A NONLINEAR PARABOLIC-HYPERBOLIC SYSTEM FOR CONTACT INHIBITION AND A DEGENERATE PARABOLIC FISHER KPP EQUATION

. We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the ﬁrst part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the diﬀerent types of cells keep spatially segregated when the initial den- sities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Ac- tually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.


1.
Introduction. In natural cell growth processes, contact inhibition of cells may occur when two different types of cells come into contact with each other (cf. [1]). A number of mathematical models (see for instance [13] [19] [20]) have been proposed for the theoretical understanding of the mechanism of contact inhibition. In [3], we have introduced a simple partial differential equation model, which describes contact inhibition between normal and abnormal cells with densities u and v. It includes the effect of pushing cells away from overcrowded regions so that each cell moves in the direction of lower cell density. In the case of one space dimension, the resulting model is given by Here α, β, γ and k are positive constants, and the initial functions are bounded and nonnegative: u 0 , v 0 ∈ L ∞ (R) and u 0 , v 0 ≥ 0 in R.
In mathematical ecology, the growth terms are regarded as of Lotka-Volterra competition type. For an introduction to the biological context we refer to [13] and the references therein.
Since u and v are nonnegative, we remark that the following inequalities hold Therefore, observe that where C + = max{1, γ}, C − = min{1, γ}, D + = max{1, α, γβ, γ/k} and D − = min{1, α, γβ, γ/k}. For several reasons the condition that w 0 is bounded away from zero is rather restrictive. First of all numerical tests are utterly important to understand the rich structure of the possible qualitative behavior of solutions, but often the simulations concern solutions with compactly supported initial data u 0 and v 0 . (See Figure 1 and also [6] for some examples.) Our first main result is Theorem 2.2, which establishes the existence of solutions if merely w 0 ≥ 0. Very recently, and independently similar results for the Neumann boundary value problem in 1D were obtained by Carrillo et al( [12]). The proofs in [12] are quite different from ours and are based on ideas from optimal transport theory. The remaining results mainly concern the singular limit k → 0. As k → 0, the solution (u k , v k ) of problem (1) converges to (u, 0), where u is the unique weak solution of the degenerate Fisher-KPP equation (see Theorem 2.3): Probably the most interesting aspect of the limit k → 0 concerns traveling wave solutions, i.e. solutions of the form (U (x − ct), V (x − ct)). Traveling wave solutions are natural candidates to describe the transient and/or large time behavior of classes of general solutions. In [6] we have shown that for all parameter values there exists a unique wave velocity for which there exists a unique (up to translation) segregated traveling wave solution, i.e. U (z)V (z) = 0 for all z ∈ R. Observe that segregated traveling wave solutions do not depend on the specific values of α and β, the coefficients related to the interaction terms uv. For example, if α = 1, k = 2 and β = 1/k = 1/2, the segregated traveling wave solutions seems to be a global attractor for a large class of solutions (see [6]).
In Figure 2 we plotted the numerical values of the wave velocity c * k when the value of k is varied (it is easy to see that c * k > 0 ⇔ 0 < k < 1, see also [6]). In view of the equation for V * k , it is not surprising that V * k → 0 as k → 0. We shall show (see Theorem 2.4 for a more general statement) that (U * k , c * k ) converges to (See Figure 2: .) It is well-known ( [9]) that c * 0 is a minimal wave velocity: the degenerate Fisher-KPP equation has a traveling wave solution if and only if c ≥ c * 0 , and if c > c * 0 the traveling wave solution is strictly positive for all z. A similar property as that of the degenerate Fisher-KPP equation holds for our system. If α = 1 and β = 1/k, it turns out ( [5]) that our system has a non-segregated or overlapping traveling wave solution for all wave velocities c > c * k > 0 (here we assume that 0 < k < 1), i.e. a solution (U, V, c) of the problem Numerical evidence also suggests that c * k is a minimal wave velocity. Moreover, we numerically discussed in [5] that the exponential decay rate of an initial function u 0 is important for the large time behavior. Actually, a pair of initial functions (u 0 , v 0 ) satisfying (u, v) → (1, 0) as x → −∞ and (u, v) → (0, k) with u 0 (x) = e −ξx as x → ∞ evolves to the traveling wave solution with velocity Roughly speaking, for an initial function u 0 with a large ξ, the solution in (1) converges to the segregated traveling wave solution, while for a moderate ξ, the solution converges to an overlapping one as time evolves. When we consider the evolution problem (1), the initial exponential decay rate of u 0 decides the large time behavior. So, at least if α = 1 and β = 1/k, the analogy with the degenerate Fisher-KPP equation is striking. We shall make this more precise by showing that, if α = 1 and β = 1/k, for all c > c * 0 the corresponding overlapping traveling wave solutions converge to the one with velocity c of the degenerate Fisher-KPP equation as k → 0 (see Theorem 2.4(ii)).
