Asymptotic behavior of solutions of quasilinear parabolic equations with convection

在本课题中，我们得到了方程μ_t-Δμ^m=μ^p在R^N中的非负解的渐近性的精确结果，其中p在Sobolev嵌入意义上是超临界的，并且p满足柯西问题具有“峰值解”的一些条件。我们国家结果大致说来如下:连续初始数据μ_0(γ)x(γ= | |)满足下一个条件:存在α∈(2 /(出版社),N)和C > 0,μ_0(γ)γ^α【小于或等于】Cγ> 1,并存在γ_0 > 0,(i)μ_0(γ)是一个不减少的函数在γ【大于或等于】γ_0,(ii)μ_0(γ)> 0在[0,γ_0),我们不需要假设条件(2)m = 1。进一步，设μ(t; μ_0)为初始数据为μ_0 （γ）的Cauchy问题的解，设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)划分为接下来的三种类型根据γ的值> 0如下:存在γ_1∈(0,∞),(I型)t_c(γμ_0)<∞即μ(t,γμ_0)炸毁在有限时间如果γ>γ_1,(II型)t_b(γμ_0)<∞,t_c(γμ_0)=∞为μ(t,γμ_0)为_∞= O (t ^ < 1 / (p - 1) >)如果γ=γ_1,(III型)t_b(γμ_0)=∞为μ(t,γμ_0)为_∞= O (t ^ 1 / (p - 1))如果0 <γ<γ_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.