Free Boundary Problem Of A Class Of Nonlinear Convection Reaction Diffusion Equation

In this paper, we consider the free boundary problem of a reaction diffusion equation with nonlinear convection term in one dimensional space. The main content is composed of the following parts.Firstly, the related research background and the main results of this paper are briefly introduced. In ecology, the free boundary problem is mainly used to describe the long time dynamics of the species in a new environment. In practice, the migration and expansion behavior of biological populations are sometimes affected by the environment. Considering the nonlinear convection term in the equation has practical significance, so this paper studies a class of nonlinear convection diffusion equation.Secondly, we obtain the local existence and uniqueness and the global existence and uniqueness by using the contraction mapping principle and estimations of parabolic equations. By considering the corresponding eigenvalue problem of the equation, and constructing the upper and lower solutions, we obtain the spreadingvanishing dichotomy, and the sufficient conditions for the occurrence of spreading or vanishing are given.Finally, using the traveling wave theory of reaction diffusion equations, we construct upper and lower solutions of parabolic equation by proper semi-waves, then we obtain some estimations of the asymptotic speed of free boundaries when spreading happens. At the same time, we discover that if the diffusion coefficient is greater than a certain value, then free boundaries at two sides have same speed when the expansion coefficient approaches infinity.