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The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. The blow-up profile does not scale as $(T-t)^{1/2}|\\log(T-t)|^{1/2},$ like in the standard nonlinear heat equa"},"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":"1506.08306","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-27T16:08:28Z","cross_cats_sorted":[],"title_canon_sha256":"ba07d627e6cfb40ebfcb75a9d54d906dfc3400befdda3cf8cd8da117d44714e9","abstract_canon_sha256":"c797f69f037842c025c00198a1784e745ee8a97242aceba0c4b6e91497b26893"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:46.833896Z","signature_b64":"IybOb0GZYhOf4vJ6Yskcvv3JyHyPS9auWxYKwe3HkjYbg/iP966j4IvP8PKXsDQvDWOFOjyoSAqJiI+YJtOVCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0cecbd59a222a6a4b96a8a771e3ae6bc8e5669314ef698ae25e5f10248bde2bd","last_reissued_at":"2026-05-18T01:37:46.833511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:46.833511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence of a Stable Blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hatem Zaag, Slim Tayachi","submitted_at":"2015-06-27T16:08:28Z","abstract_excerpt":"We consider the nonlinear heat equation with a nonlinear gradient term: $\\partial_t u =\\Delta u+\\mu|\\nabla u|^q+|u|^{p-1}u,\\; \\mu>0,\\; q=2p/(p+1),\\; p>3,\\; t\\in (0,T),\\; x\\in \\R^N.$ We construct a solution which blows up in finite time $T>0.$ We also give a sharp description of its blow-up profile and show that it is stable with respect to perturbations in initial data. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. 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