The paper concludes with two minor observations: we shall see that the case k → 0 gives some insight in the case k → ∞ (see Section 5), and we shall quantify the slope of the tangent line at k = 1, indicated in Figure 2 (see Theorem 2.4(iii)): Finally we shall list some open problems for our system, mainly motivated by the analogy with the degenerate Fisher-KPP equation.

Main results.
We first define what we mean by a weak solution of problem (1): In this article, we will prove the following results. Problem (1) possesses a solution which satisfies the segregation property: Theorem 2.2. Let α, β, γ, k > 0 and let u 0 , v 0 ∈ C(R) be such that u 0 , v 0 ≥ 0 in R. Setting w 0 := u 0 + v 0 , let either w 0 ∈ L 1 (R + ) or lim inf x→∞ w 0 (x) > 0 and w 0 ∈ W 1,∞ (M, ∞) for some M > 0, (9) and let either Then problem (1) has a solution which satisfies property (2).
As k → 0, the behavior of the solution is determined by the degenerate Fisher-KPP equation: where u is the unique weak solution of problem (4).
Observe that the definition of solution of problem (4) is similar to Definition 2.1. Also traveling wave solutions of our system converge to the corresponding ones of the degenerate Fisher-KPP equation:  (6)). Let, for k > 0, (U * k , V * k , c * k ) be the unique segregated traveling wave solution solving problem (5). Then be the unique traveling wave solution with wave velocity c solving problem (6) and the condition U (iii) c * k satisfies the convergence property (8).
3. The PDE system: Existence of solutions. In this section, we prove Theorem 2.2. We consider the sequence of smooth problems where (u 0n , v 0n ) is a sequence of smooth, nonnegative and uniformly bounded initial data, defined on R, such that u 0n → u 0 , v 0n → v 0 uniformly in R as n → ∞, {w 0n } are locally equicontinuous in R.
We will use the notations a n := and remark that the constants a and b can be finite or infinite. If lim inf x→∞ w 0 (x) > 0 and w 0 ∈ W 1,∞ (M, ∞), we assume that for some and make similar assumptions for x → −∞.
Lemma 3.1. For all R, T > 0, 0 < µ < 1 and n > R + 1 where the constant C R,T,µ does not depend on n.
Proof. We multiply the equation for w n , by ζ 2 R (x)w −µ n (x, t) and integrate by parts over (−n, n) × (0, T ), where ζ R ∈ C 1 (R) is a cut-off function satisfying: and (14) follows from the uniform boundedness of u n /w n , v n /w n and w n when we handle the term (−n,n)×(0,T ) 2ζ R w 1−µ n ζ R w nx in a suitable way. Indeed, in order to obtain (14), we use the inequality 2 (−n,n)×(0,T ) so that there exists a positive constant C(R, T, µ) such that which in turn implies (14). By [15], the functions w n are locally equicontinuous in Since r n is bounded, it converges locally and weakly along a subsequence. To pass to the limit in the equations for u n and v n , we have to prove strong convergence, at least in the set where w > 0. Observe that r n satisfies the equation where We introduce the characteristics associated to the velocity field −w nx . Below we list some of their properties.
where a n = −n 0 w 0n (s)ds and b n = n 0 w 0n (s)ds. For all a n ≤ y ≤ b n , there exists and X n has the following properties: for all T > 0 there exist positive constants ; BV loc (a n , b n ))∩H 1 ((0, T ); L 2 loc (a n , b n )). We will present the proof of Lemma 3.2 at the end of this section. Next we use the notation X n (y, 0) = X 0n (y), y ∈ [a n , b n ], and remark that X 0n is the unique solution of the initial value problem    X 0n (y) = 1 w 0n (X 0n (y)) y ∈ (0, b n ) and y ∈ (a n , 0) X 0n (0) = 0.
