TY - JOUR

T1 - On a class of nonlocal evolution equations with the p[∇u]-Laplace operator

AU - Antontsev, Stanislav

AU - Kuznetsov, Ivan

AU - Shmarev, Sergey

N1 - Funding Information: The authors would like to thank the anonymous referees for their valuable comments that helped to improve the earlier version of the paper. The authors contributed equally at all stages of the preparation of the paper. Sections 2-4 were written by the first author who was supported by the RSF grant no. 19-11-00069, Russia, and by the Portuguese Foundation for Science and Technology, Portugal, under the project UID/MAT/04561/2019. Section 1 was written by the second author supported by the Ministry of Science and Higher Education of the Russian Federation under project no. III.22.4.2, Russia. Sections 5-7 were prepared by the third author who acknowledges the support of the Ministry of Economy, Industry, and Competitiveness grant no. MTM2017-87162-P, Spain. Publisher Copyright: © 2021 Elsevier Inc. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/9/15

Y1 - 2021/9/15

N2 - We study the homogeneous Dirichlet problem for the class of singular parabolic equations ut−div(|∇u|p[∇u]−2∇u)=fin Ω×(0,T), where Ω⊂Rd, d≥2, is a smooth domain. The exponent p nonlocally depends on the gradient of the solution: p is a given function defined by p[∇u]≡p(l(|∇u|)),l(|s|)=∫Ω|s|αdx with a constant α∈(1,2]. We find sufficient conditions on the data that guarantee global in time existence and uniqueness of a strong solution of the problem. It is shown that the problem has a solution if either u0 and f, or p′(s) are sufficiently small.

AB - We study the homogeneous Dirichlet problem for the class of singular parabolic equations ut−div(|∇u|p[∇u]−2∇u)=fin Ω×(0,T), where Ω⊂Rd, d≥2, is a smooth domain. The exponent p nonlocally depends on the gradient of the solution: p is a given function defined by p[∇u]≡p(l(|∇u|)),l(|s|)=∫Ω|s|αdx with a constant α∈(1,2]. We find sufficient conditions on the data that guarantee global in time existence and uniqueness of a strong solution of the problem. It is shown that the problem has a solution if either u0 and f, or p′(s) are sufficiently small.

KW - Nonlocal evolution equations

KW - Singular parabolic equation

KW - Variable nonlinearity

UR - http://www.scopus.com/inward/record.url?scp=85103697728&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2021.125221

DO - 10.1016/j.jmaa.2021.125221

M3 - Article

AN - SCOPUS:85103697728

VL - 501

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

M1 - 125221

ER -