GLOBAL ASYMPTOTIC OF FOR NONLINEAR EQUATIONS RELATED TO A TUMOUR INVASION MODEL

. We study the global existence in time and asymptotic behaviour of solutions of nonlinear evolution equations with strong dissipation and proliferation terms arising in mathematical models of biology and medicine including tumour invasion models.

1. Introduction and notations. In this paper we study the following initial Neumannboundary value problem of nonlinear evolution equations arising from chemotaxis type of models with logistic term; u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) in Ω (1.3) where the function χ(·, ·) will be specified later in the assumption (A), D, µ are positive constants, Ω is a bounded domain in R n with smooth boundary ∂Ω and ν is the outer unit normal vector on ∂Ω.
Our purpose is to establish the existence theorem of time global solutions to (N ) and apply the result to a mathematical model from biology and medicine. The aim of this paper is in the same line as in [18]. Some of the results in this paper have been announced and some sketches or outlines of their proofs have been stated in [18]. In this paper, we improve and extend them, give the full proof, self-containedness and a broader class of applications than the one presented in [18]. The equation (1.1) is a generalisation of nonlinear evolution equations related to a tumour invasion model (CL) (see section 4) so that we study (N ) in advance.
For our convenience we first consider a special case of (N ) for µ = 0, which is written as u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x).

AKISATO KUBO, HIROKI HOSHINO AND KATSUTAKA KIMURA
We deal with (N ) in section 2. Next, by developing the results for (N ) , we get the desired result for (N ) in the section 3. In the final section, applying the arguments in sections 2 and 3 we investigate the properties of the solution to a mathematical model arising from tumour invasion. On the other hand, initial-Dirichlet boundary value problems for this type of nonlinear evolution equations have been studied by many authors (e.g. [6]- [8]).
In order to discuss the existence of the solution and its asymptotic behaviour of (N ) we seek the solution in the form of u(x, t) = a + bt + v(x, t) for positive parameters a and b. Then (N ) is rewritten as where we denote χ(v t + b, e −a−bt−v ) by χ a,b (v). Levine and Sleeman [14] obtained explicit solutions of the form: u = γt + v(γ > 0) of a simplified equation of (1.5) for n = 1 (cf. [10], [16]). In this line, the papers [10]- [13] showed the existence of the solution in the same form as above of a special case of (N ) for any spatial dimension, which arises from mathematical biology and biomedicine (see [1], [2], [17]). Let us denote an upper semicircle B r+ = B r ∩ (R × R + ) where B r is a circle of radius r at 0 in R 2 . We assume that for a constant r > 0 and (s 1 , s 2 ) ∈ B r+ there exists a positive constant c r such that for any integer m ≥ [n/2] + 3 it holds Remark 1. In the previous paper [9] additionally we need the assumption that χ(s 1 + b, s 2 ) is positive and the parameter b should be taken sufficiently large to obtain the global existence theorem in time of (N ) . In this paper it is shown that without such additional conditions we can obtain the same results as in [9].
Especially in the case of Ω = R n we denote the norm of H l (R n ) by h l,R n (t).
2. The case µ = 0. When we derive the energy estimates, we assume the regularity con- and v t ∈ L 2 ([0, ∞); W m (Ω)) and the boundedness condition (v t , e −l−v ) ∈ B r+ for l = a + bt. We first prepare some results required to derive energy estimates of (R) .
By using Lemma 2.1 we obtain the following result (see [10]).
We can also get the following result by integration by parts. Lemma 2.3. Assume that u = u(x, t) satisfies the above regularity condition with m > M ≥ [n/2] + 1. For 0 < b < b and i = 1, 2, · · · , n, it holds that Proof. For any ε > 0, we have using integration by parts for the second term Taking ε sufficiently small, finally we have the desired result. Now let us state basic results required for the estimate of the nonlinear term. The following results have been obtained by Dionne [5] for x ∈ R n . It will be shown that the corresponding results restricted the domain R n to Ω in [5] hold. Denote and ω is a multi-index for ω = (ω 1 , · · · , ω n+2 ). We define W (l) Lemma 2.4. For u ∈ C([0, T ); H l (Ω)), there exists a functionũ ∈ C([0, T ); H l (R n )) and a constant C > 0 such thatũ = u in Ω × [0, T ) and the support ofũ lies in an arbitrary bounded open set in R n whose interior containsΩ and that it holds Proof. Applying Proposition 3.4 and Theorem 3.13 in Mizohata [15] to u, we can extend u toũ ∈ C(R n × [0, T )) and (2.1) holds. Denote Du = (u t , ∇u) for simplicity. By applying Proposition 1 we obtain the following result in the same way as in [5].
By using Propositions 1, 2 and Dionne [5; Theorem 6.4] we obtain the following result.
Lemma 2.5 (Basic estimate of (R) ). We have a basic energy estimate of (R) under the regularity and boundedness conditions on Proof. In order to obtain a basic estimate of (R) we consider Since we have for any ε > 0 by using Proposition 2 and Lemma 2.2, by integrating the equality (2.4) over (0, t) and using the above estimate we get Since the last term of the right hand side of (2.5) is negligible for sufficiently small ε, we have by using Lemma 2.3 for the second term of the right hand side of (2 Taking a sufficiently large the last term of (2.6) can be negligible. Hence we have a basic energy estimate of (R) .
Lemma 2.6 (Higher order estimates for (R) ). Under the regularity and boundedness conditions on v = v(x, t) with m > M ≥ [n/2] + 1, we have the result of higher order energy estimate (R) for sufficiently large a: where we denote for any non-negative integer k, Proof. Suppose that the estimate (2.7) holds for M = k − 1 ≥ 0. Considering ∇ k v instead of v in (2.4), in the same way as in Lemma 2.5 we can obtain (2.7) for M = k. In order to show it, it is enough to prove that the following estimate holds for l = a+bt and a parameter κ > 0 by using Proposition 2 and the above assumption where the first and second terms in (2.8) are negligible for sufficiently large a and small κ > 0 respectively. In fact, by using Lemma 2.3 and taking a sufficiently large, we see that the first term of (2.8) is negligible. Hence we obtain (2.7).

