Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9501568 | Journal of Differential Equations | 2005 | 34 Pages |
Abstract
In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:ut(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),where xâR, t>0, D,d>0, r⩾0, bâC1(R) and b(0)=dK-b(K)=0 for some K>0 under monostable assumption. We show that there exists a minimal wave speed c*>0, such that for each c>c* the equation has exactly one travelling wavefront U(x+ct) (up to a translation) satisfying U(-â)=0,U(+â)=K and limsupξâ-âU(ξ)e-Î1(c)ξ<+â, where λ=Î1(c) is the smallest solution to the equation cλ-D[eλ+e-λ-2]+d-bâ²(0)e-λcr=0. Moreover, the travelling wavefront is strictly monotone and asymptotically stable with phase shift in the sense that if an initial data ÏâC(RÃ[-r,0],[0,K]) satisfies liminfxâ+âÏ(x,0)>0 and limxâ-âmaxsâ[-r,0]|Ï(x,s)e-Î1(c)x-Ï0eÎ1(c)cs|=0 for some Ï0â(0,+â), then the solution u(x,t) of the corresponding initial value problem satisfies limtâ+âsupR|u(·,t)/U(·+ct+ξ0)-1|=0 for some ξ0=ξ0(U,Ï)âR.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Shiwang Ma, Xingfu Zou,