Quasilinear Reaction Diffusion Systems with Mass Dissipation

We study quasilinear reaction diffusion systems relative to the Shigesada-Kawasaki-Teramoto model. Nonlinearity standing for the external force is provided with mass dissipation. Estimate in several norms of the solution is provided under the restriction of diffusion coefficients, growth rate of reaction, and space dimension.


Introduction
Quasilinear reaction diffusion system is given by for 1 ≤ i ≤ N, where τ = (τ i ) ∈ R N + , u = (u 1 (x, t), . . . , u N (x, t)) ∈ R N , Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, ν is the outer unit normal vector, and u 0 = (u 0 i (x)) ≡ 0 is the initial value sufficiently smooth. For the nonlinearity it is assumed that is smooth, and f = (f i (u)) : R N + → R N is locally Lipschitz continuous and quasipositive: We have, therefore, unique existence of a positive classical solution local in time. Our purpose is to extend this solution global in time. This question is posed in [11] with a positive result.
In the semilinear case when then u = (u i (x, t)) ≥ 0 is uniformly bounded and hence global in time, This result is a direct consequence of (5) for n = 1, and the cases n = 2 and n ≥ 3 are proven by [34,26] and [6,7], respectively. For the quasi-linear case of (1), however, several tools of the latter approach require non-trivial modifications, such as regularity interpolation [12] or Souplet's trick [29]. Here we examine the validity of the former approach. So far, global in time existence of the weak solution has been discussed in details. In [2,3,5,8,9] it is observed that appropriate logarithmic change of variables (1) can be transformed into a system with a symmetric and positive definite diffusion matrix. In [3], furthermore, it is shown that and D(t) stands for the energy dissipation, which induces u i log u i ∈ L ∞ (0, T ; L 1 (Ω)) and ∇ √ u i ∈ L 2 (Ω T ). This structure is used in [4,11], to derive existence of the weak solution global in time to (1) for an arbitrary number of competing population species, d i (u) = a i0 + j a ij u j with non-negative and positive constants a ij for 1 ≤ i, j ≤ N and a ij for 1 ≤ j ≤ N, respectively, under the detailed balance condition for positive constants π i , 1 ≤ i ≤ N.
The fundametnal assumption used in this approach is where c 0 , δ, C are positive constants. This assumption induces a uniform estimate of the solution in L log L norm. (2) and (9)- (10). Assume that it is bounded above and below by positive constants δ, C, Let, furthermore, f = (f i (u)) satisfy (3)-(4) and is of quadratic growth rate in the sense that it satisfies (6) and Then, it holds that sup 0≤t<T u(·, t) L log L ≤ C T (13) for u = (u i (·, t)). (2) and (9)- (10). Assume that it is of linear growth rate, with δ > 0. Assume, furthermore, the cubic growth rate of f = (f i (u)): Then (13) holds.
At this stage, the method of [34,26] ensures L q estimate of the classical solution under the cost of low space dimension. We require, however, an additional assumption to execute Moser's iteration [1]. Letting we obtain and therefore, (1) is reduced to The diffusion matrix A = (A ij (u)) is not necesarily symmetric nor positive definite.
Our assumption is where I denotes the unit matrix and δ is a positive constant.
The Shigesada-Kawasaki-Teramoto (SKT) model [28] describes separation of existence areas of competing species. There, it is assumed that N = 2, and where a ij , a i , b i , c i are non-negative constants for i, j = 1, 2 and a 10 , a 20 are positive constants.
Equalities (20) in SKT model are due to cross diffusion where the transient probability of particle is subject to the state of the target point [24,33], while equalities (21) are Lotka-Volterra terms describing competition of two species in the case of The Lotka-Volterra reaction-diffusion model without cross diffusion is the semilinear (20). For this system, any stable stationary solution is spatially homogeneous if Ω is convex [14], while there is (non-convex) Ω which admits spatially inhomogeneous stable stationary solution [22]. Coming back to the SKT model, we have several results for structure of stationary solutions to a shadow system [19,20,21,23]. There is also existence of the solution to the nonstationary SKT model global in time and bounded in H 2 norm if 64a 11 a 22 ≥ a 12 a 21 ( [36]). Obivously, Theorems 1 and 2 are not applicable to this system without total mass dissipation (4). Such f = (f i (u)), admitting linear growth term in (4), is called quasi-mass dissipative. Global in time existence of the solution without uniform boundedness is the question for the general case of quasi-mass dissipation.
We have the following theorem valid to such reaction under Theorem 4. Let d = (d i (u)) satisfy (24), and assume (3) and for f = (f i (u). Then, it holds that T = +∞ for any space dimension n.

Proof of Theorems
We begin with the following proof.
Three lemmas are needed for the proof of the other theorems.
The following lemma has been used for construction of weak solution global in time [11,4].
Proof of Theorems 1 and 2. These theorems are a direct consequence of Lemmas 5, 6, and 7.
Proof of Theorem 3. Any ε > 0 admits C ε such that See Chapter 4 of [30] for the proof. We have, on the other hand, by (6), (18) Letting with c 3 > 0. Apply the Gagliardo-Nirenberg inequality for n = 2, z r r ≤ C(r, q) z q q z r−q H 1 , 1 ≤ q < r < ∞.
Here we notice Wirtinger's inequality to deduce u p+2 p+2 = z r r ≤ C z r−1 In this inequality C on the right-hand side is independent of 1 ≤ p < ∞, beucase it then follows that 2 < r ≤ 3.
Remark 1. For system of chemotaxis in two space dimension, inequality (19) for q = 3 implies uniform boundedness of the chemical term by the elliptic regulariy, which replaces the right-hand side on (43) by a constant times 1 + u p+1 p+1 . Then Moser's iteration scheme induces (19) for q = ∞. See Chapter 11 of [30] for details. For the case of constant d i in (1), on the other hand, the semigroup estimate is applicable as in [18]. If n = 2, for example, inequality (19) for q = 2 implies that for q = ∞. Such parabolic estimate to (1) will be discussed in future.