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We show that in the defocusing case, solutions are global and have finite global $L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state.\n  These results rely on analogous statements for the two-dimensional cubic nonlinear Schr\\\"odinger equation, which are kno"},"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":"1008.2712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-08-16T17:08:36Z","cross_cats_sorted":[],"title_canon_sha256":"04e461960de3f299537f26139b15e999a8fadaf08bca281b274cbb26467d0868","abstract_canon_sha256":"cc5481a01c29ea197ad34188c5d29c018ba08163e6489a6e8fe6e4d4c5330df3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:12.734915Z","signature_b64":"yLPIZPPfq/vXnh38HobKhzkcjlmg5PWGEh5LZgERZV3kOu03Ynmqa25YtEiLtwF72s4ZudREwNVs2Xp9xKUmCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41640b5ce46ca7baad2e4649b3e76b1fa796621e1aabd33df2d9cbafb335533c","last_reissued_at":"2026-05-18T04:42:12.734254Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:12.734254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering for the cubic Klein--Gordon equation in two space dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Betsy Stovall, Monica Visan, Rowan Killip","submitted_at":"2010-08-16T17:08:36Z","abstract_excerpt":"We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \\Delta u + u \\pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\\in H^1_x$ and $u_t(0)\\in L^2_x$. 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