Stability Of Non-critical Wave In Diffusion Equation

The reaction diffusion equation(system)is a kind of nonlinear parabolic equations(systems),which plays an important role in describing the spatiotemporal model.Traveling wave solutions of such equations or systems can explain the phenomenon of finite speed propagation and finite vibration in nature.The traveling wave front,which is the monotone and bounded traveling wave solution,thus has been widely studied because of its good properties.In the qualitative study of traveling wave fronts,the stability is one of the important properties.In this paper,the stability of traveling wave fronts with non-critical wave speed(hereinafter referred to as "non-critical traveling wave fronts")of one scalar reaction-diffusion equation and reaction-diffusion systems have been studied respectively.The paper is divided into three chapters.In Chapter 1,the related researches of the scalar reaction-diffusion equation——Fisher-type equation with degenerate reaction term(hereinafter referred to as "degenerate Fisher-type")and the reaction-diffusion equation system——the Lotka-Volterra reaction-diffusion competition system are introduced.Firstly,the stability results and research methods of degenerate Fisher-type equation are introduced.Secondly,the research progress on the stability of the traveling wave fronts of the Lotka-Volterra reaction-diffusion competition system connecting the boundary equilibrias(0,1)and(1,0)under unbalanced competition is described.Finally,the main conclusions,research methods and necessary preparatory knowledge of this paper are stated.In Chapter 2,the following degenerate Fisher-type equations are considered(?)u(x,t)/(?)t=(?)2u(x,t)/(?)x2+f(u(x,t)),where f(u)=un(1-u),and n ∈(0,∞)is not necessarily to be an integer.When the difference between the initial condition and the non-critical traveling wave front at t=0 is in a given weighted Sobolev space,we prove that the solution of the initial problem converges to the non-critical traveling wave front in the form of algebraic exponent by L1energy estimates and L2-energy estimates,that is,the traveling wave front is algebraically exponentially stable in the weighted Sobolev space.Further more,using by the Fourier transform method,we obtain more accurate attenuation rate of the traveling wave front in a certain parameter range.The conclusions of this paper extend the local stability of the non-critical traveling wave fronts to the global stability,and give a novel stable form of these solutions-algebraic exponential stable form,thus enrich and promote the research on the stability of non-critical traveling wave fronts of this equation.In Chapter 3,based on the weighted energy method,the global exponential stability of traveling wave fronts connecting boundary equilibrium points(0,1)and(1,0)of the LotkaVolterra reaction-diffusion competition system under unbalanced competition is studied For any c>2(?):=c*,by improving some technical details,we find that the traveling wave front is globally exponentially stable in the weighted Sobolev space when the diffusion parameter d satisfies 0 ≤ d<2+o(1).Therefore,compared with the condition that the wave speed is much larger than c*in the existing related research results,the conclusions in this paper expand the range of wave speed that makes the traveling wave front stable.In Chapter 4,the main conclusions and methods of this paper are summarized,and the next research objectives are briefly introduced.