## Existence of entropy solutions to a doubly nonlinear integro-differential equation.(English)Zbl 1449.45021

The authors consider a class of doubly nonlinear problems with memory. They consider kernels of the type $$k(t)=t^{-\alpha}/\Gamma(1-\alpha)$$. Doing so, the time-derivatives side becomes the fractional derivative of order $$\alpha\in(0,1)$$ in the sense of Riemann-Liouville. The uniqueness of entropy solutions has been shown in a previous work. In this paper, the authors prove the existence of entropy solutions for general $$L^1$$-data and Dirichlet boundary conditions. The main idea of the existence proof is a modification of the regularization method by R. Landes [J. Reine Angew. Math. 393, 21–38 (1989; Zbl 0664.35027)].

### MSC:

 45K05 Integro-partial differential equations 47J35 Nonlinear evolution equations 45D05 Volterra integral equations 35D99 Generalized solutions to partial differential equations

Zbl 0664.35027