Qualitative Analysis Of Tumor Growth Model With Robin's Free Boundary Condition

The tumor growth model is one of the most important hot topics in the field of partial differential equations.From a biomedical point of view,the model with Robin boundary condition is more in line with this reality: the cell membrane has a certain shielding effect on nutrients and inhibitors.Compared the model with Dirichlet boundary condition or Neumann boundary condition,the model with Robin boundary condition is more complex.The study of the model with Robin boundary condition has more theoretical and practical significance,so this paper focuses on the model with Robin boundary condition.In this paper,we study two mathematical models of tumor growth with Robin's free boundary condition,and obtain qualitative analysis of the corresponding problems by strict mathematical analysis,as follows:Firstly,we study a mathematical model of necrotic tumor growth with Robin's free boundary.The model comprises a parabolic equation and an ordinary differential equation.The tumor is assumed that the nutrient concentration determines the growth of the tumor which is spherical symmetry.By strict mathematical analysis,we prove the existence uniqueness of the stationary solution of the model.Secondly,we consider the Robin's free boundary problem modeling tumor growth with angiogenesis.The model contains an ordinary differential equation describing the radius of tumor cell and two parabolic equations describing the evolution of nutrient concentration and inhibitor concentration,respectively.We study the number of the stationary solution of above problem,the existence and uniqueness of local solution and global solution,and the asymptotic behavior of the solution.