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First, we revisit the local well-posedness in $H^1(\\mathbb{R}^d)$ for (INLS) of Guzm\\'an [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. $\\mu=-1$ when $0<\\alpha<\\alpha^\\star$ where $\\alpha^\\star = \\frac{"},"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":"1710.01392","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-10-03T21:30:08Z","cross_cats_sorted":[],"title_canon_sha256":"d56164afd47501e41d95ccee57921b3217415c7b03523afc2e1fb35e5f1516f6","abstract_canon_sha256":"6b0c3a8a1efc4f1a67d9fecf9d8ae397730f44633b5fe98f4fb70e727099d91c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:36:25.016259Z","signature_b64":"hqcNwRHERi0W/9tWQxW7gHnRNKvhDj6OX9y8MujwonuX16mVD81Dl7gBeGQW8Y7mSzzZIT8RN1sHdf27Ghl/BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b86d044dee9f74a56052c98712168189e7800ba931ec1d41e2b26739683defa","last_reissued_at":"2026-07-05T01:36:25.015784Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:36:25.015784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Van Duong Dinh","submitted_at":"2017-10-03T21:30:08Z","abstract_excerpt":"In this paper, we consider the following inhomogeneous nonlinear Schr\\\"odinger equation (INLS) \\[ i\\partial_t u + \\Delta u + \\mu |x|^{-b} |u|^\\alpha u = 0, \\quad (t,x)\\in \\mathbb{R} \\times \\mathbb{R}^d \\] with $b, \\alpha>0$. First, we revisit the local well-posedness in $H^1(\\mathbb{R}^d)$ for (INLS) of Guzm\\'an [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. 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