Existence and unicity of solutions for a non-local relaxation equation

We study the following one-dimensional evolution equation: ∂u∂t(x,t)=∫A+u(x,t)λ1(ξ,t)(u(ξ,t)−u(x,t))dξ−∫A−u(x,t)λ2(ξ,t)(u(x,t)−u(ξ,t))dξ, where A+u(x,t)={ξ∈[0,1]∣u(ξ,t)>u(x,t)},A−u(x,t)=[0,1]∖A+u(x,t), and λ1λ1, λ2λ2 are non-negative functions.We prove the existence of solutions for a particular class of initial data u(x,0)u(x,0). We also prove that the solutions are unique. Finally, under additional constraints on the initial data, we give an explicit expression for the solution.