A unified theory for diffusion with memory and delay via measure-valued kernels

Abstract: We investigate a diffusion equation that incorporates historical effects, in which the memory-delay mechanism is represented by a nonnegative Borel measure $μ$ on the interval $(0, \mathfrak T]$. The diffusion component is characterised by a coercive bilinear form $a_0$, while the historical term is modelled as a measure-induced convolution operator $\mathcal K_μ$, derived from a bounded non-negative form $a_1$. This framework includes the memory-free parabolic problem ($μ=0$), distributed-memory Volterra models ($dμ(s)=k(s)\,ds$), discrete-delay models with atomic measures, and mixed kernels. For general finite measures $μ$, we establish well-posedness over finite time horizons and derive stability bounds that remain applicable in the presence of delays. Subsequently, we identify a positive-type condition under which the memory term is dissipative, leading to a refined energy inequality. For absolutely continuous kernels, we provide verifiable sufficient conditions, including complete monotonicity, and an internal-variable representation that elucidates the decay mechanism.