Well-posedness and kernel stability for diffusion equations with mixed measure-valued memory

We investigate a linear diffusion equation incorporating historical effects, characterised by a finite non-negative Borel measure on $(0,\mathfrak{T}]$. This approach accommodates both distributed memory and discrete delays within a unified weak formulation. The measure-valued framework encompasses the memory-free scenario, absolutely continuous kernels, purely atomic delay kernels, and mixed regimes. Our principal result is a finite-time well-posedness theorem for arbitrary finite measures, including kernels with atomic components. More precisely, we prove existence and uniqueness of weak solutions on $(-{\tau }_{\max },\mathfrak{T}]$ and derive stability bounds with constants depending explicitly on $\mathfrak{T}$, $\mu ((0,\mathfrak{T}])$, and the coercivity and boundedness parameters of the bilinear forms. Subsequently, we demonstrate continuous dependence on the kernel over fixed time intervals, leading to regime-consistency results such as vanishing-memory limits and concentration to a discrete delay. For a restricted dissipative subclass of absolutely continuous kernels, we identify a positive-type condition that results in an energy inequality, and we provide verifiable sufficient criteria, including complete monotonicity, along with an internal-variable representation.