Graph-space well-posedness for diffusion equations with degenerate instantaneous diffusion

We study diffusion equations with completely monotone memory when the instantaneous diffusion form is merely non-negative and may therefore lose coercivity. For a kernel whose Bernstein representing measure has finite total mass $M_0=\nu ([0,\mathrm{\infty }))$, we introduce an extended state consisting of the physical variable and its continuum of internal variables. The aggregation and constant-embedding operators are adjoint with respect to the memory energy, and the resulting cross-term cancellation makes the augmented generator $m$-dissipative. This yields a unique mild solution, Lipschitz dependence on the data, and a contraction estimate that contains no positive lower bound for the instantaneous form. The zero-prehistory trajectories form a memory graph space, in which the problem is well posed in the sense of Hadamard. If, in addition, the first Bernstein moment $M_1={\int }_{[0,\mathrm{\infty })}\lambda \text{diff}\nu (\lambda )$ is finite, the memory potential and first-moment field possess the regularity needed to identify the semigroup solution with an encoded weak formulation and to obtain explicit stability bounds. We further prove uniform norm-resolvent convergence and convergence of the associated semigroups when a coercive instantaneous contribution vanishes. Under an additional $L^2(0,\text{Tend};V)$-regularity assumption on the limiting solution, the convergence rate in the memory graph norm is $O({\epsilon }^{1\mathrm{/}2})$. These results provide a continuous stability target for structure-preserving and certified discretisations of memory-dominated diffusion.