引用本文: | 孔荣.奇型偏微分方程的柯西问题[J].国防科技大学学报,1988,10(3):77-87 ,114.[点击复制] |
Kong Rong.The Singular InitiaI Valuc Problem for a Class of Partial Differential Equations[J].Journal of National University of Defense Technology,1988,10(3):77-87 ,114[点击复制] |
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奇型偏微分方程的柯西问题 |
孔荣 |
(系统工程与应用数学系)
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摘要: |
定解条件给在奇线上的偏微分方程的各种定解问题早已有研究[1~4],多数作者使用了特殊函数作工具。本文用能量不等式组来解决一类奇型双曲型方程的柯西问题。
本文主要讨论如下问题解尚存在唯一性:
Lu≡[(ta/2?t-λ1(x,t)?x)(ta/2?t-λ2(x,t) ?x)+a(x,t)?t+b(x,t)?x+c(x,t)]u(x,t)=f(x,t)
(x,t)∈R×(0,T]
u∣t=0=φ(x),limta/2ut=ψ(x)
这是一个二阶偏微分方程,当 α>0时,?t2的系数当t=O 时变为零,因而这是一个初始值给在奇线上的柯西问题。我们假定:
(A) α为常数,0<α<1;所涉及的都是实函数;
(B) α(x,t),b(x,t),c(x,t),λj(x,t)(j=1,2)∈C1([0,T],C2(R)),且上述函数的所有可能的导数都有界;
(C) φ(x),ψ(x)∈C04(R));
(D)f(x,t)∈C((0,T],C02(R)),且sup{ta/2(∣f∣+∣fx∣+∣fxx∣}<+∞(Ⅱ)
(E)存在常数δ>0,使当(x,t)∈R×[0,T]时,有:∣λ1(x,t)-λ2(x,t)∣≥δ条件(Ⅱ)中关于实函数的假设不是必要的,作此假设仅为方便。本文主要得到:定理1:在(Ⅱ)的假设下,(Ⅰ)存在唯一弱解u,并 u∈C([0,T),H1(R))∩C1((0,T),L2(R)).为证明该定理作了一系列准备,关键是证得引理1,引理2和引理6。 |
关键词: 偏微分方程,奇型,柯西问题解 |
DOI: |
投稿日期:1987-09-01 |
基金项目: |
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The Singular InitiaI Valuc Problem for a Class of Partial Differential Equations |
Kong Rong |
(Department of Applied Mathematics and System Engineering)
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Abstract: |
In this paper we discuss a c1ass of partial differential equations with singular coefficients,the initial problem for which can be solved uniquely. We have proved the fo11owing theorem:
The problem:
Lu≡[(ta/2?t-λ1(x,t)?x)(ta/2?t-λ2(x,t)?x)+a(x,t)?t+b(x,t)?x+c(x,t)]u(x,t)=f(x,t)
(x,t)∈R×(0,T]
u∣t=0=φ(x),limta/2ut=ψ(x)
can be solved uniquely by u∈C([0,T],H1(R))∩C1((0,T],L2(R)) if the following conditions are satisfied.
(A) a is a constant and 0j(x,t)(j=1,2)∈C’([0,T], C2(R)) and all the derivatives of the above functions are bounded by the constant;
(C) φ(x),ψ(x)∈C04(R)
(D) f(x,t)∈C((0,T],C02(R)),and
sup{ta/2(∣f∣+∣fx∣+∣fxx∣}<+∞
(E) There exists a constantδ>0,such that ∣λ1(x,t)-λ2(x,t)∣≥δwhere (x,t)?R×[0,T].
In the proof,we use the particular sets of energy inequalities.A generalizatitn of the theorem is obtained. |
Keywords: Partial differential equations,Singnlar Coefficients |
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