The evolution of travelling waves in fractional order autocatalysis with decay. II. The initial boundary value problem

In this paper we examine an initial-boundary-value problem for a singular reaction diffusion equation arising from a model of isothermal chemical autocatalysis with termination. We consider the case where the reaction and termination orders, m and n, respectively, are fractional; that is, 0 < m, n < 1 ( and hence the nonlinearities are not Lipschitz continuous). We show that for m < n, the problem develops spatially uniform solutions which grow algebraically in t (time) and approach a nonzero equilibrium state as t --> infinity. For m = n, with k > 0 (k being a parameter which measures the relative strength of the termination step to that of the autocatalysis), we show that solutions approach the equilibrium state, 1-k for k < 1 and 0 for k = 1, uniformly over the reaction domain as t --> infinity. In contrast, for k > 1, we demonstrate that the problem exhibits solutions which decay to zero with contracting support in finite t. Finally, for m > n, we show that traveling waves with semi-infinite support occur when 0 < k < k(c), where k(c) varies with m and n, provided that the initial data exceeds a parameter dependent critical threshold, otherwise solutions decay to zero in finite t with contracting support.