong solution such that ut∈L2(QT),u∈L∞(0,T;W01,2(Ω)),|Dij2u|p[u]∈L1(QT).Conditions of uniqueness of strong solutions are obtained.

KW - Nonlocal equation

KW - Singular parabolic equation

KW - Strong solutions

KW - Variable nonlinearity

KW - P(U)-LAPLACIAN

KW - VARIABLE EXPONENT

KW - UNIQUENESS

UR - http://www.scopus.com/inward/record.url?scp=85084121184&partnerID=8YFLogxK

U2 - 10.1007/s41808-020-00065-x

DO - 10.1007/s41808-020-00065-x

M3 - Article

AN - SCOPUS:85084121184

VL - 6

SP - 211

EP - 237

JO - Journal of Elliptic and Parabolic Equations

JF - Journal of Elliptic and Parabolic Equations

SN - 2296-9020

IS - 1

ER -