Study On Entire Solutions Of Several Kinds Of Reaction-diffusion Equations

Entire solutions to reaction-diffusion equations are those whose time is defined on the whole real number line.Together with equilibrium solutions and traveling wave solutions,they can help people understand the global dynamic behavior of the equations.In recent decades,many scholars have constructed some entire solutions for various reaction-diffusion equations and obtained abundant results.In this paper,several kinds of reaction-diffusion equations are considered and some new entire solutions are constructed for them.In Chapter 2,we consider the Fisher-KPP equation with convection satisfying Dirichlet boundary conditions on the real semi-axis,and prove that there are two types of entire solutions to the equation.First,we construct a family of entire solutions.When time t→-∞,they converge to traveling waves with different velocities respectively under the moving frame.Secondly,when the convection is zero,we construct an entire solution derived from asymptotic flatness,i.e.when t→-∞ its slope at any horizontal set tends to zero.When the convection is to the left,we construct an entire solution with monotonically increasing time at x=∞.In Chapter 3,we consider the entire solution of a general monostable equation on the real semi-axis.A family of entire solutions is also constructed under different types of boundary conditions.When time t→-∞,they converge to traveling wave solutions with different velocities under the moving frame.In Chapter 4,we consider the lattice Fisher-KPP system and construct two kinds of entire solutions.The first type is a class of entire solutions:when t→-∞,convergence to traveling wave with different velocities under the moving frame.The second type is a convex entire solution that converges to some horizontal line under the shifting frame when t→-∞.The above work gives some new entire solutions to(continuous and discrete)reactiondiffusion equation on the real semi-axis,and describes the α-limit set,which improves the type of entire solutions of the reaction-diffusion equation.