Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models

Abstract

Abstract. We study the existence and uniqueness of traveling wave solutions for a class of two-component reaction-diffusion systems with one species being immobile. Such a system has a variety of applications in epidemiology, bio-reactor models, and isothermal autocatalytic chemical reaction systems. Our result not only generalizes earlier results of Ai and Huang (Proceedings of the Royal Society of Edinburgh 2005; 135A:663-675), but also establishes the existence and uniqueness of traveling wave solutions to the reactiondiffusion system for an isothermal autocatalytic chemical reaction of any order in which the autocatalyst is assumed to decay to the inert product at a rate of the same order.

The Traveling Wave of Auto-Catalytic Systems-Monotone and Multi-Peak Solutions

“…Moreover, v losses its monotonicity. For n = m, it was demonstrated in [15,16,26,27] that any traveling wave has the property that v is bellshaped. But, when n > m, our results show that v can oscillate a large number of times, which increases the difficulty of rigorous analysis.…”

Traveling waves solutions to general isothermal diffusion systems

“…Since we are only interested in developing a mathematical theory on traveling waves, we refer the interested reader to [1][2][3][4][5][6][7][8][9][10] for the modeling aspects. We assume H(u, v) satisfies the following conditions.…”

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

“…Indeed, Hosono [16] has shown the existence of traveling wave solutions of the system (1.1) with d 1 = 0 and the function f restricted to the form f (u) = k 1 u which corresponds to the higher-order isothermal autocatalytic chemical reaction (1.2)-(1.3) with immobile reactant. More recently, Guo and Tsai [12] have considered the other extreme case: d 1 > 0, d 2 = 0, m > 1 and a general class of functions f . Moreover, the result in Guo and Tsai [12] can be applied to show the existence of traveling wave solutions of higher-order isothermal autocatalytic chemical reactions (1.2)-(1.3) with immobile autocatalyst.…”

“…More recently, Guo and Tsai [12] have considered the other extreme case: d 1 > 0, d 2 = 0, m > 1 and a general class of functions f . Moreover, the result in Guo and Tsai [12] can be applied to show the existence of traveling wave solutions of higher-order isothermal autocatalytic chemical reactions (1.2)-(1.3) with immobile autocatalyst. On the other hand, since the methods in Hosono [16] and Guo and Tsai [12] are restricted to either d 1 = 0 or d 2 = 0, and hence, cannot be used to show the more interesting case: d 1 > 0 and d 2 > 0.…”

“…Moreover, the result in Guo and Tsai [12] can be applied to show the existence of traveling wave solutions of higher-order isothermal autocatalytic chemical reactions (1.2)-(1.3) with immobile autocatalyst. On the other hand, since the methods in Hosono [16] and Guo and Tsai [12] are restricted to either d 1 = 0 or d 2 = 0, and hence, cannot be used to show the more interesting case: d 1 > 0 and d 2 > 0. In this paper, we shall study the problem (1.4)-(1.5) for the case where d 1 > 0, d 2 > 0, m > 1, and f satisfies the same assumptions as in Huang [17] (see (A1)-(A3) below).…”