Entropy solutions to doubly nonlinear integro-differential equations

Abstract

We consider doubly nonlinear history-dependent problems of the form ∂t[k∗(b(v)−b(v0))]−diva(x,∇v)=f. The kernel k satisfies certain assumptions which are, in particular, satisfied by k(t)=t−αΓ(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.