FREE BOUNDARY PROBLEM FOR A REACTION-DIFFUSION EQUATION WITH POSITIVE BISTABLE NONLINEARITY

. This paper deals with a free boundary problem for a reaction- diffusion equation in a one-dimensional interval whose boundary consists of a fixed end-point and a moving one. We put homogeneous Dirichlet condi- tion at the fixed boundary, while we assume that the dynamics of the moving boundary is governed by the Stefan condition. Such free boundary problems have been studied by a lot of researchers. We will take a nonlinear reaction term of positive bistable type which exhibits interesting properties of solutions such as multiple spreading phenomena. In fact, it will be proved that large-time behaviors of solutions can be classified into three types; vanishing, small spreading and big spreading. Some sufficient conditions for these behaviors are also shown. Moreover, for two types of spreading, we will give sharp esti- mates of spreading speed of each free boundary and asymptotic profiles of each solution.


MAHO ENDO, YUKI KANEKO AND YOSHIO YAMADA
When f satisfies (PB), we say that f is a function of positive bistable type. Initial function u 0 satisfies u 0 ∈ C 2 ([0, h 0 ]), u 0 (0) = u 0 (h 0 ) = 0 and u 0 (x) > 0 for 0 < x < h 0 . (1.1) A free boundary problem like (FBP) was first proposed by Du and Lin [5] to describe the invasion of a new species by putting homogeneous Neumann condition at x = 0 in place of Dirichlet condition. We denote such a free boundary problem by (FBP-N). Function u(t, x) stands for the population density of the species over onedimensional habitat (0, h(t)). The free boundary x = h(t) represents the expanding front of the habitat and its dynamics is determined by the Stefan condition of the form h ′ (t) = −µu x (t, h(t)). For the ecological meaning of this condition, see [2]. Du and Lin studied (FBP-N) with logistic nonlinearity f (u) = u(a−bu), a, b > 0, and established various interesting results such as spreading-vanishing dichotomy and asymptotic behaviors of solutions as t → ∞ as well as the existence and uniqueness of global solutions. In particular, it was shown that any solution (u, h) of (FBP-N) satisfies either vanishing or spreading: vanishing means the case where lim t→∞ h(t) ≤ (π/2) √ d/a and lim t→∞ u(t, ·) C([0,h(t)]) = 0, while spreading means the case where lim t→∞ h(t) = ∞ and lim t→∞ u(t, x) = a/b locally uniformly for x ∈ [0, ∞). Since the appearance of their work, a lot of people have investigated (FBP), (FBP-N) and related free boundary problems (see, e.g. [2]- [7], [9]- [20], [22], [25]- [27] and references therein). Among them we should refer to the work of Du and Lou [6], who have discussed a similar problem to (FBP-N) (or (FBP)) by putting free boundary conditions at both ends of the interval. As one of the most important results, it was shown that the analysis of large-time behaviors of spreading solutions is closely related to the following semi-wave problem where u * is a positive equilibrium point of f such that f ′ (u * ) < 0. When f is monostable, bistable or combustion type of nonlinearity satisfying f (0) = f (1) = 0 and f (u) < 0 for u > 1, it was proved in [6] that (SWP) with u * = 1 admits a unique solution (c, q) = (c * , q * ). Their results suggest that (c * , q * ) is available to study asymptotic behavior of any spreading solution. For its sharper asymptotic estimates, see the paper of Du-Matsuzawa-Zhou [9].
Assume that f is given by which is a combination of a logistic term and a predation term called Holling Type III. This is one of the important reaction terms in population biology and discussed in detail by Ludwig, Aronson and Weinberger [21] as the spruce budworm model. It is known that, if a and b in (1.2) satisfy a certain condition, then such f possesses property (PB). In this case, we can interpret u * 1 as a low endemic state and u * 3 as an outbreak state, while u * 2 is a threshold population density. When f satisfies (PB), we recall the work of Kawai and Yamada [16] for (FBP-N). They have succeeded in the classification of solutions of (FBP-N) into four types of asymptotic behaviors: