On a free boundary problem for a nonlocal reaction-diffusion model

This paper is concerned with the spreading or vanishing dichotomy of a species which is characterized by a reaction-diffusion Volterra model with nonlocal spatial convolution and double free boundaries. Compared with classical reaction-diffusion equations, the main difficulty here is the lack of a comparison principle in nonlocal reaction-diffusion equations. By establishing some suitable comparison principles over some different parabolic regions, we get the sufficient conditions that ensure the species spreading or vanishing, as well as the estimates of the spreading speed if species spreading happens. Particularly, we establish the global attractivity of the unique positive equilibrium by a method of successive improvement of lower and upper solutions.


1.
Introduction. It is quite well understood now that the mathematical modelling of the dynamical processes in physical, biological and applied sciences requires differential equations, equations with delay, integro-differential equations, and other kinds of functional equations. For example, Volterra [28] introduced equations of the form du dt = au − bu 2 − ru t t0 φ(t − s)u(s)ds, t > 0 (1) to describe the evolution of a single population, where t 0 = 0 or −∞, u is the population size, a − bu denotes the intrinsic rate and t t0 φ(t − s)u(s)ds is a temporal convolution representing the memory rate and containing the effect of the past history on the actual population development. The initial function for (1) satisfies u(0) = u 0 if t 0 = 0 ( resp. u(t) = q(t) for t < 0 if t 0 = −∞).
In (1), the logistic term which expresses the crowding effect, is separated into two parts: a non-delay term bu and a hereditary term Redlinger [23] extended Miller's results to the situation that a diffusion term is added to (1) and supposed that the population has no interaction with the exterior, that is subject to the homogeneous Neumann boundary condition, where Ω ⊂ R N is a bounded domain with C 2 boundary and u 0 ∈ C 1 (Ω). Redlinger obtained the convergence result (lim t→+∞ u(t, x) = a b + r ∞ 0 φ(s)ds −1 uniformly for x ∈Ω) in the case that b > r by a recursively defined sequence of lower and upper solutions, and also emphasized that those conclusions still remained true if ∆u is replaced by the linear, uniformly elliptic operator with coefficient functions a ij , a i that are uniformly Hölder continuous in (0, ∞) ×Ω and satisfy a ij = a ji , a i (0, ·) ∈ C 1 (Ω) for i, j = 1, 2, · · · , n. The main theorem stated in [23] has also been proved by Schiaffino [25] if φ is nonnegative, decreasing and min x∈Ω u 0 (x) > 0, and by Yamada [33] if φ ∈ C 1 (0, ∞) is nonnegative and satisfies tφ ∈ L 1 (0, ∞). If u 0 is close to a b + r ∞ 0 φ(s)ds −1 and n < 3, Tesei [27] established the same convergence results as in [23]. Moreover, please see Schiaffino and Tesei [26] for Dirichlet boundary condition involved in problem (2). We also refer to Deng et al. [6] for a more general reaction-diffusion model on R N with a nonlocal spatial convolution (φ * u = R N φ(x − y)u(t, y)dy). Integral equations with diffusion are studied by many authors, see Cordoneanu [5] for complete references.
The aforementioned models and results showed that either in a bounded domain, or through the whole R N , the associated solutions are always positive, and eventually, converge to the corresponding positive equilibrium for all x once t is positive, regardless of the initial data and supporting area. In population biology, this means that the population may spread very quickly to the whole environment even though there is a small amount of individuals in the very early stage of its introduction, which does not match the fact that any population always spreads gradually.
Recently, free boundary problems have been studied intensively in many fields to describe a precise gradual spreading process, together with a changing underlying area. In particular, the well-known Stefan condition has been used to describe the spreading front in many applied problems. It was used to describe the melting of ice in contact with water [24], the wound healing [3] and the tumor growth [4]. In order to get a more precise prediction of the location of the spreading front of an invading species, Du et al. [9] firstly studied the spreading-vanishing dichotomy of some invasion species which is described by a diffusive logistic model in the homogenous environment of a one dimensional space. Since then, more results for reaction-diffusion equations with more general free boundaries have been obtained where d, a, b, r are positive constants, x = g(t) and x = h(t) are moving boundaries that will be determined together with u; the parameter µ > 0 can be understood as the expanding ability, i.e., the larger µ is, then the easier the population can move to a new area; bu measures the competition for local space while (φ * u)(t, x) = R φ(x − y)u(t, y)dy denotes the competition for resources in the neighborhood of an individual. The initial function u 0 satisfies and the convolution kernel function φ satisfies Problem (3)-(4) indicates that the individuals occupy the initial region [−h 0 , h 0 ] at the beginning and spread into the environment from two ends of the initial region. The one-phase Stefan conditions g (t) = −µu x (t, g(t)) and h (t) = −µu x (t, h(t)) show that the speed of the spreading frontiers are proportional to the population gradient at the front.
