Generalized Traveling Waves Of Reaction Diffusion Equations In Heterogeneous Media

It seems that many natural phenomena can be modeled by reaction-diffusion equations. Some nonlocal dispersal equations which represented by the integral op-erator have been derived from the research in many disciplines, such as biology and neural network. Anisotropy of the temporal space environment (heterogeneous me-dia) are always presented in natural diffusion phenomena, such as inhomogeneous porous structures in transport of solutes, noise effects in biology and deposition processes. Therefore, it is more meaningful and valuable in theory and practice to study such equations in heterogeneous media. One important issue is their gener-alized traveling waves, which is a global in time solution that can be viewed as a generalization of all usual notions.Firstly, we study the generalized traveling waves of reaction-diffusion equation with bistable nonlinearity in heterogeneous media. The generalized traveling waves connect the stable equilibrium 0 with 1. By solving a Cauchy problem starting at some time t=-n with the traveling wave as an initial data, then taking the limit n→+∞. So the limit function of the sequence of solutions for the correspond-ing Cauchy problem is just the generalized traveling waves. Then by constructing subsolution and supersolution coupled with mathematical induction, we establish the uniqueness and exponential stability of the generalized traveling waves. In the following, using a smooth cut-off function to construct an ignition problem, we obtain the existence of the generalized traveling waves for the monostable equations in heterogeneous media.Secondly, we investigate the generalized traveling waves of nonlocal dispersal equations in space heterogeneous media. Since the nonlinearity f(x,u) is a general heterogeneous function, so there exists a nontrivial positive steady state. According to the principal eigenvalue problem, we know that the steady state is unique. We then apply the comparison principle and sub-super solution method to prove the ex-istence of the generalized traveling waves which connecting the unstable equilibrium 0 with the nontrivial positive steady state.Finally, we consider the generalized traveling waves of time dependent nonlocal dispersal KPP equation. By modifying the degenerate parabolic equation as a reg-ular parabolic equation, and then constructing a series of auxiliary problem coupled with upper and lower solutions and iteration scheme, we prove the existence of the time global solution for the regular parabolic equation. From the limiting procedure and standard estimates, we obtain the existence of the generalized traveling waves.