vanishing, small spreading, big spreading and transition. In particular, (FBP-N) with positive bistable nonlinearity f exhibits two types of spreading phenomena; one is the small spreading of solution (u, h) with lim t→∞ u(t, ·) = u * 1 locally uniformly in [0, ∞) and the other is the big spreading of solution (u, h) with lim t→∞ u(t, ·) = u * 3 locally uniformly in [0, ∞). Moreover, it was also proved in [16] that under certain circumstances (SWP) does not have a solution, which is a big difference from previous results for other types of nonlinearity. In this sense, positive bistable f provides us with interesting and significant properties for (FBP-N). Recently, it was proved by Kaneko-Matsuzawa-Yamada [15] that, if (SWP) has no solutions, the corresponding spreading solution approaches a propagating terrace.
Our interest is to investigate the following issues for positive bistable function f : • What kind of asymptotic behaviors of solutions of (FBP) can be found?
• Are there any differences in asymptotic behaviors between (FBP) and (FBP-N)? • Is it possible to get precise estimates of when h(t) → ∞ as t → ∞?
As the first step, we will show that any solution (u, h) of (FBP) satisfies one of the following properties: Here v 1 and v 3 are bounded solutions of the following problem respectively. Note that (SP) has no bounded solutions other than v 1 and v 3 (see Proposition 3.1). In order to get better understanding on the above asymptotic behaviors, we will introduce parameter σ > 0. Let any (u 0 , h 0 ) satisfying (1.1) be fixed and consider (FBP) with (u 0 , h 0 ) replaced by (σu 0 , h 0 ). We denote such a free boundary problem by (FBP) σ . Let (u(t, x; σ), h(t; σ)) be the solution of (FBP) σ . Then it is possible to show the existence of two threshold numbers σ * 1 and σ * 2 (σ * 1 < σ * 2 ) such that the vanishing of (u(t, ·; σ), h(t; σ)) occurs for 0 ≤ σ ≤ σ * 1 , the small spreading of (u(t, ·; σ), h(t; σ)) occurs for σ * 1 < σ ≤ σ * 2 and the big spreading of (u(t, ·; σ), h(t; σ)) occurs for σ * 2 < σ. As the second step, we will derive asymptotic estimates for two types of spreading solutions. Let (u, h) be any big spreading solution of (FBP) and let (SWP) with u * = u * 3 admit a unique solution (c B , q B ). (For the existence and nonexistence of such a solution, see [16]). Then we will prove that (u, h) satisfies for any c ∈ (0, c B ). In this sense, (c B , q B ) gives a good approximation of (u, h) near the free boundary x = h(t) for large t. Moreover, we can also show that for any For any small spreading solution (u, h), it will be seen that analogous estimates as Here we should remark that there exists a small spreading solution which does not satisfy this condition. For example, when we take (u(t, x; σ * 2 ), h(t; σ * 2 )) which is a borderline solution between the small spreading and the big spreading for (FBP) σ , this solution will be proved to satisfy lim t→∞ u(t, This is a new "borderline" behavior which can not be observed in the study of (FBP-N). We have not obtained satisfactory asymptotic estimates for such small spreading solution.
This paper is organized as follows. In Section 2 we will prepare some basic results such as the existence theorem of global solutions, comparison theorem, vanishing theorem and spreading theorem. In Section 3 we study (SP) and related stationary problem by the method of the phase plane analysis. In Section 4 we will investigate large-time behaviors of solutions such as the classification of asymptotic behaviors, sufficient conditions for each behavior and the existence of threshold numbers for (FBP) σ by using parameter σ ≥ 0. Finally, in Section 5 we will derive precise asymptotic estimates for the spreading speed of the free boundary and sharp estimates for asymptotic profiles of spreading solutions with use of a semi-wave solution of (SWP), which corresponds to the spreading solution.
2. Basic properties. We first state the global existence result for (FBP).
and there exist positive constants We define spreading and vanishing of solutions under general situations.
We next give a comparison theorem for (FBP).