By Lemma 3.2(iii), Lemma A.1 in [16], and Aubin-Lions Lemma (cf. [10], Theorem II.5.16) there exist a subsequence {X nj }, which we denote again by {X n }, and a function X Here we remark that a and b are either finite or infinite. By Lemma 3.2(ii), for all t ≥ 0, the inverse function Y n (·, t) of y → X n (y, t) is well-defined for |x| < n and t ≥ 0, and it satisfies (13) and (14), Y nt is bounded in L 2 loc (R × [0, ∞)), uniformly in n. Next, we prove that Y n is locally and uniformly Hölder continuous in R × [0, ∞) and that where Y is the inverse function of y → X(y, t) for each t ≥ 0. We first remark that Y n is uniformly Lipschitz continuous with respect to x, and that Y nt is bounded in . We need to prove the uniform continuity with respect to t (at least locally).
Fix ε > 0, x 0 in a bounded interval [−l, l] and 0 < t 2 < t 1 < T . Let δ > 0 and α > 0 to be chosen below and let |t 1 −t 2 | < δ. Since is Hölder continuous with respect to t, uniformly in n; more precisely, there holds |Y n (η, if we choose δ sufficiently small. Therefore, since Y n is locally uniformly Hölder continuous in R×[0, ∞), we deduce that there exists a function Y and a subsequence of {Y n }, which we denote again by where the first term on the right-hand-side tends to zero as n → ∞ by (17) and the second one converges to zero since w n converges to w locally uniformly in R×[0, ∞).
We may assume that w 0 is not identically equal to 0 (otherwise the solution is trivial: (u, v) = (0, 0)). Consider, for t ≥ 0, the sets (3) and standard theory on the porous media equation ( [18]), By construction, the set P (y) 0 (0) = {y ∈ (a, b); w 0 (X(y, 0)) = 0} is closed, and either finite or countable: for some N ∈ {1, 2, . . . , ∞} P (y) We claim that in the (y, t) plane, the set where W = 0 is the union of at most countable vertical segments: for all y i ∈ P (y) This is an immediate consequence of the following result.
We postpone the proof to the end of this section. By local regularity results (Ch. V Theorem 3.1 in [17]), (w nj ) x → w x ∈ C loc (P + ) as j → ∞ in the set and hence it follows from (18) that This implies that X represents a regular Lagrangian flow in the sense of [2] and [14]: the equation We set R n (y, t) := r n (X n (y, t), t) for a n < y < b n , t ≥ 0. Then, by (16), Let R(y, t) be defined by Observe that )) and a.e. in (a, b) as n → ∞. Indeed by construction, the convergence is locally uniform (and hence pointwise) in the set of the points y ∈ (a, b) where w 0 (X(y, 0)) > 0, and w 0 (X(y, 0)) > 0 for a.e. y ∈ (a, b). Arguing as in section 4.5 of [4], it follows that , t), t). Since X is a regular Lagrangian flow, it follows from Proposition 4.9 in [4], which extends Proposition 3.5 in [14], that r is a distributional solution of the transport equation Observe that this is not surprising, since the product Y nx r n satisfies Arguing as in section 4.6 of [4], we prove the strong convergence of r nj to r. First we show that for any test function ϕ ∈ C ∞ (R × [0, ∞)) with bounded support Y nj x (r nj − r) 2 ϕ dxdt → 0 as j → ∞.
