I encountered the following integro-differential equation attempting to solve a system of PDEs: $$ \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2} - a c \exp\left(-b\int_0^t c dt\right). $$ I did some searching but I couldn't find anything on integro-differential equations where the integral is the argument in a non-linear function.
I don't expect there to be anything close to a solution, but do any of you know if there is at least some literature on these types of equations? Or some numerical tools that are better equipped to solve this than estimating the integral based on all the values until the previous time step?