Theorem 2.3 (Comparison theorem). Suppose thath
Proof. The proof of this theorem is the same as that of [5,Lemma 3.5]. The following result is very useful for the analysis of asymptotic behaviors of the solution of (FBP) (see [12,Theorem 2.11]).
We finally give a sufficient condition of the vanishing (see [12, Theorem 2.10]).
3. Analysis of stationary problem. To apply the results in Section 2, we study (SP) and (SP-ℓ) with nonlinearity f satisfying (PB) by making use of the phase plane analysis (see for instance [24] and Figure 1). We first give the existence of bounded nonnegative solutions of (SP) without proof.

Proposition 3.1 (Existence of bounded solutions of (SP)). Under assumption
In order to find a solution of (SP-ℓ) we consider the following initial value problem Let v = v(x; P ) be a solution of (3.1) and define ℓ = ℓ(P ) by We also define where v P = inf{v > 0 : F (v) = dP 2 /2}. Note that if one can find P * satisfying ℓ(P * ) = ℓ, then v(x; P * ) becomes a solution of (SP-ℓ). The following result gives an elementary property of ℓ(P ).
For the proof of this lemma, see [21]. Lemma 3.2 ensures the existence of a minimum of ℓ(P ) in (ω 1 , ω 3 ), namely We are thus led to the following result on the structure of solutions of (SP-ℓ) by virtue of Lemma 3.2.

Asymptotic behaviors of solutions.
In this section, we will study asymptotic behaviors of solutions of (FBP) as t → ∞.

Classification of asymptotic behaviors.
Our first main result is the classification of solutions of (FBP) in terms of their asymptotic behaviors. To prove Theorem 4.1, we will prepare a series of lemmas.
where v 1 and v 3 are given in Proposition 3.1.
where ϕ 1 is the solution to (SP-ℓ). Applying Theorem 2.4 to the solution of (FBP) with initial data where M > 0 is a constant such that M > max{ u 0 C([0,h0]) , u * 3 }. Then it follows from the standard comparison principle that (4.1) Moreover, since w 0 satisfies d(w 0 ) xx +f (w 0 ) < 0 for x ≥ 0, we see from the monotone method (see [23]) that w t (t, x) ≤ 0 for t > 0 and x > 0; that is, w(t, x) is nonincreasing with respect to t > 0 for each x > 0. Therefore, there exists a nonnegative functionv(x) such that Note w(t, x) ≥ 0 for t ≥ 0 and x ≥ 0 by the maximum principle. It can be proved thatv is a solution of (SP) (see, e.g., [12,Theorem 2.11]). Moreover since This equality together with (4.1) implies that This completes the proof.
The following result can be easily proved by virtue of Lemmas 4.2 and 4.3.

