The minimal wave speed of a general reaction-diffusion equation with nonlinear advection
In recent years, many researchers paid attention to the linear and nonlinear determinacy of the minimal wave speed of various monostable reaction-diffusion systems. Since all these systems are monotone systems, the comparison principle together with the upper-lower solutions was used as a main tool. We can refer to [the authors, SIAM J. Math. Anal. 51, No. 1, 321--345 (2019; Zbl 1407.35114)]. For the Burgers-KPP-Fisher equation, the minimal speed has already been studied by \textit{J. D. Murray} [Lectures on nonlinear differential-equation models in biology. Oxford: Clarendon Press (1977)], and numerically demonstrated later by \textit{V. Mendez} and \textit{J. Fort} [`Speed of reaction-transport processes'', Phys. Rev. E (3) 64, No. 1, Article ID 011105, 4 p. (2001; \url{doi:10.1103/PhysRevE.64.011105})], but rigorous proof is not obtained. In this paper, the authors considered a general reaction-diffusion equation with a nonlinear advection term, which is an extension of Burgers-KPP-Fisher equation. Here the convection term models the deterministic motion of substances in the direction of its gradient, while the random motion is modeled by the diffusive term. This advection reaction-diffusion equation has neither monotonicity nor variational structure. Thus, the variation method and the comparison principle cannot be applied. Hence, the study on the linear or nonlinear determinacy of the minimal speed is challenging. The authors developed the classical geometrical argument by subtly constructing invariant regions to establish the existence of monotone traveling waves, and derived a necessary and sufficient condition for the nonlinear selection mechanism via a geometrical method coupled with a perturbation argument in an abstract weighted functional space. As applications, they investigated the Burgers-Fisher model, the model with cubic reaction term and the Burger's convection, and subcritical quintic Ginzburg-Landau equation with nonlinear convection, and successfully proved some important results including in Murray's book [loc. cit.; Mathematical biology. I. An introduction. Berlin: Springer (2002)], and \textit{V. A. Sabelnikov} and \textit{A. N. Lipatnikov}'s paper [``Speed selection for traveling-wave solutions to the diffusion-reaction equation with cubic reaction term and Burgers nonlinear convection'', Phys. Rev. E (3) 90, No. 3, Article ID 033004, 9 p. (2014; \url{doi:10.1103/PhysRevE.90.033004})], and also extended the results in [\textit{W. van Saarloos}, ``Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection'', Phys. Rev. A (3) 37, No. 1, 211--229 (1988; \url{doi:10.1103/PhysRevA.37.211}); ``Front propagation into unstable states. II: Linear versus nonlinear marginal stability and rate of convergence'', ibid. 39, No. 12, 6367--6390 (1989; \url{doi:10.1103/PhysRevA.39.6367})].
scientific article; zbMATH DE number 7380413