On uniqueness of local entropy solution of a convection-diffusion type integro-differential equation

We study the uniqueness of entropy solution for a class of triply nonlinear parabolic integro-differential equations of the form: ∂t(k * (j(v) - j(v₀))) - ∇ · (a(x, ∇φ(v)) + F(φ(v))) = f in a bounded domain with homogeneous Dirichlet boundary conditions. The source term f belongs to L¹ and the memory term k * (j(v) - j(v₀)) introduces a nonlocal dependence. The functions j(v) and φ(v), assumed to be non-decreasing, further contribute to the nonlinear nature of the problem. To prove uniqueness, we apply the method of doubling variables, leading to an energy estimate that ensures the desired result.