A Class Of Reaction Diffusion Equations Describing Cancer Invasion

Partial differential equations are widely used in various fields such as medicine,chemistry,engineering and so on,which used to solve many practical problems.Through the research of their fixed solutions,the actual problems can be mathematically,which provides an effective way to solve practical problems.As a type of partial differential equations,the reaction-diffusion equation has a wide range of researches,such as the study of tumor spread and the dynamic proliferation and competition between populations.The study of the reaction-diffusion equation has been subject to domestic wide attention on foreign scholars.In this paper,firstly,we use the fixed point theorem to prove the global existence and uniqueness of the solution to the reaction-diffusion equations.We not only use the strong extremum principle to obtain the nonnegative of the solution,but also obtain the boundedness,local existence uniqueness and global existence uniqueness are obtained through further estimation of the solution.Secondly,we mainly study the local stability of the semi-trivial solutions to the reaction-diffusion equation.By linearizing the system of equations,we obtain the form of a linear equation with parameters,and then we use the principal eigenvalue method to obtain the conditions of the stability and instability of the semi-trivial solution.Finally,the global stability of the positive solutions to the reaction-diffusion equations is studied using the method of constructing Lyapunov functions.