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Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption
http://hdl.handle.net/2261/52406
http://hdl.handle.net/2261/52406687d06bf-41b1-4888-afb6-119a7cd80be1
| 名前 / ファイル | ライセンス | アクション |
|---|---|---|
jms170401.PDF (2.4 MB)
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| Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
|---|---|---|---|---|---|---|
| 公開日 | 2012-10-22 | |||||
| タイトル | ||||||
| タイトル | Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption | |||||
| 言語 | ||||||
| 言語 | eng | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | The Cauchy problem | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | diffusion equations with absorption | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | initial Dirac mass | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | very singular solutions | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | existence | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | nonexistence | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | bifurcations | |||||
| キーワード | ||||||
| 主題Scheme | Other | |||||
| 主題 | branching | |||||
| 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. |
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| 著者所属 | ||||||
| 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 2mth-order semilinear parabolic equation of the diffusion-absorp\-tion type ut = -(-Δ)mu - tα|u|(p-1)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 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) ≡ 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 self-similar form u*(x,t) = t(-β)V(y), y = x/t(1/2m), β = (1+α)/(p-1), where each V is an exponentially decaying as y → ∞ solution of the elliptic equation -(-Δ)mV + 1/2my・∇V + βV - |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+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. 323-358, 発行日 2011-03-29 |
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| 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Scheme | NDC | |||||
| 主題 | 415 | |||||
| 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 | ||||||
| 2007-11-05 | ||||||
