On a nonlinear elliptic–parabolic integro-differential equation with L1-data

We generalize the concept of entropy solutions for parabolic equations with L1-data and consider a class of nonlinear history-dependent degenerated elliptic– parabolic equations including problems with a fractional time derivative such as (delta)/(delta)f (b(v)-b(v(omega)))-div a(x,Dv)=f with Dirichlet boundary conditions and initial condition, where 0<(gamma)=<1. We show uniqueness of entropy solutions for general L1-data by Kruzhkovʹs method of doubling variables. Moreover, existence in the nondegenerated case, i.e. b(identical)id, is shown by using the concept of generalized solutions of a corresponding abstract Volterra equation.