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We show an upper bound for any blow-up solution of (1). Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, namely $u'' =|u|^{p-1}u\\log^a (2+u^2)$ Unlike the pure power case ($g(u)=|u|^{p-1}u$) the difficulties here are due to the fact t","authors_text":"Hatem Zaag, Mohamed Ali Hamza","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-28T06:47:18Z","title":"The blow-up rate for a non-scaling invariant semilinear wave equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.12059","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1d5b2369a4fc0fbb8297138688392c89f46006c8b7e98462fb2a5318afa03c27","target":"record","created_at":"2026-05-17T23:41:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3eecf61ebbb21256dbb62b28c6cc8df1a7ef006d0e31646500fbfeebe7cbd636","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-28T06:47:18Z","title_canon_sha256":"5c71bd1a9d1639bf168924018c43d32012b0eaac44291e6b735b44261d8eee9a"},"schema_version":"1.0","source":{"id":"1906.12059","kind":"arxiv","version":1}},"canonical_sha256":"d976a5c29f3d4132eedd00eb000ff3dcff5743829eb5c61011bddb94434d2d0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d976a5c29f3d4132eedd00eb000ff3dcff5743829eb5c61011bddb94434d2d0d","first_computed_at":"2026-05-17T23:41:59.391370Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:59.391370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K8w0w4AnTYU0Bk5/gLCWMo01EQB2bIDy3Va+mM450ehD2sM9MMHbuDeZHY7zdkYNvd4lVNxP/Gr2aqRFAhR9AA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:59.391827Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.12059","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d5b2369a4fc0fbb8297138688392c89f46006c8b7e98462fb2a5318afa03c27","sha256:709689a1ac85676055888ed75aa771ae7ad3c45887d979842c36d230bb9f7f29"],"state_sha256":"8431a54c7882b5e5fcd97d5e1caf4a8ba3d226c32588afbb836b97240f6ffcf4"}