The existence of minimum speed of traveling wave solutions to a non-KPP isothermal diffusion system

The reaction-diffusion system at=axx−abn,bt=Dbxx+abn, where n≥1 and D>0, arises from many real-world chemical reactions. Whereas n=1 is the KPP type nonlinearity, which is much studied and very important results obtained in literature not only in one dimensional spatial domains, but also multi-dimensional spaces, but n>1 proves to be much harder. One of the interesting features of the system is the existence of traveling wave solutions. In particular, for the traveling wave solution a(x,t)=a(x−vt), b(x,t)=b(x−vt), where v>0, if we fix limx→−∞⁡(a,b)=(0,1) it was proved by many authors with different bounds v⁎(n,D)>0 such that a traveling wave solution exists for any v≥v⁎ when n>1. For the latest progress, see [7]. That is, the traveling wave problem exhibits the mono-stable phenomenon for traveling wave of scalar equation ut=uxx+f(u) with f(0)=f(1)=0, f(u)>0 in (0,1) and, u=0 is unstable and u=1 is stable. A natural and significant question is whether, like the scalar case, there exists a minimum speed. That is, whether there exists a minimum speed vmin>0 such that traveling wave solution of speed v exists iff v≥vmin? This is an open question, in spite of many works on traveling wave of the system in last thirty years. This is duo to the reason, unlike the KPP case, the minimum speed cannot be obtained through linear analysis at equilibrium points (a,b)=(0,1) and (a,b)=(1,0). In this work, we give an affirmative answer to this question.