On the asymptotic behavior for a nonlocal diffusion equation with an absorption term on a lattice

In this paper, we consider the following initial value problem $$U_i'(t) = \sum_{j\in B} J_{i-j}(U_J(t) - U_i(t)) - U_i^p(t),~t\geq 0,~i\in B,$$ $$ U_i(0)=\varphi_i>0;~i\in B.$$ where $B$ is a bounded subset of $Zd^$, $ p > 1$, $J_h = (J_i)_{i \in B}$ is a kernel which is nonnegative, symmetric, bounded and $\sum_{j \in Z^d} J_j = 1$. We describe the asymptotic behavior of the solution of the above problem. In this paper, we consider the following initial value problem.

and it is also important to have informations about the asymptotic behavior of continuous solutions if we have a continuous nonlocal problems or discrete solutions if the problems are represented by discrete equations.The solution u(x, t) can be interpreted as the density of a single population at the point x at time t and J(x−y) as the probability distribution of jumping from location y to location x.Then the convolution (J * u)(x, t) = R N J(y − x)u(y, t)dy is the rate at which individuals are arriving to position x from all other places and −u(x, t) = − R N J(y − x)u(x, t)dy is the rate at which they are leaving location x to travel to all other sites (see [21]).For the discrete case where (J * U ) i = j∈Z d J i−j U j , the component of the solution U h (t) = (U j (t)) j∈B in i, namely U i (t) can be interpreted as the density of a single population at the point i at time t and J i−j can be interpreted as the probability distribution of jumping from location i to location j.Then the convolution (J * U ) i (t) is the rate at which individuals are arriving to position i from all other places and −U i (t) is the rate at which they are leaving location i to travel to all other sites (see [21]).
In this paper, we are interested in the asymptotic behavior of the solution of ( 1)- (2).For local diffusion problems, the asymptotic behavior of solutions has been the subject of investigations of several authors (see [3], [9]- [12], [26], [27] and the references cited therein).For nonlocal problems, in the continuous case, the authors in [14] and [31] have studied the asymptotic behavior of solutions.For our problem, in the case where there is no absorption term, the authors in [25] have studied the asymptotic behavior of the solution when B = Z d .Our paper is written in the following manner.In the next section, we prove the local existence and the uniqueness of the solution.Finally in the last section, we show that the solution U h of ( 1)-( 2) tends to zero as t approaches infinity and describe its asymptotic behavior as t → +∞.

Local existence and uniqueness
In this section, we shall establish the existence and the uniqueness of the solution U h (t) of ( 1)-( 2) on (0, T ) for small T .Let t 0 > 0 be fixed and define the function space Theorem 2.1.Assume that ϕ h ∈ Y t0 .Then R maps X t0 into X t0 and R is strictly contractive if t 0 is appropriately small relative to ϕ h ∞ .
Proof.A straightforward computation reveals that then Therefore if (3) holds, then R maps X t0 into X t0 .Now, we are going to prove that the map R is strictly contractive.
Use Taylor's expansion to obtain where β i is an intermediate value between V i and Z i .We deduce that and the proof is complete.2 It follows from the contraction mapping principle that for appropriately chosen t 0 ∈ (0, 1), R has a unique fixed point U h (t) ∈ Y t0 which is a solution of (1)-(2).To extend the solution to [0, ∞), we may take as initial data U h (t 0 ) ∈ Z d and obtain a solution in [0, 2t 0 ].Iterating this procedure, we get a solution defined in [0, ∞).

Asymptotic behavior of solutions
In this section, we show that the solution U h (t) of ( 1)-( 2) tends to zero as t approaches infinity.We also give its asymptotic behavior as t → ∞.Before starting, let us prove the following lemma which is a version of the maximum principle for discrete nonlocal problems.
Proof.Let T 0 < ∞ and let λ be such that b Introduce the vector Z h (t) = e λt U h (t) and let m = min t∈[0,T0] Z h (t) inf where Z h (t) inf = min 0≤i≤I Z i (t).Then, there exists t 0 ∈ [0, T 0 ] such that m = Z i0 (t 0 ) for a certain i 0 ∈ B. We get Z i0 (t 0 ) ≤ Z i0 (t) for t ≤ t 0 and Z i0 (t 0 ) ≤ Z j (t 0 ) for j ∈ B, which implies that Using the first inequality of the lemma, it is not hard to see that It follows from ( 4)-( 6) that (b i0 (t 0 ) − λ)Z i0 (t 0 ) ≥ 0, which implies that Z i0 (t 0 ) ≥ 0 because b i0 (t 0 ) − λ > 0. We deduce that U h (t) ≥ 0 for t ∈ [0, T 0 ], which leads us to the result.2 Another version of the maximum principle for discrete nonlocal problems is the following comparison lemma.
Then, we have It follows that But, this contradicts the first strict differential inequality of the lemma and the proof is complete.2 Remark 3.1.If we modify slightly the proof of Lemma 3.2, it is not hard to see that U h (t) > 0 for t ≥ 0 where U h (t) is the solution of ( 1)-( 2).
Introduce the function where C 0 = ( 1 p−1 ) 1 p−1 and λ = 1 p−1 , which is crucial for the asymptotic behavior of solutions.We have µ(0) = 0 and µ (0) = 1.We deduce that µ(ε) > 0 and µ(−ε) < 0 for ε small enough.The lemma below shows that the solution U h of the problem (1)-(2) tends to zero as t approaches infinity.Lemma 3.3.Let U h (t) be the solution of (1)- (2).Then, we have Let k > 1 be so large that Obviously kZ p i < (kZ i ) p which implies that Hence, we have We deduce that and the proof is complete.2 Now, let us give the asymptotic behavior of the solution U h .We have the following result.
Proof.From Remark 3.1, we know that U h (t) > 0 for t ≥ 0. Introduce the vector W h (t) such that