A kernel of the form φ σ (x) = 1 σ φ( x σ ) can be used to quantify the nonlocal interaction and if σ → 0, then φ σ * u → u, and problem (3) reduces to the classical logistic model described in Du et al. [9]. We are mainly interested in comparing the behavior of our nonlocal model (3) with the corresponding local one (i.e. φ(x) = δ(x)). Caused by the nonlocal nature of the nonlinear term, the lack of an order-preserving property is the main difficulty in our analysis. Due to this, many key techniques including the general comparison principle for handling similar problems collapse here, while many of our results such as the long time behaviors of solutions and the estimates of the spreading speed rely on comparison principles. We introduce some comparison principles over suitable parabolic regions to overcome those difficulties, and get the expected attractivity of the positive steady state by the comparison principle (Lemma 3.1) and the method of successive improvement of lower and upper solutions.
The organization of this paper is as follows. In Section 2, we prove the general existence and uniqueness result, which implies in particular that problem (3)-(4) has a unique positive solution that is defined for all t > 0, the method is inspired by [9,7,11]. In Section 3, we establish the spreading-vanishing dichotomy, that is, the asymptotic behavior of solutions (u, g, h). Section 4 is concerned with the estimate of the spreading speed.
2. Global positive solutions of (3)-(4). We firstly prove the following local existence and uniqueness result by the contraction mapping theorem and then use a priori estimates to show that the solution is defined for all t > 0. The proof can be done by modifying the arguments of Du et al. [9]. For readers' convenience, we give a detailed proof here.
Proof. As in [34], we first straighten the free boundaries by the transformation which changes x = h(t) and x = g(t) to fixed lines s = h 0 and s = −h 0 respectively, and there holds If we set A = A(g(t), h(t), s), B = B(g(t), h(t)) and u(t, x) = u(t, h(t)−g(t) = v(t, s), then and the free boundary problem (3) becomes the following Clearly, H T and G T are bounded and closed convex sets of C 1 ([0, T ]). For any h(t) ∈ H T and g(t) ∈ G T , Then the transformation from (t, x) to (t, s) is well defined. Define Furthermore, for any h 1 , h 2 ∈ H T and g 1 , wherec 1 ,c 2 andc 3 are some positive constants. Applying standard L p theory and then the Sobolev embedding theorem [19], we can find that for any v ∈ V T , the following initial boundary value problem Let (h,g) be the unique solution of Thenh ,g ∈ C In what follows, we define a map F : It is clear that (v, g, h) ∈ D is a fixed point of F if and only if it solves (6). According to (10) and (11), we see that Now, we are in the position to prove that F is a contraction mapping on D, and hence it admits a unique fixed point in D. (10) and (12) that SettingṼ =ṽ 1 −ṽ 2 , we then find that for 0 < t < T and −h 0 < s < h 0 ,Ṽ (t, s) satisfies Using L p estimates for parabolic equations and Sobolev embedding theorem again, we have where C 3 is a constant which depends on C 1 , C 2 and C * , where C * denotes the uniform bound of the coefficients A 1 , A 2 , B 1 and B 2 . Moreover, C 3 depends on the elliptic constant d( 8 9 ) 2 . From (11), we find that Combining (7), (13) and (14), we obtain that provided that It follows that F is a contraction mapping on D for our choice of T . Now, F admits a unique fixed point (v, g, h) in D. Moreover, by the Schauder's estimate, we get additional regularity for (u, g, h) as a solution of (6), namely, In other words, (v, g, h) is the unique classical solution of problem (6). This completes the proof.
To show that the local solution obtained in Theorem 2.1 can be extended to all t > 0, we need the following estimate.
in which M 1 and M 2 are positive constants independent of T 0 .
in which M > 0 is a constant that will be chosen later. For any (t, x) ∈ Ω h M , we havē In the region Ω g M := (t, x) ∈ R 2 : 0 < t < T 0 , g(t) < x < g(t) + M −1 , we compareū(t, x) and u(t, x) as above and get −g (t) ≤ M 2 . This completes the proof. Proof. It follows from the uniqueness of the solutions of (3) that there is some T max > 0 such that [0, T max ) is the maximal time interval where the solution exists. It remains to show T max = ∞. We get the conclusion by deriving a contradiction. Suppose that T max < ∞, then as in Theorem 2.2, there exist positive constants M 1 and M 2 independent of T max such that According to the standard L p estimates, the Sobolev embedding theorem and the Hölder estimates for parabolic equations, for some fixed γ * ∈ (0, T max ), we can find a positive constant M 3 depending on γ * , M 1 and M 2 such that u(t, ·) such that the solution of problem (3) with initial time T max − τ 2 can be extended uniquely to T max − τ 2 + τ , which contradicts to the definition of T max . Thus, our result follows. Theorem 2.2 establishes the monotonicity of the free boundaries g(t) and h(t).
