Heterogeneous Viscoelastic Wave Propagation with Exponential Memory: Variational Formulation, Spectral Perturbation Theory, and Essential Spectrum Invariance

Many natural and engineered materials exhibit hereditary constitutive behavior in which the present stress depends upon the accumulated deformation history. Mathematical models describing such media typically incorporate memory through convolution operators whose analysis requires infinite-dimensional state spaces. While the theory of homogeneous viscoelastic equations with fading memory is well established, comparatively less attention has been devoted to the operator-theoretic consequences of spatially heterogeneous exponential-memory perturbations acting directly within the constitutive law. This paper develops a functional-analytic framework for a class of linear viscoelastic wave equations whose relaxation kernel possesses bounded spatial modulation and exponential temporal decay. Introducing the Dafermos history variable transforms the hereditary evolution equation into an autonomous first-order system posed on a weighted Hilbert history space. Under boundedness, ellipticity, and positivity assumptions on the material coeficients, the associated evolution operator is realized as the generator of a strongly continuous semigroup. The heterogeneous contribution is represented as a bounded perturbation of a homogeneous reference operator. Within the assumptions established in this paper, the perturbation framework is then used to analyze both the essential and discrete spectral components. Under the hypotheses proved in the manuscript, the essential spectrum is shown to be invariant under the admissible heterogeneous perturbations through an application of Weyl-type perturbation arguments. Isolated spectral components are subsequently examined using Riesz spectral projectors, yielding continuity of finite-dimensional spectral subspaces for suficiently small perturbations. Finally, a finite-element continuation framework is introduced to investigate the numerical evolution of isolated eigenvalues as the heterogeneous perturbation parameter varies. These computations are presented as numerical investigations that complement the analytical theoryrather than as mathematical proofs. The resulting framework provides a rigorous basis for studying heterogeneous memory efects within linear viscoelastic wave propagation.