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

Abstract

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 $M_0=\hat{I}½([0,\infty ))$, 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 $M_{1}=\int_{[0,\infty)}λ\,\diffν(λ)$ 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 $L^{2}(0,\Tend;V)$-regularity assumption on the limiting solution, the convergence rate in the memory graph norm is $O({\epsilon }^{1/2})$. These results provide a continuous stability target for structure-preserving and certified discretisations of memory-dominated diffusion.

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