To this end, we prove that r nj Y nj x converges weakly to rY x as j → ∞ and Y nj x c nj converges weakly to Y x c as j → ∞, where c n := r n (1 − r n )G(r n , w n ) and c := r(1 − r)G(r, w). Let ϕ(x, t) be a smooth test function with bounded support. Then, by the strong convergence of R nj and X nj as j → ∞, On the other hand, let ξ be the weak limit of r nj Y nj x (up to subsequences). Then Next, let χ be the weak limit of Y nj x c nj . Taking the limit in But we already know that We repeat this procedure, replacing r nj by r 2 nj . Since the strong convergence of R nj implies the strong convergence of R 2 nj , On the other hand, letξ be the weak limit of r 2 nj Y nj x . Then, Therefore,ξ = r 2 Y x and r 2 nj Y nj x converges weakly to r 2 Y x as j → ∞. Finally we consider Y nj x (r nj − r) 2 = Y nj x r 2 nj + Y nj x r 2 − 2Y nj x r nj r. We deduce from the weak convergences above that, for any test function ϕ ∈ C ∞ (R × R + ) with bounded support, which, together with the inequaliy above and the locally uniform convergence of w nj , implies that and hence we have that r nj → r in L 2 loc (R × [0, ∞)) as j → ∞. Combining this result and w nj → w in C loc (R × [0, ∞)), we obtain that This permits to pass to the limit in the equations for u nj and v nj and we have found a solution of Problem (1).
Arguing as in section 4.7 in [4], the segregation property (2) follows immediately from the equation for R(y, t) (see (19)). That is, Since R and W are uniformly bounded, it follows from Gronwall's inequality that R(y, t)(1−R(y, t)) = 0 for all t > 0 and a.e. in (a, b). Hence we have u(x, t)v(x, t) = 0 for all t ≥ 0 and a.e. in R.
To complete the proof of Theorem 2.2 we prove the two Lemmata in this section.
Proof of Lemma 3.2. Before proving the existence of X n (y, t), we prove the properties (i) − (iii) as a priori estimates.
(i): The function q n (y, t) = w n (X n (y, t), t)X ny (y, t) satisfies q n (y, 0) = 1 and q nt = F n q n , where F n is the uniformly bounded function , and the result follows from Gronwall's Lemma (see also (5) in [4]).
(ii) follows at once from (i) and the uniform boundedness of w n .
This implies the uniform boundedness of X n in L ∞ ((0, T ); BV loc (R)). By (23) and part (i) of this lemma, uniform boundedness of X nt (y, t) in L 2 loc (R × [0, T ]) is equivalent with uniform boundedness of √ w n w nx in L 2 loc (R × [0, T ]), and hence, by (14), X n is uniformly bounded in H 1 ((0, T ); L 2 loc (R)). To complete the proof of (iii) we have to consider the cases that w 0 ∈ L 1 (R + ) satisfies (10), and, respectively, that w 0 ∈ L 1 (R − ) satisfies (9). It is enough to combine the ingredients of the proofs in the previous cases, and we leave the details to the interested reader.
Finally we observe that for all y the function t → X n (y, t) is well-defined since, by part (iii), it is a priori bounded in [0, T ].
Proof of Lemma 3.3. By Lemma 3.2, 0 ≤ Y x ≤ Cw and hence Y does not depend on x in P i 0 . Let t i = max{t ≥ 0; (x, t) ∈ P i 0 } and let (x i , t i ) ∈ P i 0 . Without loss of generality we may assume that t i > 0 and x i = 0. Let x 1 > 0 and let µ 0 < µ 1 < 1. For all τ ∈ (0, t i ], On the other hand, since µ 0 < µ 1 < 1 and sup 4. Singular limits: The degenerate KPP equation. In this section we prove Theorems 2.3 and 2.4. First we consider the singular limit problem of the PDEsystem (1).
Proof of Theorem 2.3. Let T > 0. In view of Lemma 3.1, it follows that where C is independent of k. Let R > 0 and let ζ R be a cut-off function which satisfies (15).
and γ k R×(0,T +1) Proof. Multiplying the equation for w k , formally, by ζ 2 R w µ k with µ > 0 and integrating by parts over R × (0, T + 1), we obtain that Here, we can estimate the first term of the right hand side by Young's inequality as follows: 2 Therefore, thanks to this inequality and Lemma 3.1, we obtain where the constant C R,T,µ does not depend on k. This implies (24). Simply, multiplying the equation for w k by ζ 2 R and using the integration by parts over R×(0, T +1), we obtain that The use of (24), Lemma 3.1 and the similar estimate This leads (25).