The following result follows as a similar result in [11,34], which implies that g(t) and h(t) are both finite or infinite at the same time. We omit the proof for brevity. Theorem 2.4. Suppose that (K) holds and let (u, g, h) be the solution of problem (3), then −2h 0 < g(t) + h(t) < 2h 0 for all t > 0.
As a concluding remark, we discuss the relations and differences among our free boundary problem (3), the corresponding Cauchy problem and the initial-boundary value problem with fixed boundary.
Recalling the initial-boundary value problem (2) stated in section 1, Redlinger [23] (see also [25]) has showed that the positive uniform equilibrium a b+r is globally asymptotically stable in the case that φ * u is a temporal convolution and b > r.
Moreover, it is concluded in Deng et al. [6] that the positive solution u(t, x) of the following Cauchy problem converges to a b+r uniformly in x provided that u 0 (x) > 0 and b > r. Indeed, once b > r, any nontrivial initial population u 0 (x) will spread successfully regardless of its initial size, supporting area and expanding ability. For our free boundary problem (3), we show that Our results show that for the initial supporting area h 0 , there exists a sharp criteria h * , for h 0 ≥ h * , the new species successful spreading in the long run. While for the case that h 0 < h * , whether spreading happens or not depending on the expanding ability µ even though the initial function u 0 is nontrivial.
Moreover, if we chose φ(x) = δ(x), then problem (3) becomes the classical logistic model with double free boundary, which has been studied by Du et al. [9], and our results correspond to those in [9].
3. Long time behaviors of (u, g, h). In this section, we mainly analyze the asymptotic behavior of positive solution (u, g, h) of problem (3) obtained in section 2, and our conclusions here are mainly based on the comparison principles that are set over some suitable parabolic regions. In what follows, we discuss the comparison principle for problem (3), please refer to Deng et al. [6] for similar results while in R N . The proof of Lemma 3.1 is motivated by Lemma 2.6 in [10].
Letting T * ≤ t * and assume that there exists (t 1 , x 1 ) ∈ D T * such that U (t 1 , x 1 ) = τ < 0, then the negative minimum of w 1 denoted by w 1,min satisfiesw 1,min = min (t,x)∈D T * U 1+γt+x 2 ≤ τ 1+γt1+x 2 1 < 0. In addition, if we assume that w 1 attains its minimum at some point (t,x) ∈ D T * , then (w 1 ) t | (t,x) ≤ 0, ∆w 1 | (t,x) ≥ 0 and (w 1 ) x | (t,x) = 0. Therefore, by the first inequality of (19), there holds Due to the inequality satisfied by u in (16), it follows from the maximum principle Then, if we choose T * = min t * , 1 rM * and γ sufficiently large such that a contradiction occurs. If there exists (t 2 , x 2 ) ∈ D T * such that V (t 2 , x 2 ) = τ < 0, we obtain a similar contraction. For the case that g(t * ) = g(t * ) (resp. g(t * ) = g(t * )), we can deduce a similar contradiction by the same arguments as above. Therefore, U ≥ 0 and V ≥ 0 in D T * , which further implies that u ≥ u and v ≥ v in D T * .
The following comparison principle is an immediately result of Lemma 3.1.
Then the unique solution (u, g, h) of problem (3) satisfies Below are two necessary lemmas that will be used later.
with g(u) is Lipschitz and decreasing in u, g(0) > 0, g(u) < 0 for u > K > 0 (i) , where u * (x) depending on l is the unique positive stationary solution of (23).
