Entropy solutions to doubly nonlinear integro-differential equations

We consider doubly nonlinear history-dependent problems of the form partial derivative(t)[k*(b(v) - b(v(0)))] - div a( x, del v) = f. The kernel k satisfies certain assumptions which are, in particular, satisfied by k(t) = t(-alpha)/Gamma(1-a), i.e., the case of fractional derivatives of order alpha is an element of (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. (C) 2019 Elsevier Ltd. All rights reserved.