Traveling Wave Slutions Of Reaction-Diffusion Equation With Gradient Structure

It is well known that many natural phenomena can be described by reaction-diffusion equation, so it has become one of the most important research fields of mathematics. In the study of reaction-diffusion equations, traveling wave solution is one of the important branches. Recently, some scholars have found that many wave equations from reaction-diffusion equations possess gradient structure. Thus they investigate traveling wave solutions of reaction-diffusion equations by variational method based on the gradient structure, and have broken the original framework based on the maximum principle. It is worth noting that variational method can describe the researched object more finely and can be easily extended to gradient diffusion system for which the maximum principle does not hold. On the other hand, when spatial variable is high-dimensional, the theory of nonplanar traveling wave solutions of reaction-diffusion equation is relatively less than that of the planar traveling wave solution in the one-dimensional case, moreover, the corresponding wave equation is elliptic equation in high-dimensional case. In recent years, a lot of nonlocal dispersal equations described by integral operator have been derived from many disciplines, such as biology, material science, neural network. Hence, it has important realistic and theoretical significance to study traveling wave solutions of high-dimensional reaction-diffusion equations and mixed dispersal equation of random and nonlocal with gradient structure. Furthermore, there are some difficult problems.Firstly, wc study traveling wave solutions of scalar reaction-diffusion-advection equations with gradient structure under mixtures of Dirichlet and Neumann bound-ary conditions in infinite cylinder. By virtue of the comparison principle and sub-super solution method, when the wave velocity is larger than the minimal wave ve-locity, the existence of traveling wave solution is proved. Furthermore, for reaction-diffusion-advection equations with bistable and ignition nonlinearity, we obtain the existence, uniqueness (up to translation) and monotonicity of traveling wave solu- tions by converting the existence of traveling wave solutions into the existence of constraint minimizers. Moreover, since the wave velocity related to the infimum, we find the monotone dependence of wave speed on the nonlinearity term and the domain. Finally, the influence of advection on traveling waves is concerned.Secondly, we discuss traveling wave solutions of gradient diffusion systems-Ginzburg-Landau type problem. To this aim, we first give a variational formulation for a general parabolic equation. Then we apply this formulation to Ginzburg-Landau-type problems in infinite cylinder. In particular, we consider traveling wave solutions of a scalar reaction-diffusion-advection equation by this variational formu-lation. At last, by the energy method coupled with the comparison principle, the global exponential stability of traveling wave solutions for scalar reaction-diffusion-advection equation is established.Finally, we study traveling waves of the mixed evolution equation of random dispersal and nonlocal dispersal with gradient structure and degenerate bistable and ignition nonlinearities. By the variational method, the existence of traveling waves is reduced to the existence of constraint minimizers. Thus, the existence, uniqueness (up to translation) and monotonicity of traveling wave solutions are obtained by considering the constraint minimization problem.