Entropy solutions to doubly nonlinear integro-differential equations. (March 2020)

• Record Type: Journal Article
• Authors: Sapountzoglou, Niklas
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: 45K05 -- 47J35 -- 45D05 -- 35D99; Nonlinear Volterra equation -- Doubly nonlinear -- Entropy solution; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111656
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider doubly nonlinear history-dependent problems of the form ∂ t [ k ∗ ( b ( v ) − b ( v 0 ) ) ] − div a ( x, ∇ v ) = f . The kernel k satisfies certain assumptions which are, in particular, satisfied by k ( t ) = t − α Γ ( 1 − α ), i.e., the case of fractional derivatives of order α ∈ ( 0, 1 ) is included. We show existence of entropy solutions in the case of a nondecreasing b . An existence result in the case of a strictly increasing b is used to get entropy solutions of approximate problems. Kruzhkov's method of doubling variables, a comparison principle and the diagonal principle are used to obtain a.e. convergence for approximate solutions. A uniqueness result has been shown in a previous work.