Corollary 4.4. Let (u, h) be the solution of (FBP) with initial data
where v 1 and v 3 are given in Proposition 3.1.
In order to prove Theorem 4.1, we will make use of the zero number arguments developed by Angenent [1]. Denote by Z I (w) the number of zero points of a continuous function w in an interval I ⊂ R. We should recall the following results which are extensions of Angenent that w(t, x) is a continuous function defined for t ∈ (t 1 , t 2 ) and x ∈ I(t) and that it satisfies in the classical sense, where c is a bounded function of t ∈ [t 1 , t 2 ] and x ∈ I(t). If w(t, −ξ(t)) = 0 and w(t, ξ(t)) = 0 for t ∈ (t 1 , t 2 ), then the following properties hold true: (i) Z I(t) (w(t, ·)) < ∞ for any t ∈ (t 1 , t 2 ) and it is non-increasing in t; (ii) If w(s, x) has a degenerate zero x 0 ∈ (−ξ(s), ξ(s)) at some s ∈ (t 1 , t 2 ) , then Z I(s1) (w(s 1 , ·)) > Z I(s2) (w(s 2 , ·)) for any s 1 ∈ (t 1 , s) and s 2 ∈ (s, t 2 ). Lemma 4.6. Let I ⊂ R be an open interval and let {w n (t, x)} ∞ n=1 be a sequence of functions which converges to w(t, x) in C 1 ((t 1 , t 2 ) × I). Assume that for every t ∈ (t 1 , t 2 ) and n ∈ N, the function x → w n (t, x) has only simple zeros in I and that w(t, x) satisfies an equation of the form (4.4) in (t 1 , t 2 ) × I. Then for every t ∈ (t 1 , t 2 ), either w(t, x) ≡ 0 in I, or w(t, x) has only simple zeros in I.
We will prove the following convergence property of the solutions of (FBP) by using these zero number arguments and basic properties of the structure of ω-limit set. where v * is a bounded positive solution of (SP).

Proof.
Let ω(u) be an ω-limit set of u(t, ·) in the topology of L ∞ loc ([0, ∞)), that is, for every w ∈ ω(u) there exists a sequence 0 < t 1 < t 2 < · · · < t n < t n+1 < · · · → ∞ such that (4.5) By local parabolic regularity estimates, we can replace the topology of L ∞ loc ([0, ∞)) by that of C 2 loc ([0, ∞)). Since ω(u) is a compact, connected and invariant set, for any w ∈ ω(u) there exists an entire orbit {W (t, x)} t∈R with W (0, x) = w(x). This fact implies that for every w ∈ ω(u) there exists W (t, x) satisfying This convergence can be also replaced by the topology of C 1,2 loc (R×[0, ∞)) on account of parabolic regularity. Let We will investigate intersection points between W (t, x) and v(x). Let v 1 and v 3 be functions given in Proposition 3.1. Lemma 4.2 gives (4.7) Since . Therefore, by the phase plane analysis (see Figure 1), it is seen that either (i) v(x) > 0 for x > 0, or (ii) there exists a positive number R such that v(R) = 0 and v(x) > 0 for x ∈ (0, R).

MAHO ENDO, YUKI KANEKO AND YOSHIO YAMADA
First, we consider the case (i). Letû(t, x) be an odd extension of u(t, x) for t ∈ (0, ∞) and Thenû is a classical solution of wheref is defined byf Similarly, we also denote byv an odd extension of v over (−∞, ∞). Note thatv is also a classical solution of .
. Therefore, for any t ∈ R,Û (t + t n , ·) has only simple zeros provided that n ∈ N is sufficiently large. Moreover, by (4.6), satisfies a parabolic equation of the form (4.4) for any (t, x) ∈ R 2 , it follows from Lemma 4.6 that, for every t ∈ R, eitherŴ (t, has only simple zeros in R. However we see that the latter case never occurs becauseŴ (t, 0) −v(0) =Ŵ x (t, 0) −v x (0) = 0 at t = 0. Therefore, W (t, x) ≡v(x) in R. Since the right hand side is not dependent on t, Thus any w ∈ ω(u) is equal to v which is a bounded positive solution of (SP). We will next exclude the case (ii). Assume that (ii) holds true. Since lim t→∞ h(t) = ∞, there exists a positive number T such that h(t) ≥ R for t ≥ T . By virtue of U (t, R) = 0 for t > T , we can repeat the previous argument with t ∈ (0, ∞) and replaced by t ∈ (T, ∞) and x ∈ [−R, R], respectively. Then it is possible to show that, for every t ∈ R, eitherŴ (t, x) −v(x) ≡ 0 for all x ∈ (−R, R), orŴ (t, x) −v(x) has only simple zeros in (−R, R). In the former case, we see that w(x) ≡ v(x) for x ∈ [0, R), which contradicts (4.7). On the other hand the latter case contradicts the fact thatŴ (t, x) −v(x) has a degenerate zero x = 0 at t = 0. In this way we conclude that the case (ii) never occurs. The proof is complete.
. By the phase plane analysis, v * must coincide with v 1 or v 3 . The proof is complete.

