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This result completes our previous study \\cite{Tu-Lin} on the sub-Strauss type exponent $p<p_S(n+\\mu)$. Our novelty is to construct the suitable test function from the modified Bessel function. This approach might be also applied to the other type damping wave equations."},"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":"1711.00223","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-11-01T06:44:54Z","cross_cats_sorted":[],"title_canon_sha256":"2a717f3bd294163bccd61a76ae8ae2dc0ec6511edc979bafbe7a0f9ef140a606","abstract_canon_sha256":"6f0490424513cb64f072020b458ef1ad3bebcdd68ecfec07f674fb88fa3ca722"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:34.765402Z","signature_b64":"KQb7fx4Dy50Ir7uoWq874tIQxvg1MTgQJDhuJ1mGwQ7lCQbATd8WZ6FCkOWcF5gZAoosVsNEdYgxsinp7RXeBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1535d8394e0efd805522b36269251e2852bb6b6ee5c1d06f3e9b94e9fe143e99","last_reissued_at":"2026-05-18T00:31:34.764602Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:34.764602Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Life-Span of Semilinear Wave Equations with Scale-invariant Damping: Critical Strauss Exponent Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiayun Lin, Ziheng Tu","submitted_at":"2017-11-01T06:44:54Z","abstract_excerpt":"The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. 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