A free boundary problem with nonlocal diffusion and unbounded initial range

We consider a free boundary problem with nonlocal diffusion and unbounded initial range, which can be used to model the propagation phenomenon of an invasion species whose habitat is the interval (-∞,h(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\infty ,h(t))$$\end{document} with h(t) representing the spreading front. Since the spatial scale is unbounded, a different method from the existing works about nonlocal diffusion problem with free boundary is employed to obtain the well-posedness. Then we prove that the species always spreads successfully, which is very different from the free boundary problem with bounded range. We also show that there is a finite spreading speed if and only if a threshold condition is satisfied by the kernel function. Moreover, the rate of accelerated spreading and accurate estimates on longtime behaviors of solution are derived.

(1. 1) In this model, the invasive species, whose density is denoted by v, initially occupies the spatial domain (s 0 ,s 0 ) with initial density v 0 (x). Since the species will move instinctively for survival as time goes on, its habitat will evolve naturally and further is represented by (s 1 (t), s 2 (t)). The spreading frontiers s i (t) are assumed to satisfy the Stefan boundary condition, and in other words they are expanding at a rate proportional to the population of the invasive species across them. Du and Lin [1] found that the species either spreads successfully in the sense that  where the species will always establish itself in the whole space, please see [2]. When giving a nontrivial and supported compact initial function to (1.2), one can see from [3,4] that 2 √ Da is the spreading speed of the level set of solution to (1.2).
Similarly to the Cauchy problem (1.2), if the species spreads successfully in (1.1), Du and Lin [1] where k 0 is uniquely given by the semi-wave problem Dq − kq + q(a − bq) = 0 in (0, ∞), For more results about propagation modeled by reaction-diffusion equations, which has a long history, one can refer to [5][6][7][8] and the references therein, and for the results derived from reaction-diffusion equations with free boundary, one can refer to the expository article [9].
As is well-known to us, long-distance dispersal usually happens, such as wind-dispersed seeds, which is inappropriate to be modeled by the Laplacian operator, also called random diffusion or local diffusion operator, please see, e.g., [10,11]. Therefore, a class of new operators which can take long-distance dispersal into account are derived, please see [12] for the details of the derivation. Such kind of operator is usually referred to nonlocal diffusion operator, and one of them often takes the following form where d is the dispersal rate, and J(x) is the kernel function which is usually nonnegative, bounded and unit integrable. The dynamics of (1.2) with Dv xx replaced by (1.3) have already been known, please see [13][14][15][16][17] and the references therein. Recently, Cao et al. [18] proposed the nonlocal diffusion version of (1.1). More precisely, they considered the following problem where nonlinear term f is of the Fisher-KPP type, i.e., for some u * > 0, and f (u)/u is strictly decreasing in u > 0.
Similarly to (1.1), Cao et al. proved that the dynamics of (1.4) was also govern by a spreadingvanishing dichotomy in [18]. When spreading occurs, the spreading speed was later obtained by Du et al. [19]. Particularly, there are two differences worth emphasizing between the dynamics of (1.1) and (1.4). Firstly, from [18], if the dispersal rate d is no more than the growth rate f (0), then spreading always happens. While in [1], no matter how small the dispersal rate d is , vanishing always can occur if the initial habitat is suitably small. Secondly, when spreading happens, the spreading speed of (1.1) is always finite, while that of (1.4) can be infinite, which is usually called the accelerated spreading, if J does not satisfy a threshold condition.
Motivated by the above works, we here consider the case where the species occupying the initial habitat (−∞, h 0 ) propagates with the strategy of nonlocal diffusion as in [18], and h(t) stands for its spreading front. Therefore, same as the derivation of the problem (1.4), the concerned model can be written as where f (u) satisfies the above (F), and u 0 (x) meets The kernel function J satisfies |x|u 0 (x)dx < ∞ and J are compactly supported, Cortázar et al. [30] proved that the solution (u, h) of (1.5) satisfies However, our main result shows that under the assumptions on f , J and u 0 (x), the solution (u, h) of (1.5) has very different longtime behaviors. Throughout this paper, we always suppose that conditions (F), (H) as well as (J) hold. For convenience, we first give some notations. For any given h 0 ∈ R and T > 0, we define In the following, D T h represents the closure of D T h , A := max{ u 0 L ∞ ((−∞,h0)) , u * }, and we say s(t) = o(r(t)) if lim t→∞ s(t)/r(t) = 0. We use C, C 1 and C 2 to represent the generic positive constants, which may vary with the place they appear. Below are our main results.
Next we show the longtime behavior of solution (u, h) to (1.5). We start by giving a proposition that is crucial to the later discussions. Moreover, the following more accurate estimates hold.
(1) If (J1) holds, then where c 0 is uniquely determined by the semi-wave problem (1.6). Moreover, if we further suppose that J is compactly supported and f ∈ C 2 , then (5) The free boundary h(t) will not spread faster than exponential, that is,

