摘要
文章主要考察一类非线性波动方程utt+uxxxx+λu=σ(ux)x,λ>0的柯西问题解的存在性和唯一性。当σ(ux)x=-β(|ux|pux)x,β>0,p>0时,通过构造稳定集(位势井)W={u∈H2(M)|‖uxx‖2+λ‖u‖2<2(p+2)pd}和不稳定集V={u∈H2(M)|‖uxx‖2+λ‖u‖2>2(p p+2)d},得到了W和V在上述方程的流下是不变的,并证明了如果初始能量E(0)≤d,那么当初值u0∈W-时,问题存在惟一整体解u∈C1([0,∞);H2);当初值u0∈V时,问题的解在有限时刻T1∈(t1,t1+p4φφ′/((tt11)))发生爆破。
This paper considers the existence and uniqueness of the solution for the Cauchy problem of the nonlinear wave equation utt＋uxxxx＋λu=σ（ux）x,λ0.For σ（ux）x=-β（｜ux1｜pux）x,β0,p0,we define the stable set（potential well） W and unstable set V respectively by W={u∈H2（M）｜‖uxx‖2＋λ‖u‖22（p＋2）pd}and V={u∈H2（M）｜‖uxx‖2＋λ‖u‖22（p p＋2）d},moreover,we show the invariance of the set W and V under the flow of the problem.In addition,we prove the existence and uniqueness of the global solution to the problem as the initial value u0∈W-and blow-up in finite time T1∈（t1,t1＋4/（t1）p′（t1）） as the initial value u0∈V if the initial energy E（0）≤d.
出处
《重庆师范大学学报（自然科学版）》
CAS
2010年第6期48-51,共4页
Journal of Chongqing Normal University:Natural Science
关键词
非线性波动方程
柯西问题
位势井
整体解
爆破
nonlinear wave equation
Cauchy problem
potential well
global solution
blow-up
