TRAVELING WAVE TO NON-KPP ISOTHERMAL DIFFUSION SYSTEMS: EXISTENCE OF MINIMUM SPEED AND SHARP BOUNDS

摘要

The reaction-diffusion system u(t) = u(xx) - ug(v), v(t) = Dv(xx) + ug(v), where g is an element of C-1,(gamma)[0, infinity) boolean AND C-2(0, infinity) -> [0, infinity), g(0) = g'(0) = 0, g(v) > 0 in (0, infinity), 0 < gamma < 1, and D > 0, arises from many real-world chemical reactions. The traveling wave of the underlying system is very important in understanding the various applications it models. Whereas the case of g'(0) > 0 is the KPP type nonlinearity, which is much studied and has very important results obtained in the literature not only in one-dimensional but also multidimensional settings, the present case of non-KPP has more interesting features. Several significant issues on traveling wave still need to be addressed. In particular, for a traveling wave solution, one outstanding open question is whether there exists a minimum speed c(min) > 0, which depends on D, the function g(v), and the boundary condition at -infinity, such that a traveling wave solution of speed c exists iff c >= c(min). We settle this question by providing an affirmative answer. We also derive, with fixed function g, sharp bounds c(*)(D) and c*(D) > 0 such that a traveling wave solution with speed c exists for any c >= c(*), but no traveling wave with speed c exists if c < c(*). Our estimates show that the non-KPP case has very different features from that of the KPP counterpart.