Remark 1.2.
We can improve the conclusion (5) in Theorem 1.2 if extra assumptions are imposed on the initial value u 0 (x) and kernel J. More precisely, the assumptions we need are as follows: By the above assumptions and [31, Theorem 1.1], we can obtain the decay rate of L p norm of solution v to the later problem (3.12), for every p ∈ (1, ∞) and some C > 0. L. Li, X. Li and M. Wang ZAMP Then we have the following improvement of the conclusion (5) in Theorem 1.2: We give an example of kernel J: Obviously, this kernel J satisfies (J) and the above assumption (ii) with α = B = 1 and all q ∈ (1, ∞), but doesn't meet (J1).
This paper is arranged as follows. In Sect. 2, we show the maximum principle and comparison principle. In Sect. 3, we give the proofs of the above two theorems. Note that the spatial range of (1.5) is unbounded. Therefore, a new method, which is different from that of [18], is adopted to prove the well-posedness of (1.5). Concretely, we construct a sequence of free boundary problems defined in bounded domain, then prove that the solutions of this sequence converge to a solution of (1.5), and at last show the uniqueness by a comparison argument and the contraction mapping principle. Then a brief discussion is given in Sect. 4.

The maximum principle and comparison principle
This section is devoted to show the maximum principle and comparison principle.
A free boundary problem with nonlocal diffusion Page 7 of 23 192 Proof. For small ε > 0 and B > d Obviously, this is a contradiction to the definition of T 0 . Thus, for any sequence {ε n } with ε n > 0 and lim Since w is bounded below and continuous in [0, Due to the choice of σ, we have t n > 0, and thus, On the other hand, for the later discussion, we now show that Combining this with (2.1), we have w t (t n , y n ) ≥ c(t n , y n )w(t n , y n ) + εe Btn ≥ − c ∞ ε n + εe Btn ≥ 1 2 ε for all large n, . If there exists (t * , x * ) ∈ (0, T ] × (s 0 , s(t)) such that z(t * , x * ) = 0, then by continuity of z we can find ax ∈ (s 0 , x * ] such that z(t * ,x) = 0 and z(t * , x) > 0 for x ∈ [s 0 ,x). Clearly, z t (t * ,x) ≤ 0. Hence, by (J), we see This contradiction indicates that z > 0 in (0, T ] × (s 0 , s(t)). The proof is complete.

Proofs of Theorems 1.1 and 1.2
This section is devoted to the proofs of Theorems 1.1 and 1.2. The proof of Theorem 1.1 will be carried out by four steps.

