一类多维非线性波动方程组的Cauchy问题

The Cauchy Problem of a System for a Class of the Nonlinear Wave Equations with Several Variables

作者: 专业:基础数学 导师:陈国旺 年度:2010 学位:硕士  院校: 郑州大学

Keywords

System of nonlinear wave equations with several variables, Cauchy problem, Local solution, Global solution, Blow-up of solution

        本文共分三章.第一章为引言,给出了本文要研究的方程模型的来历,一些已有的结果和要用到的记号.在第二章中,我们研究如下n维非线性广义波方程组的Cauchy问题utt(x,t)-σΔu(x,t)-Δutt(x,t)=Δf(v(x,t)),x∈Rn,t>0, (1)utt(x,t)-Δutt(x,t)=△g(v(x,t)),x∈Rn,t>0, (2)u(x,0)=u0(x),ut(x,0)=u1(x),x∈Rn, (3)v(x,0)=v0(x),vt(x,0)=v1(x),x∈Rn, (4)其中u(x,t)和v(x,t)是未知函数,△为n维Laplace算子,σ>0为常数,f(y)和g(y)是给定的非线性函数,u0(x),u1(x),v0(x)和v1(x)是定义在Rn上的初值函数.为此,我们将Cauchy问题(1)-(4)写成以下矢量形式Wtt-△Wtt=ΔF(u,v),x∈Rn,t>0, (5)W(x,0)=W0(x),Wt(x,0)=W1(x),x∈Rn, (6)其中然后利用压缩映射原理证明Cauchy问题(5),(6),在空间C2([0,∞);Hs(Rn)×Hs(Rn))(s>n/2)存在惟一的整体广义解和在空间C2([0,∞);Hs(Rn)×Hs(Rn))(s>2+n/2)存在惟一的整体古典解,主要结果如下:定理1假定s>n/2,W0,W1∈Hs×Hs,f∈C[s]+1(R),f(0)=0,g∈C[s]+1(R),g(0)=0,则Cauchy问题(5),(6)有惟一的局部广义解W∈C2([0,T0),Hs×Hs),其中[0,T0)是解存在的最大时间区间.进一步地,若则T0=∞.现在,我们证明Cauchy问题(5),(6)解的延拓条件(7)转化为以下的解的延拓条件(8),即证明以下定理.定理2假设s>n/2,W0,W1∈Hs×Hs,f∈C[s]+1(R),f(0)=0,g∈C[s]+1(R),g(0)=0,则Cauchy问题(5),(6)有惟一的局部广义解W∈C2([0,T0);Hs×Hs),其中[0,T0)是解存在的最大时间区间,进一步地,若则T0=∞.定理3假设s≥3/2+n/2,W0,W1∈Hs×Hs,Λ-1W1∈L2×L2,f∈C[s]+1(R),f(0)=0,g∈C[s]+1(R),g(0)=0,z(v)=∫0vg(y)dy≥0,若存在γ,满足使得其中A,B>0为常数,则Cauchy问题(5),(6)有惟一整体广义解W∈C2([0,∞),Hs×Hs).注1若s>2+n/2,则Cauchy问题(5),(6)的整体广义解W(x,t)是整体古典解.定理4设W0,W1∈Hs×Hs,v1,Λ-1v1∈L2,g∈C(R),z(v0)∈L1,f(v)=v,且9(y)y≤2(1+2a)z(y),(?)y∈Rn,其中a>0为常数,如果满足下面三个条件之一:(1)E(0)<0,(2)E(0)=0,〈Λ-1v1,Λ-1v0〉+<v1,v0〉>0,其中则Cauchy问题(5),(6)的广义解或古典解W(x,t)在有限时刻爆破.第三章证明Cauchy问题(5),(6)在空间C([0,∞);(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2))(m≥0)中存在惟一的整体广义解和在空间C3([0,∞);(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2))(m≥2+n/p)有惟一的整体古典解.主要结果如下:定理5假设W0,W1∈(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2),f,g∈Cm+l(Rn),且f(0)=0,g(0)=0,那么Cauchy问题(5),(6)有惟一局部广义解W∈C2([0,T0);(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2))(m≥0),其中[0,T0)是解存在的最大时间区间,进一步地,若则T0=∞.定理6假设(1)W0,W1∈(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2),f,g∈Cm+1(Rn)(m≥0).(2)f,g∈Cm+1(Rn),f(0)=0,g(0)=0,且z(v)>0,若存在γ满足使得其中A,B>0为常数,则Cauchy问题(5),(6)有惟一整体广义解W∈C3([0,∞);(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2))和Λ-1ut∈C([0,∞);L2).引理1假设定理6的条件成立,f,g∈Ck+m+1(R),其中k≥0为任意常数,则Cauchy问题(5),(6)的广义解W(x,t)∈Ck+3+1([0,T];(Wm-l,p∩L∞∩L2)×(Wm-l,p∩L∞∩L2)),((?)T>0),0≤l≤m.定理7设引理1的条件成立,且k=0,l=0,m>2+n/p,则Cauchy问题(5),(6)有惟一整体古典解W(x,t)∈C3([0,T];(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2)),即W(x,t)∈C3([0,T];(CB2(Rn)∩L∞∩L2)×(CB2(Rn)∩L∞∩L2)),其中CB2(Rn)由那些C2(Rn)中在Rn上所有有界的函数组成.
    This paper consists of three chapters. The first chapter is the introduction. The origin of the model equation of the study in this paper, some known results and the notations are given.In the second chapter,we study the following Cauchy problem for a system of the n-dimensional nonlinear wave equations. utt(x,t)-σΔu(x,t)-Δutt(x,t)=Δf(v(x,t)), x∈Rn, t>0, (1) vtt(x,t)-Δvtt(x,t)=Δg(v(x,t)), x∈Rn, t>0, (2) u(x,0)=u0(x), ut(x,0)=u1(x), x∈Rn, (3) v(x,0)=v0(x), vt(x,0)=v1(x), x∈Rn, (4)where u(x,t) and v(x,t) are the unknown functions;Δdenotes the n-dimensional Laplace opertor andσ>0 is a constant; f(y) and g(y) are given nonlinear functions; u0(x), u1(x), v0(x), and v1(x) are given initial value functions defined on Rn. For this purpose, the problem (1)-(4) can be written the following vector form: Wtt-ΔWtt=ΔF(u,v), x∈Rn, t>0, (5) W(x,0)=W0(x), Wt(x,0)=W1(x), x∈Rn, (6) whereThen, using the contraction mapping principle,we prove the Cauchy