This paper is devoted to solving a degenerate parabolic integrodifferential equation with the Robin boundary condition. We begin with solving the equation without the integral delay term. For that purpose we introduce some new unknown function following Favini and Yagi  and construct the fundamental solution to the equation to be satisfied by it by the method of Kato and Tanabe . Using this fundamental solution we transform the original problem to an easily solvable integral equation for the time derivative of the new unknown function.
Angelo FAVINI. Alfredo LORENZI. Hiroki TANABE. "Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: -theory." J. Math. Soc. Japan 61 (1) 133 - 176, January, 2009. https://doi.org/10.2969/jmsj/06110133