Proof of Theorem 1.1.. Step 1: The construction of a sequence of free boundary problems defined in bounded domain. For any given
). Thus, this step is complete.
Step 2: The monotonicity of (u l , h l ) on l. For any l 1 < l 2 < h 0 , denote the corresponding solution of (3.1) by (u i , h i ) for i = 1, 2, respectively. Then (u 1 , h 1 ) satisfies It then follows from a comparison argument that h 1 (t) > h 2 (t) and u 1 (t, x) > u 2 (t, x) for t > 0 and x ∈ [l 2 , h 2 (t)]. The step 2 is thus finished.
Step 3: The existence of solution to (1.5). For any l < h 0 , let (u l , h l ) be the unique global solution of (3.1). Consider the following problem Thus, for any 0 < T < ∞, Then we can define h ∞ (t) = lim A free boundary problem with nonlocal diffusion Page 11 of 23 192 By the dominated convergence theorem, letting l → −∞ we have On the other hand, for any (t, Using the dominated convergence theorem again, we have Differentiating the above equation by t leads to Therefore, (u ∞ , h ∞ ) satisfies the first equation of (1.5) for (t, x) ∈ (0, T ] × (−∞, h 0 ]. When x ∈ (h 0 , h l (t)) for l ≤ l 1 , there exists a s ∈ (0, t) such that h l1 (s) = x < h l (s) for any l < l 1 . Thus, for any l < l 1 , we have Similarly to the above, we obtain Thus, (u ∞ , h ∞ ) satisfies the first equation of (1.5) for (t, Obviously, t x is continuous in x < h ∞ (T ). By the above analysis, for any x ∈ (h 0 , h ∞ (T )), we see u(·, x), u t (·, x) ∈ L ∞ ((t x , T )). Thus, we can define u ∞ (t x , x) = lim t→tx u ∞ (t, x). Next we show u ∞ (t x , x) = 0 L. Li, X. Li and M. Wang ZAMP for x ∈ (h 0 , h ∞ (T )), which implies u ∞ (t, h ∞ (t)) = 0 for t ∈ (0, T ). Obviously, for any t ∈ (t x , T ) Moreover, since x ∈ (h 0 , h ∞ (T )), there is a l 2 such that x ∈ (h 0 , h l (T )) for any l < l 2 . Thus, for such l we can find a t l ∈ (0, T ) such that h l (t l ) = x. Since h l is increasing in l, t l is decreasing in l. Thus, we can definet = lim (t x ,t). Therefore, there is a large L > 0 such that t l > t 1 for any l < −L. Furthermore, for such l, we have that 0 = u l (t 1 , x) → u ∞ (t 1 , x) > 0 as l → −∞. This contradiction implies that lim For some t close to T , we have By the dominated convergence theorem again, we see which, compared with (3.4), immediately yields u ∞ (t, h ∞ (t)) = 0 for t ∈ (0, T ). Thanks to (1.5) and u ∞ (t, h ∞ (t)) = 0 for t ∈ (0, T ), we have that for any x ∈ (h 0 , h ∞ (T )), which indicates that lim with u ∞ (t, x) := 0 for t ∈ [0, T ] and x = h ∞ (t). By our previous analysis, we have . Then it follows from the fundamental theory of ODEs that Therefore, (u ∞ , h ∞ ) solves (1.1). Moreover, for any Step 4: The uniqueness of solution to (1.5). Now we show the uniqueness of solution to problem (1.5). To this aim, we first give some estimates for the solution of (1.5). Let (u, h) and U be a solution of (1.5) and (3.3), respectively. By a comparison consideration, we have u(t, x) ≤ U (t, x) with t ≥ 0 and x ≤ h(t). As before, we have Furthermore, by (J) there exist small ε 0 and δ 0 such that J(x) ≥ δ 0 for |x| ≤ ε 0 . Thus, there is small T 1 depending only on initial data such that when T ≤ T 1 , one has Noticing that (u, h) solves (1.5) and letting L( which arrives at Combining the above estimates with the third equation of (1.5), we have Therefore, for T ∈ (0, T 1 ], we have Now we verify the uniqueness of solution of (1.1). Let (u 1 , h 1 ) and (u 2 , h 2 ) be the solution of (1.1) for any T > 0 and i = 1, 2. In the following, we first show that there is a small T 2 < T 1 , which relies only on initial data, such that h 1 (t) ≡ h 2 (t) for t ∈ [0, T 2 ]. Thus, we can define Clearly, T 2 ≤ T * . On the other hand, using Lemma 2.1 we have u 1 (t, x) ≡ u 2 (t, x) for t ∈ [0, T 2 ] and (−∞, h 1 (t)]. If T * = ∞, then the uniqueness is obtained. Assume that T * < ∞. By continuity and Lemma 2.1, we have h 1 (T * ) = h 2 (T * ) and u 1 (T * , x) ≡ u 2 (T * , x) for x ∈ (−∞, h 1 (T * )]. Thus, we can think of T * as the initial time, and then argue as above to derive that there is a δ > 0 such that h 1 (t) ≡ h 2 (t) for t ∈ [T * , T * + δ], which obviously contradicts the definition of T * . Then the uniqueness is obtained. L. Li, X. Li and M. Wang ZAMP Therefore, to our purpose, it remains to show that there is a small T 2 < T 1 depending only on initial data such that h 1 (t) ≡ h 2 (t) for t ∈ [0, T 2 ]. Recall that (u 1 , h 1 ) and (u 2 , h 2 ) satisfy (1.5). Then for t ∈ (0, T ], we have Hence, choosing T sufficiently small arrives at (3.6) Now we estimate max with (i) The estimate of I 1 . Recalling the equation satisfied by u 1 and u 2 , we have (ii) The estimate of I 2 . Notice h 1 (t) ≥ h 2 (t) and monotonicity of h i . For every x ∈ (h 0 , h 2 (t)), from (3.5) there are the unique t 1 and t 2 ∈ (0, T ) such that h 1 (t 1 ) = h 2 (t 2 ) = x.
Therefore, letting T small enough leads to where C 1 depends only on initial data. Then substituting the above inequality into (3.6) and choosing T appropriately small, we have Li, X. Li and M. Wang ZAMP which clearly implies h 1 (t) = h 2 (t) for every t ∈ [0, T ]. So the uniqueness is proved. The properties of solution (u, h) clearly follow from the above arguments. Therefore, we complete the proof of Theorem 1.1.
Next we prove Theorem 1.2 by using some properly upper and lower solutions as well as some comparison arguments.