Dynamical Analysis on Traveling Wave of a Reaction-Advection-Diffusion Equation with Double Free Boundaries

JI Changqingand YANG Pan

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001,China.

Abstract. This paper investigates a reaction-advection-diffusion equation with double free boundaries. The stationary solution of the system is studied by phase plane analysis. Then,the scale logarithm change sequence method is introduced to show the exact heteroclinic of the system with corresponding parameters. Moreover,a complete description of the types of traveling wave solutions is given with different advection term coefficients.

Key Words: Phase plane analysis;stationary solution;heteroclinic orbit;traveling wave solution.

1 Introduction

Reaction-diffusion equation, a fundamental class of nonlinear partial differential equations, plays an important role in the fields of ecology, epidemiology, materials science,neural network etc. The classical reaction-diffusion equation

was studied by scholars to describe the spreading of biological species or chemical substances [1–5]. Traveling wave solution, as a focal point in reaction-diffusion system, is regarded as the stationary solution of this system[6–9]. Wang and Xiong[10]proposed the explicit front wave solution of system

In 2010,Du and Lin[11]estimated the solutionuand the free boundaryh(t)corresponding to the equation(1.1)with Fisher-KPP type of nonlinearity and a free boundary. Furthermore,whenf(u)is a generalized Fisher-KPP type of nonlinearity

Hu and Zou [12] achieved the traveling wave solutions and predicted the speed and manner of the extinction of a species in a shifting habitat representing a transition to a devastating environment.

In reality, due to the influence of various environmental factors, the migration and expansion of biological population show certain directionality, which can be described by the advection term in the reaction-diffusion equation. Therefore, many academics discussed the following equation

In 2015, Gu et al. [13] researched the stationary solutions and traveling wave solutions of system (1.4) with a Fisher-KPP type of nonlinearity. Moreover, the traveling wave dynamical analysis on a specific case of equation(1.4)

In our work,we first give the global existence and uniqueness of the solution. Based on the phase plane analysis,we calculate the heteroclinic orbit between the equilibrium points by using scale logarithm change sequence(SLS)method[16,17]. This is a method for continuous-time autonomous dynamical systems, which can find exact orbits as opposed to approximate one.In addition,the existence of periodic traveling wave solutions whenβ=0 and other types of traveling wave solutions whenβ>0 are shown.

The rest of this paper is arranged as follows. In Section 2, the global existence and uniqueness of the solution are proved.Stationary solutions and travelling wave solutions are studied in Section 3. In Section 4,we give a brief discussion.

2 Existence and uniqueness

In this section, we give the global existence and uniqueness of system (1.6). Before that,we list some conditions[18]for the reaction termf(t,x,u):=um(1?u)(u?α)(m∈?)as follows

(F1)fis Lipschitz continuous inu: For any givenτ,l,k>0,f ∈L∞((0,τ)×(0,l)×(0,k))and there exists a constantM1(τ,l,k)such that

Under the above conditions,we give the following theorems.

Theorem 2.1.There exists a positive number T such that for any initial value u0∈Φ(u0),problem(1.6)has a unique solution(u,g,h),which satisfies

whereT0is called the maximal existence time of the solution of problem (1.6). By the idea of Theorem 1.2 in[18], we obtainT0=∞. This means that the global existence and uniqueness of the solution is achieved.

3 Stationary solutions and travelling waves

First,we use phase plane analysis to study the stationary solutions of system(1.6)

Each solutionq(z)of(3.2)corresponds to a trajectory,whenp/=0,which has a slope

Forf′(0)=?α<0 ifm=1,andf′(0)=0 ifm>1,the equilibrium point(0,0)will produce different types. By using Theorem 2.1 in [19] and Theorem 1 in Section 2.11 of [20], we obtain that (0,0) is a saddle-node ifmis even; (0,0) is an unstable node ifmis an odd number not less than 3. For eachm ≥2, there existsc0>0 such that system (3.2) has heteroclinic orbit ifβ ≥c0by the analysis in [19]. According to the above analysis, the parity ofmresults in several different phase diagrams, see Fig. 1-Fig. 3. The specific analysis is given as below.

Case 3. Fig. 3(a) is the phase diagram form=3,β=0,α=0.4, the system has three equilibrium pointsE31,E32andE33,which are an unstable node,an unstable node and a

Figure 1: Phase diagram of system(3.2) when m=1. (a) β=0, (b) β= (c) β=0.5, (d) β=4.

Whenβ=0, we find system (3.2) has periodic solutions, see Fig. 1(a), Fig. 2(a) and Fig. 3(a),which is consistent with the results in[15]. According to the above phase plane analysis,we know the system has heteroclinic orbits between the equilibrium points(0,0)and(α,0),(α,0)and(1,0),(1,0)and(0,0)whenβ/=0. In the following,we will discuss the exact heteroclinic orbits of system(3.2). Take the group parameters in Fig. 1(b)as an example,there exists a heteroclinic orbit between the equilibrium points(0,0)and(1,0).

Figure 2: Phase diagram of system(3.2) when m=2. (a) β=0, (b) β=, (c) β=0.5, (d) β=4.

Denote this heteroclinic orbit asS=(q(t),p(t)),we have

Now, we use the SLS method to calculate this type of heteroclinic orbit. Ift<0, we use the following logarithmic transformation

whereT1is a positive real constant, called scaling factor. Obviously,τ=etT1∈(0,1).System(3.2)is transformed into the following equations

Figure 3: Phase diagram of system(3.2) when m=3. (a) β=0, (b) β=, (c) β=0.5, (d) β=4.

In the case of other values of the parameters in system (3.2), we can still use the SLS method to find the exact heteroclinic orbit.

Different types of orbits represent different solutions in Fig. 1-Fig. 3. Now we give kinds of bounded solutions of system(3.2),see Fig. 4 in details.

Figure 4: (a) β=0, (b) β∈(0,c0), (c) β∈(c0,∞).

(i) Constant solutions:q≡0,q≡αandq≡1.

(ii) Periodic solutions:q(?∞)=q(∞),see Γ1in Fig. 4(a).

Theorem 3.1.Suppose that β>0,m∈,we have

(i) there exists c1(β)∈(0,c0+β)such that the problem

has a unique solution U1(x??c1(β)t).

Combining the analysis in[13]and[19], we have the following two theorems corresponding to the tadpole-like solution and the finite interval length solution respectively.

has a unique solution WLθ=W(x?c3(β)t),which corresponds to the finite interval length solution,where Lθ is the width of the support of WLθ.

4 Conclusion

This paper is devoted to the dynamical analysis on traveling wave of a reaction-diffusion equation with double free boundaries. Based on the phase plane analysis,the types of equilibrium points and the existence of heteroclinic orbits are studied,and the SLS method is used to solve the exact heteroclinic orbit of the system with corresponding parameters.Moreover, we show the existence of periodic traveling wave solutions whenβ=0 and other types of traveling wave solutions whenβ>0.

The current research only considered the dynamical analysis on the traveling waves of the system,but it is known that the long-time behavior theory and numerical analysis of the solutions are more meaningful and difficult to handle. These issues are of interest and will be considered in our forthcoming works.

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and suggestions that significantly improve the initial version of this paper.