In order to obtain a strong convergence of {w k }, we claim that In order to do that, we first prove the following Lemma: Proof. The proof of (27) is immediate: Here we have used the estimate (24) with µ = 1. Next we consider (28). Using (15), we have In the first term of the right hand side, we integrate by parts: The second and third terms are estimated by using (25) as follows: which completes the proof of (28) and Lemma 4.2.
Passage to the limit. It follows from (25) that v k → 0 strongly in L 2 loc (R × [0, ∞)) as k → 0. Moreover, we deduce from the boundedness of {w k } that there exists a function w ∈ L ∞ (R × [0.∞)) and a subsequence {w kj } such that   (R × [0, ∞)) and a.e. in R × [0, ∞) and weakly in L 2 loc ([0, ∞); H 1 loc (R)). Applying Lebesgue's dominated convergence theorem, we also deduce that [0, ∞)). Taking the difference between the equations for w and v, we deduce that for all ϕ ∈ C 1 (R × [0, ∞)) such that ϕ vanishes for large |x| and t. We deduce from the convergence properties above and Lebesgue's dominated convergence theorem that as k j → 0. Moreover, there also holds that Next, we consider the remaining term. as k j → 0. Setting u := w, we deduce that u satisfies the weak form for all ϕ ∈ C 1 (R × [0, ∞)) such that ϕ vanishes for large |x| and large t. Thus u coincide with the unique weak solution of the initial value problem and the whole sequence {u k } converges to u as k → 0. Next we consider the singular limit problem for traveling wave solutions. We begin with the case of segregated traveling wave solutions.
Proof of Theorem 2.4(i). Let 0 < k < 1 and let (U * k , V * k , c * k ) be the unique segregated traveling wave solution solving problem (5).
We can prove from the phase plane analysis that The proof is given in the appendix. Let R > 0 and let ζ R be a cut-off function satisfying (15). Multiplying the equation for W * k by ζ 2 R W * k and integrating by parts, we have Using the estimates we obtain that Here we have used that c * k and W * k are uniformly bounded (see also Appendix). Since (W * k ) 3/2 is uniformly bounded in H 1 loc (R), there exist a function U 3/2 ∈ H 1 loc (R) and a subsequence {k j } such that (W * kj ) 3/2 → U 3/2 weakly in H 1 loc (R) and W * kj → U in C loc (R) as k j → 0.
In addition, since W * k → 0 in L 3 loc ([0, ∞)), Since W * k is decreasing in R (see [6]), U is also decreasing in R. Observe that, since W * k ((W * k ) + c * k ) is decreasing in (−∞, 0] in the equation for W and since it vanishes at z = 0( [6] and see also (35) 0). On the other hand, since Finally, passing to the limit j → ∞ in the equation for W * kj , it follows that U ∈ C(R) satisfies Since U is decreasing, U (−∞) = 1, and we have proved that U = U * 0 , the unique traveling wave solution with minimal wave velocity c * 0 and interface at z = 0. Since the limit U is uniquely defined, it does not depend on the specific subsequence {k j }.
Next we prove the convergence of overlapping traveling wave solutions.
Proof of Theorem 2.4(ii). Let 0 < k < 1 be so small that c > c * k . We have shown in [5] that there exists an overlapping traveling wave solution, (U By translation invariance we assume that U The function W k := U k + V k satisfies Let R > 0. Multiplying the equation for W k by ζ 2 R W k , where ζ R is a cut-off function satisfying (15), integrating by parts and using similar estimates to (31) and (32), we obtain that , there exist a function U 3/2 ∈ H 1 loc (R) and a subsequence k j → 0 such that W 3/2 kj → U 3/2 weakly in H 1 loc (R) as k j → 0 and hence W kj → U in C loc (R) as k j → ∞. Passing to the limit in the equation for U k , we find that U is a solution of To show that U is the unique traveling wave solution with wave velocity c solving problem (6) which satisfies U (0) = 1 2 , it is enough to prove that U (−∞) = 1 and U (∞) = 0. Since W k is decreasing in R (see [5]), U is also decreasing in R. So, U (−∞) = 1 and U (∞) = 0.
