Asymptotic behavior of solutions of quasilinear parabolic equations with convection

对流拟线性抛物型方程解的渐近行为

• 批准号: 11640182
• 负责人: SUZUKI Ryuichi
• 金额: $ 1.15万
• 依托单位: Kokushikan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640182/
• 关键词: quasilinear parabolic equation; asymptotic behavior; blow-up of solutions; complete blow-up; supercritical; blow-up; asymptotic behavior; parabolic equation; quasilinear

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.

在我们的项目中，我们获得了cauchy问题的非负溶液的渐近词，即权益μ_t-δμ=μ=μ=μ=μ=μ^p在r^n p ik ik超临界的意义上，就sobolev嵌入并满足了一些连接，例如cauchy问题“峰值解决方案”。 c> 0使得μ_0（γ）γ^α[γ> 1小于或等于] C，并且存在γ_0> 0，因此（i）μ_0（γ）在γ[大于或等于或等于]γ_0和（ii）μ_0（γ）> 0在[0，γ_0]μ_0中是cauchy问题，其溶液的初始数据μ_0（γγ）的时间为μ（t;γμ_0））。 （μ_0（γ）*0）将类型分为γ> 0的类型，如下所示：存在γ_1∈（0，0）（type I）T_C（γμ__0）如果γ_1，（II型）t_b（γμ__0）<∞，t_c（γμ_0）=∞和μ（t;γμ_0），则在有限的情况下打击。如果γ=γ_1，（III型）t_b（γμ_0）=∞和μ（t;γμ_0）'_∞= O（t^-1/（p-1））如果0 <γ_1。

项目成果

R.Suzuki: "Asymptotic behavior of solutions of quasilinear parabolic equations with slowly decaying initial data"Adv.Math.Sci.Appl.. 9. 291-317 (1999)

R.Suzuki：“具有缓慢衰减初始数据的拟线性抛物方程解的渐近行为”Adv.Math.Sci.Appl.. 9. 291-317 (1999)

R.Suzuki: "Complete blow-up for quasilinear degenerate parabolic equations"Proc.Royal Soc.Edinburgh. 130A. 877-908 (2000)

R.Suzuki：“拟线性简并抛物线方程的完全放大”Proc.Royal Soc.Edinburgh。