Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model

We consider nonlinear evolution equations with logistic term satisfying initial Neumann-boundary condition and show global existence in time of solutions to the problem in arbitrary space dimension by using the method of energy. Applying the result to a mathematical model of tumour invasion, we discuss the property of the rigorous solution to the model. Finally we will show the time depending relationship and interaction between tumour cells, the surrounding tissue and matrix degradation enzymes in the model by computer simulations. It is seen that our mathematical result of the existence and asymptotic behaviour of solutions verifies our simulations, which also confirm the mathematical result visibly.


Introduction
In this paper we consider the initial Neumann-boundary value problem of nonlinear evolution equations with logistic term, arising from tumour invasion models with proliferation and re-establishment: (NE) ,0 , ,0 in ( 3) It is noticed that we have

which enables us to use
Poincare's Inequality.
where a and b are positive parameters. (NE) is rewritten by the following problem.
We will show the global existence in time of solutions of (RP), which gives our desired result of (NE).
Applying the above result to the following mathematical model of tumour invasion proposed by Chaplain and Lolas [1], we have a rigorous mathematical understanding to tumour invasion for the key variables n , m and f .
, n n x t = is the density of tumour cells, In the right hand side of (4) the second, third and fourth terms mean chemotaxis, haptotaxis and proliferation of tumour cell respectively. Also the second term of the right hand side of (5) describes the re-establishment of ECM. We consider an initial boundary value problem for (C-L) satisfying Chaplain and Anderson [2], corresponding to the case of While most tumours are asymptomatic at this stage, it is still possible for cells to escape and migrate to the lymph nodes and for more aggressive tumours to invade.
In our previous papers [3] [4], we consider only the case of 2 0 µ = and

Existence Theorem of (NE)
By deriving the energy estimate of (RP) (see [3]- [9]) and considering the iteration scheme we obtain existence of solutions to (RP) by the standard argument to show the convergence of solutions of the iteration scheme.

We begin with
in order to obtain a basic estimate of (RP). Then we have for It is noticed that the following estimate is obtained in [4] Then for the nonlinear term we have by using (8) where we used Dionne [10] for the estimate of nonlinear terms and ε is sufficiently small positive constant. Therefore we have by integrating the both sides of (7) over ( ) 0, t and using (9) Since the last term of the right hand side of (10) is negligible for sufficiently small ε , we have by integration by parts with respect to t Taking a sufficiently large for the second term of the right hand side of (11) is neg- instead of ν in the above procedure, we obtain the following estimate of higher order.
Lemma 1 (Energy estimate of (RP)) Assume that ( ) where we denote for any non-negative integer k M The local existence in time of ( ) ij f t is shown by the theory of ordinary differential and standard argument of convergence for Then we obtain the following result of (NE) by using the above result of (RP).

Application to a Tumour Invasion Model
In the last several decades, a number of mathematical models describing the procedure of tumour growth have been the remarkable subject of research (cf. [1] [2] [11]- [19], further references therein). Especially our main concern in this section is mathematical models of avascular tumour growth proposed by Chaplain et al. (see [1] [2]). They are considered mainly by three components in the process of tumour invasion, tumour cells, ECM (extracellular matrix) and MDEs (matrix degradation enzymes) without the effect of proliferation of tumour cell. Anderson and Chaplain [2] has been developed by Chaplain and Lolas [1] additionally considering into chemotaxis, proliferation of tumour cells and re-establishment of ECM.
Their mathematical approach to above models mainly depends on numerical analysis. In this paper first we show the rigorous mathematical result of (C-L) and then computer simulations, of which the validity is guaranteed by our mathematical result.
On the other hand, there are many mathematical models which can be found in the literature describing tumour angiogenesis. In [20] Levine and Sleeman applied the mathematical model of Othmer and Stevens [21] for the understanding of tumour angiogenesis, which arises in the theory of reinforced random walk. Anderson and Chaplain [12] proposed a model of tumour angiogenesis taking account of endothelial tip-cell migration.

Existence Theorem of (C-L) with n x 0 =
The Equations (16) and (17) are essentially regarded as the same type of equation as (1).
Hence the energy estimates of u and ν are obtained and combining these estimates we obtain the desired estimate (cf. [
Then applying the same argument as used for Theorem 2 to the above mathematical model, we obtain global existence in time and asymptotic behaviour of the solutions to our mathematical model.
Our main result for 0 n x = is as follows.
in the same way as in Section 2 ( )

Conclusions
In order to obtain the global existence in time and asymptotic profile of solutions of a mathematical model of tumour invasion proposed by Chaplain and Lolas, we investigate nonlinear evolution equations with logistic term related to our mathematical models as an initial Neumann-boundary value problem. We could show the global existence in time of rigorous mathematical solutions to the initial boundary value problem for the model in arbitrary space dimension by using the energy inequalities. Applying the result to our model we show global existence in time of mathematical solutions of the model.
By Figures 1-7, it is recognized that our rigorous mathematical result of the existence and asymptotic behaviour of smooth solutions verifies our computer simulations and confirms the pattern form of each component of the model in the graphs respectively. Then we can gain the understanding of the process of tumour invasion more in details.