数学进展
數學進展
수학진전
ADVANCES IN MATHEMATICS
2005年
1期
43-53
,共11页
粘性冲击波%存在性%几何奇异摄动%全局稳定
粘性遲擊波%存在性%幾何奇異攝動%全跼穩定
점성충격파%존재성%궤하기이섭동%전국은정
viscous shock waves%existence%geometric singular perturbation%global stability
本文首先利用几何奇异摄动方法,证明了粘性系数充分小时一类非凸粘性平衡律方程的粘性冲击波的存在性,推广了原来在非线性项严格凸的条件下得到的结果.进而,利用谱分析和上下解方法,证明了对固定的小的粘性系数此类波是全局渐近指数稳定的,推广了反应扩散方程中经典的全局稳定性结果.
本文首先利用幾何奇異攝動方法,證明瞭粘性繫數充分小時一類非凸粘性平衡律方程的粘性遲擊波的存在性,推廣瞭原來在非線性項嚴格凸的條件下得到的結果.進而,利用譜分析和上下解方法,證明瞭對固定的小的粘性繫數此類波是全跼漸近指數穩定的,推廣瞭反應擴散方程中經典的全跼穩定性結果.
본문수선이용궤하기이섭동방법,증명료점성계수충분소시일류비철점성평형률방정적점성충격파적존재성,추엄료원래재비선성항엄격철적조건하득도적결과.진이,이용보분석화상하해방법,증명료대고정적소적점성계수차류파시전국점근지수은정적,추엄료반응확산방정중경전적전국은정성결과.
In this paper, by geometric singular perturbation method, we first prove the existence of a class of viscous shock wave solutions to a scalar balance law for sufficiently small viscosity, which extends the previous results in the strictly convex nonlinearity case to nonconvex case. Further, by detailed spectral analysis and sub- and super-solution method, we show that viscous shock waves are globally asymptotically stable which extends the classical global stability result for bistable reaction-diffusion equation.
