2000 Fiscal Year Final Research Report Summary
Asymptotic behavior of solutions of quasilinear parabolic equations with convection
| Project/Area Number |
11640182
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kokushikan University |
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Principal Investigator |
SUZUKI Ryuichi Kokushikan University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (00226573)
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| Co-Investigator(Kenkyū-buntansha) |
HAMADA Toshihiko Wakayama National College of Technology, Department of Mechanical Engineering, Associate Professor, 機械工学科, 助教授 (20280430)
FUKUDA Isamu Kokushikan University, Faculty of Engineering, Professor, 工学部, 教授 (40103642)
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| Project Period (FY) |
1999 – 2000
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| Keywords | quasilinear parabolic equation / asymptotic behavior / blow-up of solutions / complete blow-up / supercritical |
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| Research Abstract |
In our project, we obtain the precise results about the asymptotic behavior of nonnegative solutions of the Cauchy problem for equation μ_t-Δμ^m=μ^p in R^N where p is supercritical in the sense of Sobolev embedding and p satisfies some conditions such that the Cauchy problem has "peaking solutions". We state the results roughly speaking as follows :
Let the continuous initial data μ_0 (γ)(γ=|x|) satisfy the next conditions : There exist α ∈ (2/(p-m), N) and C>0 such that μ_0 (γ) γ^α【less than or equal】C for γ>1, and there exists γ_0>0 such that (i) μ_0 (γ) is a nondecreasing function in γ【greater than or equal】γ_0 and (ii) μ_0 (γ)>0 in [0, γ_0], where we do not need to assume the condition (ii) in the case m=1. Further, let μ(t ; μ_0) be the solution of the Cauchy problem with the initial data μ_0 (γ), and let t_b (μ_0) and t_c (μ_0) be the blow-up time and the complete blow-up time of the solution, respectively. Then, μ(t ; γμ_0))(μ_0 (γ)*0) is classified into the next three types according to the value of γ>0 as follows : There exists γ_1 ∈(0, ∞) such that (Type I) t_c (γμ_0)<∞ i.e. μ(t ; γμ_0) blows up in finite time if γ>γ_1, (Type II) t_b (γμ_0)<∞, t_c (γμ_0)=∞ and ‖μ(t ; γμ_0)‖_∞=O (t^<-1/(p-1)>) if γ=γ_1, (Type III) t_b (γμ_0)=∞ and ‖μ(t ; γμ_0)‖_∞=O(t^-1/(p-1)) if 0<γ<γ_1.
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