{"_buckets": {"deposit": "a2af046d-4b92-468e-8d44-bb0fc63529db"}, "_deposit": {"id": "40049", "owners": [], "pid": {"revision_id": 0, "type": "depid", "value": "40049"}, "status": "published"}, "_oai": {"id": "oai:repository.dl.itc.u-tokyo.ac.jp:00040049"}, "item_4_biblio_info_7": {"attribute_name": "\u66f8\u8a8c\u60c5\u5831", "attribute_value_mlt": [{"bibliographicIssueDates": {"bibliographicIssueDate": "2011-03-29", "bibliographicIssueDateType": "Issued"}, "bibliographicIssueNumber": "4", "bibliographicPageEnd": "358", "bibliographicPageStart": "323", "bibliographicVolumeNumber": "17", "bibliographic_titles": [{"bibliographic_title": "Journal of mathematical sciences, the University of Tokyo"}]}]}, "item_4_description_5": {"attribute_name": "\u6284\u9332", "attribute_value_mlt": [{"subitem_description": "As a basic model, the Cauchy problem in RN\u00d7R+ for the 2mth-order semilinear parabolic equation of the diffusion-absorp\\-tion type ut = -(-\u0394)mu - t\u03b1|u|(p-1)u, with p \u003e 1, \u03b1 \u003e 0, m \u2267 2, with singular initial data u0(x) \u2262 0 such that u0(x) = 0 for any x \u2260 0, is studied. The additional multiplier h(t) = t\u03b1 \u2192 0 as t \u2192 0 in the absorption term plays a role of a time-dependent non-homogeneous potential that affects the strength of the absorption term in the PDE. Existence and nonexistence of the corresponding very singular solutions (VSSs) is studied. For m = 1 and h(t) \u2261 1, first nonexistence result for p \u2267 p0 = 1 + 2/N was proved in the celebrated paper by Brezis and Friedman in 1983. Existence of VSSs in the complement interval 1 \u003c p \u003c p0 was established in the middle of the 1980s. The main goal is to justify that, in the subcritical range 1 \u003c p \u003c p0 = 1 + {2m(1+\u03b1)}/N, there exists a finite number of different VSSs of the self-similar form u*(x,t) = t(-\u03b2)V(y), y = x/t(1/2m), \u03b2 = (1+\u03b1)/(p-1), where each V is an exponentially decaying as y \u2192 \u221e solution of the elliptic equation -(-\u0394)mV + 1/2my\u30fb\u2207V + \u03b2V - |V|(p-1)V = 0 in RN. Complicated families of VSSs in 1D and also non-radial VSS patterns in RN are detected. Some of these VSS profiles Vl are shown to bifurcate from 0 at the bifurcation exponents pl = 1 + {2m(1+\u03b1)}/(l+N), where l = 0,1,2,...", "subitem_description_type": "Abstract"}]}, "item_4_publisher_20": {"attribute_name": "\u51fa\u7248\u8005", "attribute_value_mlt": [{"subitem_publisher": "Graduate School of Mathematical Sciences, The University of Tokyo"}]}, "item_4_source_id_10": {"attribute_name": "\u66f8\u8a8c\u30ec\u30b3\u30fc\u30c9ID", "attribute_value_mlt": [{"subitem_source_identifier": "AA11021653", "subitem_source_identifier_type": "NCID"}]}, "item_4_source_id_8": {"attribute_name": "ISSN", "attribute_value_mlt": [{"subitem_source_identifier": "13405705", "subitem_source_identifier_type": "ISSN"}]}, "item_4_subject_15": {"attribute_name": "\u65e5\u672c\u5341\u9032\u5206\u985e\u6cd5", "attribute_value_mlt": [{"subitem_subject": "415", "subitem_subject_scheme": "NDC"}]}, "item_4_text_16": {"attribute_name": "Mathematical Reviews Number", "attribute_value_mlt": [{"subitem_text_value": "MR"}]}, "item_4_text_17": {"attribute_name": "Mathmatical