一类拟线性抛物方程组几个定性问题之分析-analysis of some qualitative problems for a class of quasilinear parabolic equations.docxVIP

一类拟线性抛物方程(组)几个定性问题之研究摘要本文讨论一类具源项的拟线性抛物方程(组)的几个定性问题，如解的存在性、渐近性及解的生命跨度等. 本文主要包含以下两部分内容.mI. 讨论一维拟线性方程 ut = 1 (um)xx + aup 的第二初边值问题. 证明如下结论：(1) 全局光滑解的存在性；(2) 解的爆破时间的估计；(3) 当 t → ∞ 时，解的渐近性质；(4) 当 m → 1, a → 0 时，方程的解 u(x, t, m, a) 在 L2 空间中逼近于对应线性方程ut = uxx,的解 u(x, t, 1, 0) ，并给出显示的误差估计，即存在仅与 T 有关的正常数 C, C? 满足∫ T ∫ 1|u(x, t, m, a) ? u(x, t, 1, 0)|2dx ≤ C |m ? 1| + C? a2.00II. 讨论耦合方程组的Cauchy问题?p?? ut = vuxx,(x, t) ∈ R1× (0, T ),vt = uq vxx,(x, t) ∈ R1 × (0, T ),?? u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ R1,其中 0 p, q 1. 证明如下结论：(1) Cauchy问题正则解的存在性；(2) 当 p → 0, q → 0 时，方程组第二初边值问题?? ut = vpuxx,(x, t) ∈ (?l, l) × (0, T ),?? vt = uq vxx,(x, t) ∈ (?l, l) × (0, T ),? ux|x=±l = 0, vx|x=±l = 0,t ∈ (0, T ),?? u(x, 0) = u0(x), v(x, 0) = v0(x),x ∈ (?l, l),的解 (u, v) 在 L2 空间中逼近于线性方程组?? ut = uxx,(x, t) ∈ (?l, l) × (0, T ),?? vt = vxx,(x, t) ∈ (?l, l) × (0, T ),? ux|x=±l = 0, vx|x=±l = 0,t ∈ (0, T ),?? u(x, 0) = u0(x), v(x, 0) = v0(x),x ∈ (?l, l),I集美大学硕士学位论文一类奇异扩散方程(组)解关于非线性性质的连续依赖性的解，同时证明方程组的解关于初值的连续依赖性，即有如下估计：∫ l ∫ Tu)2dxdt+∫ l ∫ T(v ?v)2dxdt ≤ C1p+C2q +T ||u0 ?u0||L2(l,l) +T ||v?v0||L2(l,l).?l 0(u? ?l 0?0?关键词: ：拟线性 , 耦合, 线性逼近, 连续依赖性The qualitative properties on a parabolic system with sourcesAbstractIn this paper, some qualitative questions of a quasilinear parabolic equations with sources are discussed, such as the existence, asymptotic behavior, the life span of solutions and so on.mThe singular di?usion ut = 1 uxx + aup with second initial boundary value is consid-ered. it proved that:the existence of global smooth solution；the estimated of blow-up time；the asymptotic properties as t → ∞；；the solutions u(x, t, m, a) converge to their corresponding solution u(x, t, 1, 0) of thelinear equation obtained:ut = uxx in L2?as m → 1, a →∫ T ∫ 10, and the explicit error estimate is|u(x, t, m, a) ? u(x, t, 1, 0)|2dx ≤ C |m ? 1| + C? a2.00Consider?p?? ut = vuxx,(x, t) ∈ R1× (0, T ),vt = uq vxx,(x, t) ∈ R1 × (0, T ),?? u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ R1.where 0 p, q 1, we pro