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Prior work by Li\u00a0et\u00a0al.\u00a0(Physica D 423:132916, 2021) has demonstrated the existence of such waves for two classes of regularizations, including viscous relaxation (see Li et al. in Physica D 423:132916, 2021). Their analysis uses geometric singular perturbation theory: for sufficiently small values of a parameter <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varepsilon &gt; 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b5<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> characterizing the \u2018strength\u2019 of the regularization, the waves are constructed as perturbations of a singular heteroclinic orbit. Here we show rigorously that these waves are spectrally stable for the case of viscous relaxation. Our approach is to show that for sufficiently small <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varepsilon &gt;0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b5<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the \u2018full\u2019 eigenvalue problem of the regularized system is controlled by a reduced <jats:italic>slow eigenvalue problem<\/jats:italic> defined for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varepsilon = 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b5<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In the course of our proof, we examine the ways in which this geometric construction complements and differs from constructions of other reduced eigenvalue problems that are known in the wave stability literature.\n<\/jats:p>","DOI":"10.1007\/s00332-023-09941-x","type":"journal-article","created":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T13:02:19Z","timestamp":1689339739000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Spectral Stability of Shock-fronted Travelling Waves Under Viscous Relaxation"],"prefix":"10.1007","volume":"33","author":[{"given":"Ian","family":"Lizarraga","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Marangell","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,14]]},"reference":[{"key":"9941_CR1","first-page":"167","volume":"410","author":"J Alexander","year":"1990","unstructured":"Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of travelling waves. 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