Asymptotic behavior of solutions of quasilinear parabolic equations with convection

本文得到了方程μ_t-Δμ^m=μ^p在R^N中的Cauchy问题非负解的渐近性态的精确结果，其中p在Sobolev嵌入意义下是超临界的，且p满足某些条件使得Cauchy问题有“峰解”.我们把结果概括如下：设连续的初始数据μ_0（γ）（γ=| X|）满足以下条件：存在α ∈（2/（p-m），N）和C>0，使得对于γ>1，μ 0（γ）γ^α[小于或等于]C，并且存在γ 0>0，使得（i）μ 0（γ）是关于γ[大于或等于]γ 0的非减函数，（ii）μ 0（γ）>0关于[0，γ 0]，其中，在m=1的情况下，我们不需要假设条件（ii）。此外，设μ（t ; μ_0）为初值为μ_0（γ）的Cauchy问题的解，t_B（μ_0）和t_c（μ_0）分别为解的爆破时间和完全爆破时间。然后，μ（t ; γμ_0）（μ_0（γ）*0）按γ>0分为以下三种类型：存在γ_1 ∈（0，∞）使得（Ⅰ型）t_c（γμ_0）<∞，即当γ>γ_1时，μ（t ; γμ_0）在有限时间内爆破;（Ⅱ型）t_B（γμ_0）<∞，t_c（γμ_0）=∞，且μ（t ;（Ⅲ型）t_B（γ μ_0）=∞，且当0<γ<γ_1时，τ μ（t ; γ μ_0）τ_∞=O（t^-1/（p-1））.

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.