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Here the function $f(u,u_t)$ stands for the power nonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. We are interested in investigating $L^{1}$ estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.02706","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-08-08T10:11:01Z","cross_cats_sorted":[],"title_canon_sha256":"ead97e2ba27303063e7f4a9c3ca58aa42e4bb3665c88d5cefe60e163aa4c9a38","abstract_canon_sha256":"f9e2c5c85a7486931f611d228d58f8ab9c58a3dc646d502075cf0707a89616fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:55.543542Z","signature_b64":"MFiGhlmS5PPHcZAwJabsvtGNVsfPqpJMIrX+mZnX8lGOEmRcEBSVizdJEkSxi6w2fHjjx9DGTORh/wO2U03fAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67ecc9804d03169a4f02363cae6b12a1b44b31d14e85fd945a443dce9dbb6de1","last_reissued_at":"2026-05-18T00:03:55.543040Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:55.543040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $\\sigma$-evolution models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Reissig, Tuan Anh Dao","submitted_at":"2018-08-08T10:11:01Z","abstract_excerpt":"We study the following Cauchy problems for semi-linear structurally damped $\\sigma$-evolution models: \\begin{equation*} u_{tt}+ (-\\Delta)^\\sigma u+ \\mu (-\\Delta)^\\delta u_t = f(u,u_t),\\, u(0,x)= u_0(x),\\, u_t(0,x)=u_1(x) \\end{equation*} with $\\sigma \\ge 1$, $\\mu>0$ and $\\delta \\in (0,\\frac{\\sigma}{2})$. Here the function $f(u,u_t)$ stands for the power nonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. 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