Choosing > 0 small and T 3 > T 0 large, we may get u(t, x) ≥ w for all t ≥ T 3 and x ∈ [−l, l]. Thus, we can obtain that u(t, 0) ≥ θ := w(0) for all t ≥ T 3 . In addition, we have u(t, h(t)) = u(t, h(t)) = 0. Now we choose 0 < δ 1 and T > max It follows from Lemma 3.1 that u(t, x) ≤ u(t, x) over Ω T . In addition, u(t, h(t)) = u(t, h(t)) yields that Then, by taking t → +∞, we arrive at − 1 µ h (∞) = 0 ≤ δ l h∞ ω 1 (l) < 0, a contradiction. Hence, we conclude that It follows from Theorems 3.6 and 3.5 that if h ∞ − g ∞ < ∞, then h ∞ − g ∞ < 2h * and lim t→+∞ u(t, ·) C([g(t),h(t)]) = 0, that is vanishing happens. Following theorem establishes the spreading or persistence of the species in the case that h ∞ = −g ∞ = ∞. Proof. It follows from Theorem 2.2 that there exists a large T 0 such that u(t, x) ≤ a b +¯ 1 for all t ≥ T 0 and x ∈ [g(t), h(t)], where¯ 1 > 0 is small. We denote p 1 = a b for brevity. Fix l p1 such that l p1 > max h 0 , π 2 d a−r(p1+¯ 1) , then we can find some Letting T lp 1 = max{T 0 , t lp 1 }, by the comparison principle, we get Note that we have used p 1 +¯ 1 and u lp ] as a pair of upper and lower solutions to (3). Since l p1 > π 2 d a−r(p1+¯ 1) , it follows from Lemma 3.4 that u lp 1 (t, x) → u it is easy to see that q 1 < a b+r < p 1 . Again, we can find a large T lq 1 > 0 and a small 1 > 0 such that u(t, x) ≥ q 1 − 1 for all t ≥ T lq 1 and x ∈ [−l q1 , l q1 ], where l q1 is the one such that l q1 ≥ π 2 d a−r(q1− 1 ) . In addition, the comparison principle yields that ] are a pair of lower and upper solutions to (3). And hence, by letting 1 → 0 + immediately deduce that lim sup t→∞ u(t, x) ≤ p 1 − r b q 1 . By putting p 2 = p 1 − r b q 1 , we then find that a b+r < p 2 < p 1 . Continuing above procedure, we can similarly get that lim inf t→∞ u(t, x) ≥ p 1 − r b p 2 . Letting q 2 = p 1 − r b p 2 , then 0 < q 1 < q 2 < a b+r . Note that above procedure leads to a larger lower bound and a smaller upper bound for u(t, x), respectively. This approach is a similarly use of the method of successive improvement of lower and upper solutions. To arrive at our conclusion, define sequences {q n } ∞ n=2 and {p n } ∞ n=2 by following q n = p 1 − r b p n and p n = p 1 − r b q n−1 .
It follows from the comparison principle that q n − n ≤ u(t, x) ≤ u lq n in [T lq n , ∞) × [−l qn , l qn ], where u lq n > 0 satisfying following It is obvious that 0 < q 1 < q 2 < · · · < q n < a b+r < p n < · · · < p 2 < p 1 , then there exist q * > 0 and p * > 0 such that q n → q * and p n → p * as n → ∞, respectively. If we can prove that p * = q * = a b+r , then by letting n → ∞ in q n ≤ lim inf t→∞ u(t, x) ≤ lim sup t→∞ u(t, x) ≤ p n directly leads to lim t→∞ u(t, x) = a b+r . In deed, q * and p * satisfying following equations which indicates that p * = q * = a b+r . Then lim t→∞ u(t, x) = a b+r follows. Now we can deduce the following spreading-vanishing dichotomy.
Theorem 3.8. Let (u, g, h) be the solution of problem (3). Then the following alternative holds: Either (i): spreading: h ∞ = −g ∞ = ∞ and lim t→+∞ u(t, x) = a b+r uniformly in any bounded subset of R; or (ii): vanishing: h ∞ − g ∞ < 2h * and lim t→+∞ u(t, x) C([g(t),h(t)]) = 0. Theorem 3.6 and the strictly monotonicity of h (t) and For h 0 < h * , we give two lemmas to illustrate the dependence of the spreading or vanishing of the species on the expanding ability µ. Lemma 3.9. Assume that h 0 < h * , then there exists µ * > 0 depending on u 0 (x) such that spreading happens, that is Proof. We argue indirectly. To obtain the formula of µ * , supposing on the contrary that vanishing happens for all µ > 0 if h 0 < h * . Then the positive solution (u(t, x), h(t), g(t)) of (3) satisfies h ∞ − g ∞ < ∞ for all µ > 0.
The idea below mainly based on the proof of Lemma 3.7 in Du et al. [9]. For all t ≥t and x ∈ [g(t), h(t)], direct calculation yields that d dt .
Integrating fromt to t deduces that Then by letting t → ∞ immediately leads to − h(t) Since h ∞ − g ∞ ≤ 2h * and h(t) − g(t) > 2h 0 , then the above inequality implies that Note that u(t, x) here depends on µ.
It follows from the comparison principle that u(t, x) ≥ w(t, x) in [t, ∞) × [−h 0 , h 0 ]. Particularly, there is u(t, x) ≥ w(t, x) in [−h 0 , h 0 ], where w(t, x) is the unique positive solution to the following initial-boundary value problem It is obvious that w(t, x) is independent of µ, as well as w(t, x), which in turn indicates that As a result, inversely, if µ > µ * , then h ∞ = −g ∞ = ∞. Then the conclusion follows.