Remark 2.
The estimate (2.7) implies that for any fixed r by taking E a,M [v](0) sufficiently small we have (v t , e −l−v ) ∈ B r+ . As will be seen, considering an iteration scheme of the problem, (2.7) also guarantees global existence in time of the solution satisfying the condition (A) and the regularity of the solution.
Now we state our result of (N ) .
Proof. The proof will be shown in the same manner as in [9]- [13]. We take an iteration scheme and derive the energy estimate of it.
. Taking account of Remark 2 the energy estimate, we find that (2.7) guarantees the estimate with a uniform upper bound of each problem (R) (i+1) for i = 1, 2, · · · .
We determine f ij (t) by Galerkin method and by applying (2.7) to the following system of ordinary equations with initial data, for j = 1, 2, · · · we obtain the global smooth solution in time. ( Also the energy estimate enables us to get the solution of (R) by considering Actually, we consider the following problem.
In order to obtain the estimate of (R) (i+1)−(i) we only deal with the last term of P i − P i−1 as follows: for θ > 0 and m ≥ M ≥ [n/2] + 2 Then by the same method as derived the energy estimate of (R) , we have for sufficiently large a by taking account of (2.10) and Lemma 2.3 where C a depends on sup M and e −a , C a → 0 as a → ∞. Here we took θ sufficiently small to neglect the last term of (2.10). Hence taking a sufficiently large we see C a < 1. By the standard argument we see that the solutions {v i } converges strongly such that for m ≥ [n/2] + 3 (2.12) The proof of (2.9) is shown in the same way as in Theorem 2 of [10].
3. The case µ = 0. In this section we consider the case of µ = 0 in (1.1) developing the way used in the previous section. Put u(x, t) = a + t + v(x, t), then (N ) is rewritten as In order to derive the estimate of (R), in the same way as used in section 2, it is enough to deal with the inner product from the proliferation term For the smooth solution of our problem v(x, t) satisfying |v t | 1 and regularity and boundedness conditions we have in the same way as derived (2.7) for m > M ≥ [n/2] + 3 and sufficiently large a For sufficiently large a and small r, there is a solution Proof. We consider the following iteration scheme of (R) The time global smooth solution of (R) (i+1) is obtained successively in the same way as in the proof of Theorem 2.7. Since it is enough to derive the estimate of (R) (i+1)−(i) for we only deal with the inner product from the proliferation term where C depends on v it M . Since we may assume that v it M is small enough for m ≥ M > [n/2] + 2, neglecting the last term of the right hand side of the above inequality, we have finally M and e −a . Taking a sufficiently large and initial data sufficiently small we obtain the desired result.
Remark 5. In [10]- [13] our solution is in the form of u(x, t) = bt + v(x, t) for sufficiently large b > 0, but in [9] we can get the solution in more general form u(x, t) = a+bt+v(x, t) for sufficiently large a, b > 0. Since in section 2 we obtain the existence theorem for sufficiently large a and any b > 0, it enables us to deal with (1.1) for µ = 0 in the section 3.

4.
Application to a tumour invasion model. The following is a mathematical model of tumour invasion by the Chaplain-Lolas [3]. Integrating the both sides of (4.4) over (0, t) we have Since our aim is to seek solutions {n, f, m} of (CL), we may substitute t 0 nds and t 0 mds by a n + t + u and a m + b m t + v respectively for new unknown functions u, v and the parameters a n , a m , b m > 0. Setting a m η + a n µ 2 = a and b m η = b it is reduced to wheref = t 0 f ds. Then substituting f (x, t) by the right hand side of (4.5), (4.1) and (4.3) are reduced to the following two equations respectively ∂ 2 ∂t 2 u = d n ∆u t − γ∇ · ((1 + u t )∇(f 0 (x) · e −a−bt−ηv−µ2(u+f ) )) +µ 1 (u t + 1)(−u t − f 0 (x) · e −a−bt−ηv−µ2(u+f ) ) (4.6) ∂ 2 ∂t 2 v = d m ∆v t + αu t − βv t + α − bβ. (4.7) They are essentially regarded as the same type of equation as (1.1). Here the initial data are written as u t (0) = n 0 − 1, v t (0) = m 0 − b m . Setting Θ = a + bt + ηv + µ 2 (u +f ), (4.6) is rewritten as .
If the energies of initial data are sufficiently small, by Lemma 4.2 i) we obtain the estimate of the first term of (CL) (i)−(i−1) for sufficiently large a Concluding remark. In section 3 for a more general form of the solution u(x, t) = L a (t) + v(x, t) instead of u(x, t) = a + t + v(x, t) we can deal with (N ) for µ = 0 where L a (t) = t 0 l(τ )dτ + a and l(t) satisfies the logistic equation. In the same way we can discuss the mathematical model in section 4. This result will be published somewhere soon.