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For any given positive integer $k\\geq2$, we construct a solution $u(t,x)$ with $k$ interfaces, which has the form $$ u(t,x)\\approx\\sum_{j=1}^k(-1)^{j+1}\\omega\\big(x-\\gamma_j(t)\\big)-\\frac{1+(-1)^k}{2}\\qquad \\text{as}\\ t\\rightarrow +\\infty, $$ where $\\omega$ is the solution to the Allen-Cahn equation $$ \\o"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2303.17288","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2023-03-30T10:53:13Z","cross_cats_sorted":[],"title_canon_sha256":"3a29c831b2bd851632c068297ba808a10f052693c806745a178916532acae5c3","abstract_canon_sha256":"fec83a7b6915f8b3f1d915d1ed60c622102ff6827d2f5ae71963a187db3d54b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:56:32.488571Z","signature_b64":"4NK2XI7LxhnvoFj6PltTHJj1Mq/efr9a2GVes6bPdN3tCMWaRTKFodDd9jL1/k7CKeIBgqIssqXEykn8wON0Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1e1cb2b5a08e9c1c7f164b2e5909c2bdb5d8acd03b584a446fb700fa3b6b853","last_reissued_at":"2026-07-05T05:56:32.488178Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:56:32.488178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple-interface solutions of one dimensional generalized parabolic Cahn-Hilliard equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chao Liu, Jun Yang","submitted_at":"2023-03-30T10:53:13Z","abstract_excerpt":"We consider one dimensional generalized parabolic Cahn-Hilliard equation $$ u_t=-\\partial_{xx}\\big[\\partial_{xx}u-W'(u)\\big]+W''(u)\\big[\\partial_{xx} u -W'(u)\\big], \\qquad \\forall\\, (t,x)\\in [0,+\\infty)\\times {\\mathbb R}, $$ where the function $W(\\cdot)$ is the standard double-well potential. For any given positive integer $k\\geq2$, we construct a solution $u(t,x)$ with $k$ interfaces, which has the form $$ u(t,x)\\approx\\sum_{j=1}^k(-1)^{j+1}\\omega\\big(x-\\gamma_j(t)\\big)-\\frac{1+(-1)^k}{2}\\qquad \\text{as}\\ t\\rightarrow +\\infty, $$ where $\\omega$ is the solution to the Allen-Cahn equation $$ \\o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.17288","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.17288/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2303.17288","created_at":"2026-07-05T05:56:32.488244+00:00"},{"alias_kind":"arxiv_version","alias_value":"2303.17288v1","created_at":"2026-07-05T05:56:32.488244+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.17288","created_at":"2026-07-05T05:56:32.488244+00:00"},{"alias_kind":"pith_short_12","alias_value":"4HQ4WK22BDU4","created_at":"2026-07-05T05:56:32.488244+00:00"},{"alias_kind":"pith_short_16","alias_value":"4HQ4WK22BDU4DR7R","created_at":"2026-07-05T05:56:32.488244+00:00"},{"alias_kind":"pith_short_8","alias_value":"4HQ4WK22","created_at":"2026-07-05T05:56:32.488244+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP","json":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP.json","graph_json":"https://pith.science/api/pith-number/4HQ4WK22BDU4DR7RMSZOLEE4FP/graph.json","events_json":"https://pith.science/api/pith-number/4HQ4WK22BDU4DR7RMSZOLEE4FP/events.json","paper":"https://pith.science/paper/4HQ4WK22"},"agent_actions":{"view_html":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP","download_json":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP.json","view_paper":"https://pith.science/paper/4HQ4WK22","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2303.17288&json=true","fetch_graph":"https://pith.science/api/pith-number/4HQ4WK22BDU4DR7RMSZOLEE4FP/graph.json","fetch_events":"https://pith.science/api/pith-number/4HQ4WK22BDU4DR7RMSZOLEE4FP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP/action/storage_attestation","attest_author":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP/action/author_attestation","sign_citation":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP/action/citation_signature","submit_replication":"https://pith.science/pith/4HQ4WK22BDU4DR7RMSZOLEE4FP/action/replication_record"}},"created_at":"2026-07-05T05:56:32.488244+00:00","updated_at":"2026-07-05T05:56:32.488244+00:00"}