Sufficient conditions for asymptotic behavior.
In this subsection we will give some sufficient conditions for (I)-(III) of Theorem 4.1. We first introduce a sufficient condition for the vanishing which can be proved in the same way as [11,Theorem 2.2]. then the solution (u, h) of (FBP) satisfies the vanishing.
The following result gives a sufficient condition for the spreading when h 0 < π √ d/f ′ (0): ( then the solution of (FBP) satisfies the spreading.
We next discuss the case u 0 C([0,h0]) > u * 1 . Let (u, h) be a solution of the following free boundary problem : (4.10) Then (u, h) is a lower solution to (FBP). Applying Theorem 2.3 yields h(t) ≤ h(t) and u(t, x) ≤ u(t, x) for t ≥ 0 and 0 ≤ x ≤ h(t). In view of u 0 C([0,h0]) = u * 1 , the preceding result implies the spreading of (u, h) and, therefore, (u, h), provided that ∫ h0 we complete the proof.
We will show a sufficient condition for the small spreading of solutions. Let u = u(t, x) be the solution of the following problem where M > 0 is a constant satisfying max{u * 1 , u 0 C([0,h0]) } < M < u * 2 . Since u 0 satisfies d(u 0 ) xx + f (u 0 ) < 0, it is possible to prove the monotone decreasing convergence of u(t, x) as t → ∞ to a solution of (SP) in the same way as the proof of Lemma 4.2. Moreover, since u 0 ∈ (u * 1 , u * 2 ) for x ≥ 0, we see that lim t→∞ u(t, x) = v 1 (x) locally uniformly for x ≥ 0. Since u is an upper solution to (FBP), it follows from the comparison principle that Combining this fact and (4.11), we conclude which implies the small spreading of u(t, ·) as t → ∞.
Finally, we will give a sufficient condition for the big spreading.
Using Theorem 4.8, Lemma 4.12 and (4.12), one can see that σ * 1 given in (4.13) is the threshold number which separates the vanishing and the spreading: Theorem 4.13. Let (u(t, x; σ), h(t; σ)) be the solution of (FBP) σ with initial data (σϕ, h 0 ) for σ > 0. Then (u(t, x; σ), h(t; σ)) satisfies the vanishing for every σ ≤ σ * 1 and the spreading for every σ > σ * (4.15). For the proof of this theorem, see [6,Theorem 5.2] or [16,Theorem 3.7]. Next we will show that σ * 2 defined as (4.14) is the threshold number which separates the small spreading and the big spreading: Remark 4. If we consider (FBP-N), then the transition occurs at σ = σ * 2 when σ * 1 and σ * 2 are defined by (4.13) and (4.14) (see, [16,Theorem 3.8]). This fact means that the transition is a borderline behavior between the small spreading and big spreading in the case of zero Neumann boundary condition at x = 0.

Remark 5.
The notion of small spreading in Theorem 4.1 is defined by lim t→∞ h(t) = ∞ and lim t→∞ u(t, x) = v 1 (x) in [0, R] for any R > 0. It may be classified into two sub-cases; (i) lim inf t→∞ u(t, ·) C([0,h(t)]) < u * 2 , (ii) lim inf t→∞ u(t, ·) C([0,h(t)]) ≥ u * 2 . In particular, case (ii) implies that u(t, x) has a peak at x = x * (t) satisfying u(t, x * (t)) ≥ u * 2 for sufficiently large t. This is an interesting phenomenon, but we have no further information on this kind of small spreading. The phenomenon of case (ii) may correspond to the "transition", which is a borderline solution between small spreading and big spreading for solutions of (FBP-N).