Mathematical analysis of tumour invasion with proliferation model and simulations

In this paper it is shown that the global existence in time and asymptotic profile of the solution of a mathematical model of tumour invasion with proliferation proposed by Chaplain et al. For this purpose we consider a related nonlinear evolution equation with strong dissipation and proliferation corresponding to our mathematical model and the initial Neumann-boundary value problem for the evolution equation. We prove the global existence in time of solutions to the problem for the nonlinear evolution equation in arbitrary space dimension by the method of energy. In this paper our main mathematical approach is heavily based on energy estimates. Applying our mathematical result of the problem to the tumour invasion model we will discuss the existence and property of solutions to the model which gives us a rigorous mathematical understanding of the model. Finally we will show the time depending relationship and interaction between tumour cells, the tissue and degradation enzymes by computer simulations of the model. It is seen that our mathematical result of the existence and asymptotic behaviour of solutions guarantees the validity of computer simulations and implies the pattern figure of each component of the model respectively.