Subject Classification", "attribute_value_mlt": [{"subitem_text_value": "35K55(MSC2010)"}, {"subitem_text_value": "35K40(MSC2010)"}, {"subitem_text_value": "35K65(MSC2010)"}]}, "item_4_text_33": {"attribute_name": "\u539f\u7a3f\u53d7\u9818\u65e5", "attribute_value_mlt": [{"subitem_text_value": "2007-11-05"}]}, "item_4_text_34": {"attribute_name": "\u8cc7\u6e90\u30bf\u30a4\u30d7", "attribute_value_mlt": [{"subitem_text_value": "Departmental Bulletin Paper"}]}, "item_4_text_4": {"attribute_name": "\u8457\u8005\u6240\u5c5e", "attribute_value_mlt": [{"subitem_text_value": "Department of Mathematical Sciences, University of Bath"}]}, "item_creator": {"attribute_name": "\u8457\u8005", "attribute_type": "creator", "attribute_value_mlt": [{"creatorNames": [{"creatorName": "Galaktionov, V. A."}], "nameIdentifiers": [{"nameIdentifier": "92500", "nameIdentifierScheme": "WEKO"}]}]}, "item_files": {"attribute_name": "\u30d5\u30a1\u30a4\u30eb\u60c5\u5831", "attribute_type": "file", "attribute_value_mlt": [{"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "2017-06-14"}], "displaytype": "detail", "download_preview_message": "", "file_order": 0, "filename": "jms170401.PDF", "filesize": [{"value": "2.4 MB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_free", "mimetype": "application/pdf", "size": 2400000.0, "url": {"label": "jms170401.PDF", "url": "https://repository.dl.itc.u-tokyo.ac.jp/record/40049/files/jms170401.PDF"}, "version_id": "2c32449f-fff0-4277-88e9-8e32bd8eb2eb"}]}, "item_keyword": {"attribute_name": "\u30ad\u30fc\u30ef\u30fc\u30c9", "attribute_value_mlt": [{"subitem_subject": "The Cauchy problem", "subitem_subject_scheme": "Other"}, {"subitem_subject": "diffusion equations with absorption", "subitem_subject_scheme": "Other"}, {"subitem_subject": "initial Dirac mass", "subitem_subject_scheme": "Other"}, {"subitem_subject": "very singular solutions", "subitem_subject_scheme": "Other"}, {"subitem_subject": "existence", "subitem_subject_scheme": "Other"}, {"subitem_subject": "nonexistence", "subitem_subject_scheme": "Other"}, {"subitem_subject": "bifurcations", "subitem_subject_scheme": "Other"}, {"subitem_subject": "branching", "subitem_subject_scheme": "Other"}]}, "item_language": {"attribute_name": "\u8a00\u8a9e", "attribute_value_mlt": [{"subitem_language": "eng"}]}, "item_resource_type": {"attribute_name": "\u8cc7\u6e90\u30bf\u30a4\u30d7", "attribute_value_mlt": [{"resourcetype": "departmental bulletin paper", "resourceuri": "http://purl.org/coar/resource_type/c_6501"}]}, "item_title": "Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption", "item_titles": {"attribute_name": "\u30bf\u30a4\u30c8\u30eb", "attribute_value_mlt": [{"subitem_title": "Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption"}]}, "item_type_id": "4", "owner": "1", "path": ["312/6865/6919/6920", "9/504/6868/6921/6922"], "permalink_uri": "http://hdl.handle.net/2261/52406", "pubdate": {"attribute_name": "\u516c\u958b\u65e5", "attribute_value": "2012-10-22"}, "publish_date": "2012-10-22", "publish_status": "0", "recid": "40049", "relation": {}, "relation_version_is_last": true, "title": ["Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption"], "weko_shared_id": null}