The aim of the present paper is to state and discuss the well-posedness of a new evolution inclusion governed by the subdifferential of a function $\varphi $ perturbed both by a Carathéodory mapping and by an integral forcing term. The integrand of the forcing term depends on two time-variables. We prove the existence and uniqueness of local and global solution, assuming that $\varphi $ is primal lower regular and the perturbation terms satisfy some standard conditions. The results are applied to the study of non-regular electrical circuits with tranmission line, Zener diode and inductor. A second application is concerned with semilinear heat equation with memory. These applications illustrate an additional novelty of our paper.
