Asymptotic behavior of solutions of quasilinear parabolic equations with convection
Project/Area Number 
11640182

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Kokushikan University 
Principal Investigator 
SUZUKI Ryuichi Kokushikan University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (00226573)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
1999 – 2000

Project Status 
Completed (Fiscal Year 2000)

Budget Amount *help 
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)

Keywords  quasilinear parabolic equation / asymptotic behavior / blowup of solutions / complete blowup / supercritical / blowup / asymptotic behavior / parabolic equation / quasilinear 
Research Abstract 
In our project, we obtain the precise results about the asymptotic behavior of nonnegative solutions of the Cauchy problem for equation μ_tΔμ^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/(pm), 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 blowup time and the complete blowup 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/(p1)>) if γ=γ_1, (Type III) t_b (γμ_0)=∞ and ‖μ(t ; γμ_0)‖_∞=O(t^1/(p1)) if 0<γ<γ_1.

Report
(3 results)
Research Products
(7 results)