problem (5), (6) has a unique global generalized solution in C2([0,∞); Hs(Rn)×Hs(Rn)), and a unique global classical solution in C2([0;∞); Hs(Rn)×Hs(Rn)). The main results are as follows:Theorem 1 Suppose that s>n/2, W0,W1∈Hs(Rn)×Hs(Rn),f∈C[s]+1(R),f(0)= 0,g∈C[s]+1(R),g(0)=0. Then the Cauchy problem (5), (6) admits a unique local generalized solution W∈C2([0,T0), Hs×Hs), where [0,T0) is the maximal time interval of the existence of the solution. Moreover, if then T0=∞.Now, we prove that the extension condition of the solution (7) for the Cauchy problem (5), (6) transforms the extension condition of the solution (8) below, i.e, we prove the following theorem.Theorem 2 Suppose that s>n/2, W0,W1∈Hs×Hs,f∈C[s]+1(R),f(0)=0,g∈C[s]+1(R),g(0)=0. The the Cauchy problem (5), (6) admits a unique local generalized solution W∈C2([0,T0),Hs×HS), where [0,T0) is the maximal time interval of the existence of the solution. Moreover, if then T0=∞.Theorem 3 Suppose that s≥3/2+n/2, W0, W1∈Hs×Hs,Λ-1W1∈L2×L2,f∈C[s]+1(R),f(0)=0,g∈C[s]+1(R),g(0)=0,z(v)=∫0vg(y)dy≥0. If there existγsatisfies such that where A, B>0 are constants, then the Cauchy problem (5) and (6) admits a unique global generalized solution W∈C2([0,∞), Hs×Hs). Theorem 4 Suppose that W0, W1∈Hs×Hs, v1,Λ-1v1∈L2, g∈C(R), z(v0)∈L1,f(v)=v, and whereα>0 is a constant. Then the generalized solution or the classical solution W(x, t) of the Cauchy problem (5), (6) blows up in finite time if one of the following conditions holds(1) E(0)<0,(2) E(0)=0,<Λ-1v1,Λ-1v0>+<v1,v0>>0,In Chapter 3, we prove that the Cauchy problem (5), (6) has a unique global gen-eralized solution in C3([0,∞); (Wm,p∩L2)×(Wm,p∩L∞∩L2)) when m≥0 and a unique global classical solution in C3([0,∞); (Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2)) when m>2+n/p. The main results are as follows:Theorem 5 Suppose that W0, W1∈(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2),f,g∈Cm+1(R)(m≥0), andf(0)=0,g(0)=0. Then the Cauchy problem (5) and (6) admits a unique local generalized solution W∈C2([0,T0);(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2))(m≥0), where [0,T0) is the maximal time interval of the existence of the solution. Moreover, if then T0=∞.Theorem 6 Suppose that(1)WO, W1∈(Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2),f,g∈Cm+1(Rn)(m≥0). (2)f, g∈Cm+1(R),f(0)=0,g(0)=0, and z(v)>0, ifγsatisfies such that where A,B>0 are constants, the Cauchy problem (5), (6) admits a unique global generalized solution W∈C3([0,∞); (Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2)).Lemma 1 Suppose that the conditions of Theorem 6 hold and f, g∈Ck+m+1(R), where k≥0 is arbitrary integer. Then the generalized solution W(x,t) belongs to Ck+3+l([0,T]; (Wm-l,p∩L∞∩L2)×(Wm-l,p∩L∞∩L2)), (VT>0),0≤l≤m.Theorem 7 Suppose that the conditions of Lemma 1 hold and k=0, l=0, m> 2+n/p, then the Cauchy problem (5), (6) has a unique global classical solution W(x, t)∈C3([0,T]; (Wm,p∩L∞∩L2)×(Wm,p∩L∞∩L2)), i.e., W(x,t)∈C3([0,T]; (CB2(Rn)∩L∞∩L2)×(CB2(Rn)∩L∞∩L2)), where CB2(Rn) consists of all those functions in C2(Rn) that are bounded in Rn.
        

一类多维非线性波动方程组的Cauchy问题

摘要4-8
Abstract8-11
1 引言13-19
2 Cauchy问题(1.30),(1.31)在C~2([0,∞);H~s(R~n)×H~s(R~n))空间中整体解的存在性,惟一性以及解的爆破19-33
    2.1 局部解的存在和唯一性19-26
    2.2 Cauchy问题(1.30),(1.31)整体解的存在和唯一性26-29
    2.3 Cauchy问题(1.30),(1.31)解的爆破29-33
3 Cauchy问题在C([0,∞);(W~(m,p)∩L~∞n L~2)×(W~(m,p)∩L~∞∩L~2))空间的整体解的存在性和惟一性33-46
    3.1 局部解的存在和唯一性33-40
    3.2 整体解的存在和唯一性40-46
参考文献46-48
致谢48-49
个人简历、在学期间发表的学术论文与研究成果49
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