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)].


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


Zbl 0664.35027