Since the limit U is uniquely defined, it does not depend on the specific subsequence {k j }.
Finally we prove the behavior of the wave velocity of segregated traveling wave solutions as k → 0.
Proof of Theorem 2.4(iii). We only consider the limit k → 1 + . The proof for the limit k → 1 − is similar.
So let k > 1 and let (U * k , V * k , c * k ) be the unique segregated traveling wave with its interface at z = 0. Let w * k ∈ (1, k) be the value of W * k = U * k + V * k at z = 0. Observe that c * k < 0 if k > 1. We set , and consider p k as a function of s (this is possible since z → W * k (z) is strictly increasing if k > 1). Then Observe that (k − 1)p k (s)(p k (s) + 1) 1 + s(k − 1) → 0 as k → 1 + .

5.
Remarks and open problems. In this paper, we have considered the existence of a solution with segregation property of Problem (1) for general initial data and singular limit problems to reveal a relation between Problem (1) and the degenerate Fisher-KPP equation. Though we considered the case k → 0, this gives some insight in the case k → ∞. Actually, the change of variablesṽ = u/k,ũ = v/k,t = γt, x = γ/kx,γ = 1/γ,k = 1/k,α = βk andβ = αk gives which corresponds to the system (1).
In the first part, we proved the existence of a solution with segregation property for initial data merely w 0 = u 0 + v 0 ≥ 0 in one space dimension. However, for arbitrary space dimensions, the problem is still open. In the second part, we treated two types of traveling wave solutions, say segregated and overlapping traveling waves. Recently, the existence of other traveling waves for large and small gamma, namely partially overlapping traveling wave solutions, and standing waves which possess extremely rapid decay tails was proven in [7] [8]. From these results, Problem (1) possesses a surprising variety of mathematical structures depending on the parameter values even though we focus on traveling wave solutions. It appears that these reflect the parabolic-hyperbolic nature of Problem (1), which are completely different from those of a parabolic-parabolic system such as a reaction-diffusion system. As argued in this paper and in [5], in a special case, the structure of traveling wave solutions of Problem (1) is similar to that of the degenerate Fisher-KPP equation, noting that the first equation in Problem (1) coincides with a degenerate Fisher-KPP equation if we set v = 0, whereas setting u = 0 in the second equation of Problem (1) also yields a Fisher-KPP equation. We propose to study in future work the complete structure of the set of traveling wave solutions. Appendix.
Remark A.6. The upper bound of the velocity of the segregated traveling wave solution coms from Lemma A.5.
Lemma A.7. For any 0 < c < 1/ √ 2 the function h U given in Lemma A.5 satisfies the following (i) and (ii): (i) h U is monotone decreasing for any c > 0, (ii) lim c→0 h U (c) = 1 and lim c→1/ Proofs of Lemmata A.5, A.7, A.8 and A.9 are essentially reduced into the study of the dynamcal system ϕ ψ = cψ G(ϕ, ψ, c; k, γ) For the sake of convenience, we display the phase portrait of (40) in Figure 3.
The proofs are the same as those in [6].
Lemma A. 10. Let h V (c) = h V (c; k, γ) as in Lemma A.8. Then, it is satisfied that for any k > 0 and γ > 0, k < h V (c) < k + c k γ .
Remark A. 11. Lemma A.10 shows Corollary A.3.  Proof of Lemma A.10. Let (V 1 , V 2 ) be the solution of (38) with c > 0. Then, By integrating the above inequality with respect to V 1 ∈ (k, h V (c)) and combining it with k < h V (c), we are led to Lemma A.10. is a desired unique solutions of (35) and (36).
Since h U is continous with respect to c, we conclude by letting j → ∞ that h U (σ) = 0.
It implies that σ = 1/ √ 2. Finally, since σ is independent of the choice of the subsequence {k j }, we obtain a desired result. Thus it complete the proof.