On Existence of Entropy Solution for a Doubly Nonlinear Differential Equation with L1-Data

---

Résumé

We consider a class of doubly nonlinear history-dependent problems having a convection term and a pseudomonotone nonlinear diffusion operator associ-
atedanequationofthetype ∂t (k∗(b(v)−b(v0 )))−div(a(x,Dv)+F(v))= f where the right hand side belongs to L1 . The kernel k belongs to the large
class of  kernels. In particular, the case of fractional time derivatives of order α∈(0,1) is included. Assuming b nondecreasing with L1-data, we
prove existence in the framework of entropy solutions. The approach adopted for the proof is based on a several step approximation method and by using a result in the case of a strictly increasing b.

Mots-clés

Fractional Time Derivative, Nonlinear Volterra Equation, Doubly Nonlinear, Entropy Solution