| Many problems arised in physics, chemistry and biology can be described by mathemat-ical models, and most of these models can be classified as reaction diffusion equations. For examples, population genetics in biological dynamics investigated the change of the amount of humans and their diffusion situations. Similarly, migration kinetics of species describes the change of numbers when a new species invades in. In chemicals, diffusion situation in chemical reactions such as bunsen flames can be described by reaction diffusion equations. As we all know, reaction diffusion equations have a kind of important solution, namely, trav-eling wave solutions(TW solution) with the following form:u=u(x·e+ct). TW solutions are generally used to describe the propagation of states, such as propagation of sound and heat. Because of its value in use in practical life, study of TW solutions is of great significance.In this article, our purpose is to summarize TW solution problems of reaction diffusion equations of the following form Based on the different forms of f(u), equations can be divided into the following three kind of equations1)KPP type:2)Bistable type:there existsμ∈(0,1), such that3)Ignition temperature type: f is Lipschitz continued in [0,1], differentiable in u=1, and satisfyEquations of different types may have planar TW solutions or non-planar TW solutions. Level sets of planar waves are plane, while level sets of non-planar TW solutions are curved, like conical or else. When x∈Rn, n≥2, planar waves have the following form in whichγ∈SN-1 is a direction vector, namely the direction of the propagation of waves, c is the wave speed. Non-planar TW solutions have the following form in which y and c have the same meaning as before. This paper mainly introduced TW solution problems of the three type of equations mentioned above.In 1937, Fisher [3]and Kolmogorov-Petrovskii-Piskounov [4] independently proposed a kind of equations of the following form ut=uxx+u(1-u), x∈R, which is called Fisher equations or KPP equations later. They studied the existence of TW solutions of KPP equa-tions, proved there exists critical wave speed c*= 2(?), for any c≥c*, KPP equation exists wave frontsΦ(x-ct) connecting 1 and 0,when x→±∞, TW solutions respectively converge to 0 and 1 exponentially. Later many works concerning the KPP type equations appeared. Especially many articles focused on the entire solutions of KPP equations, which means t∈R. Firstly Aronson [2] and Berestycki [22] proved the existence of 2-dimensional manifold of entire solutions. Some a priori estimates, based on the maximum principle and on comparisons with some solutions of the linear heat equations are used in Hamel and Nadirashvili [23]'s proof of building four other manifolds of different dimension of entire solutions, with a 5-dimensional manifold, a 4-dimensional manifold and two 3-dimensional manifolds. Moreover, Hamel [21] proved the existence of an infinite-dimensional manifold of entire solutions.A kind of n degree Fisher equations with the form ut=uxx+un(1-u), n is not necessarily an integer, arises in isothermal autocatalytic chemical reactions(see [41][42][43]). When n=2, Leach [44] showed that there exists TW front connecting two steady states u=1 and u=0 iff the wave speed c≥(?)/2. By using detailed spectral analysis, Yaping Wu [45] proved that each traveling wave with speed c>c*(n) is locally exponentially stable to perturbations in some exponentially weighted space. For more generalized n degree Fisher equation ut=uxx+un f(u),n>1, Xin [46] proved there exists a TW solutionΦ(x-ct) connecting u=1 and u=0 iff c≥c*, in which c*> 0.For one dimensional degenerate KPP equation ut=(um)xx+u(1-u), (x,y)∈R×R+, under a definition of weak solution, Atkinson [15] and Gilding [16] proved the existence of its solution.For Bistable type equations, Fife and Mcleod [20] studied its one dimensional form and showed there is a unique wave speed ca=(1-a)/(?) and a unique wave frontΦ(x-ct) connecting 1 and 0, also they obtained global asymptotic exponentially stability by using comparison principle. Under different conditions and the meaning of limit, bistable type equations have different form of TW solutions, that is, planar TW solution and conical TW solution. Hamel [26] gave a proof of the existence of a conical TW solution of bistable type, in which some conditions are necessary(see§4.4.1).For the stability of bistable type equtions, Fife and Mcleod [20] firstly proved the fol-lowing one dimensional result:Let u0(y) be a one dimensional Cauchy datum, satisfying then there exists y0∈R andγ>0 such that, if u(t, y) is the solution of bistable equation emanating from u0, we have uniformly in y∈R.Later, by using the theory of semigroups, Xin [38] showed that if u∈R and n≥4, then stability in one space dimension implies multidimensional stability, with perturbations decaying like t-(n-1)/4. It can be easily extended to systems of equations with the diffusion matrix being a scalar multiple of the identity. For the case of n=2,3, Levermore and Xin [39] again studied the problem in the scalar case. They achieve a partial result, in that the wave is shown to be stable in compact domains moving with the wave. However, the proof of the result depends quite heavily on the maximum principle and energy methods, and is therefore not applicable to most systems. Kapitula [40] proved the methods used in Xin [38] can be extended to n=2,3, with the following given result:if TW solutions are exponentially stable in one dimensional space, then it is also stable in multidimensional space when x∈Rn, n≥2.Hamel [26] proved the existence of conical TW solutions, then studied the stability of bistable type equation. Hamel [27] also proved the asymptotic property of bistable type equation with solutions of cylindrical symmetry, using a centre manifold-like argument, dis-tinguished the case N=2 from N≥3. as the consequence of this, the uniqueness of solution can be obtained.Especially for Allen-Cahn equtions of bistable type, many works are concerned with its property of planar wave and conical wave. Matano [7] studied stability of planar waves in the Allen-Cahn equations. Motano specified the class of initial perturbations under which the planar wave are asymptotically stable. The asymptotic stability of planar waves is studied in Kapitula [40], Levermore [39] and Xin [38] in various topologies for perturbations that decay to zero as│x│+│y│→∞.However, in these papers, perturbations are always assumed to be sufficiently small. By constructing suitable subsolutions and supersolutions, Matano [7] proved that planar waves are asymptotically stable under any-possibly large-initial perturbations that decay at space infinity. He also showed that the planar waves are asymp-totically stable under almost periodic perturbations with the same constructing method used above.More precisely, the perturbed solution converges to a planar wave as t→∞. The ex-istence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations. For conical wave of Allen-Cahn equations, Taniguchi [9] showed the existence of solutions, and in Taniguchi [8], the uniqueness and asymptotic stability are studied. He characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. More-over he characterize the pyramidal traveling front in another way, that is, to write it as a combination of two-dimensionalⅤ-form waves on the edges.Ignition temperature type equations arises in combustion theory with a typical model of premixed bunsen flame. Translate the conical shape of the flame into some conditions on the function, the following conical condition is obtained by in whichα>0 denotes the angles of the flame. Under this condition, Bonnet and Hamel [28] proved the existence of solution of the bunsen flame model in two dimensional space. In multidimensional space, Hamel [24] showed that ignition temperature type equation has no solution whenα<π/2 as long as N≥3. However, the existence of solutions in RN under some more weak conical conditions are possible. This is an open question. Hamel and Monneau [24] showed that in spaces of any dimension, as long asα>π/2, solutions are not existed. As for its asymptotic stability, Hamel [25] studied the stability of solutions of bunsen flames in two dimensional space, and proved that when the initial value satisfy certain condition, the solution is asymptotically stable.Lastly, the generalized fronts of ignition temperature type equations are discussed. A definition of generalized front was first given by Matano and later formalized by [34]. Berestycki and Hamel [35] proposed an alternative notion of generalized traveling fronts. Under their definition, Mellet and Roquejoffre [35] proved the existence of generalized fronts of ignition temperature type equations, Mellet, Nolen [37] showed their uniqueness and sta-bility.
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