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

We study the following one-dimensional evolution equation: frac(∂ u, ∂ t) (x, t) = ∫_{A+ u (x, t)} λ₁ (ξ, t) (u (ξ, t) - u (x, t)) d ξ - ∫_{A- u (x, t)} λ₂ (ξ, t) (u (x, t) - u (ξ, t)) d ξ, where A⁺ u (x, t) = {ξ ∈ [0, 1] {divides} u (ξ, t) > u (x, t)}, A^- u (x, t) = [0, 1] {set minus} A⁺ u (x, t), and λ₁, λ₂ are non-negative functions. We prove the existence of solutions for a particular class of initial data 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.