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The additional multiplier h(t) = t\u03b1 \u2192 0 as t \u2192 0 in the absorption term plays a role of a timedependent nonhomogeneous 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 selfsimilar form u*(x,t) = t(\u03b2)V(y), y = x/t(1/2m), \u03b2 = (1+\u03b1)/(p1), where each V is an exponentially decaying as y \u2192 \u221e solution of the elliptic equation (\u0394)mV + 1/2my\u30fb\u2207V + \u03b2V  V(p1)V = 0 in RN. Complicated families of VSSs in 1D and also nonradial 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": "20071105"}]}, "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. 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Vast Multiplicity of Very Singular SelfSimilar Solutions of a Semilinear HigherOrder Diffusion Equation with TimeDependent Absorption
http://hdl.handle.net/2261/52406
687d06bf41b14888afb6119a7cd80be1
名前 / ファイル  ライセンス  アクション  

jms170401.PDF (2.4 MB)


Item type  紀要論文 / Departmental Bulletin Paper(1)  

公開日  20121022  
タイトル  
タイトル  Vast Multiplicity of Very Singular SelfSimilar Solutions of a Semilinear HigherOrder Diffusion Equation with TimeDependent Absorption  
言語  
言語  eng  
キーワード  
主題  The Cauchy problem  
主題Scheme  Other  
キーワード  
主題  diffusion equations with absorption  
主題Scheme  Other  
キーワード  
主題  initial Dirac mass  
主題Scheme  Other  
キーワード  
主題  very singular solutions  
主題Scheme  Other  
キーワード  
主題  existence  
主題Scheme  Other  
キーワード  
主題  nonexistence  
主題Scheme  Other  
キーワード  
主題  bifurcations  
主題Scheme  Other  
キーワード  
主題  branching  
主題Scheme  Other  
資源タイプ  
資源  http://purl.org/coar/resource_type/c_6501  
タイプ  departmental bulletin paper  
著者 
Galaktionov, V. A.
× Galaktionov, V. A. 

著者所属  
Department of Mathematical Sciences, University of Bath  
抄録  
内容記述タイプ  Abstract  
内容記述  As a basic model, the Cauchy problem in RN×R+ for the 2mthorder semilinear parabolic equation of the diffusionabsorp\tion type ut = (Δ)mu  tαu(p1)u, with p > 1, α > 0, m ≧ 2, with singular initial data u0(x) ≢ 0 such that u0(x) = 0 for any x ≠ 0, is studied. The additional multiplier h(t) = tα → 0 as t → 0 in the absorption term plays a role of a timedependent nonhomogeneous 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) ≡ 1, first nonexistence result for p ≧ p0 = 1 + 2/N was proved in the celebrated paper by Brezis and Friedman in 1983. Existence of VSSs in the complement interval 1 < p < p0 was established in the middle of the 1980s. The main goal is to justify that, in the subcritical range 1 < p < p0 = 1 + {2m(1+α)}/N, there exists a finite number of different VSSs of the selfsimilar form u*(x,t) = t(β)V(y), y = x/t(1/2m), β = (1+α)/(p1), where each V is an exponentially decaying as y → ∞ solution of the elliptic equation (Δ)mV + 1/2my・∇V + βV  V(p1)V = 0 in RN. Complicated families of VSSs in 1D and also nonradial 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+α)}/(l+N), where l = 0,1,2,...  
書誌情報 
Journal of mathematical sciences, the University of Tokyo 巻 17, 号 4, p. 323358, 発行日 20110329 

ISSN  
収録物識別子タイプ  ISSN  
収録物識別子  13405705  
書誌レコードID  
収録物識別子タイプ  NCID  
収録物識別子  AA11021653  
日本十進分類法  
主題  415  
主題Scheme  NDC  
Mathematical Reviews Number  
MR  
Mathmatical Subject Classification  
35K55(MSC2010)  
Mathmatical Subject Classification  
35K40(MSC2010)  
Mathmatical Subject Classification  
35K65(MSC2010)  
出版者  
出版者  Graduate School of Mathematical Sciences, The University of Tokyo  
原稿受領日  
20071105 