{
  "abstract": "We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several ``resonances\" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-λI)^{-1}||_H$ on ~$i\\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $λ$, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\\mathcal R(I-S(T))\\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17712",
  "author_total": 3,
  "authors": [
    {
      "name": "Boris Muha"
    },
    {
      "name": "Sebastian Schwarzacher"
    },
    {
      "name": "Justin T. Webster"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/drwjax699yr3b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17712",
  "fulltext_url": null,
  "id": "drwjax699yr3b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17712",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7213520511",
  "pdf_url": "https://arxiv.org/pdf/2609.17712",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/drwjax699yr3b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2609.17712",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17712"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Range Failure, Resolvent Growth, and Resonance in Partially Dissipative Systems",
  "title_text": "Range Failure, Resolvent Growth, and Resonance in Partially Dissipative Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/drwjax699yr3b",
  "venue": "arXiv",
  "versions": []
}