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

TL;DR

This paper establishes well-posedness and stability for diffusion equations with measure-valued memory, unifying distributed and discrete delay effects, and explores regime limits and energy properties.

Contribution

It proves finite-time existence, uniqueness, and stability of solutions for general measure-based memory kernels, including atomic and continuous cases, with explicit bounds.

Findings

Existence and uniqueness of weak solutions for arbitrary finite measures.

Stability bounds depending explicitly on measure and parameters.

Continuous dependence and regime limits such as vanishing-memory and discrete delay.

Abstract

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.