Abstract
This paper is concerned with a class of nonlocal dispersal problem with Dirichlet boundary conditions. We analyze the limit of solutions when the dispersal kernel is rescaled. Our main results reveal that the solutions of Dirichlet heat equation can be approximated by the nonlocal dispersal equation. The investigation also shows that the nonlocal dispersal equation is similar to the convection–diffusion equation by taking another special kernel function.
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1 Introduction
Let \(K:{\mathbb {R}}^N\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) be a nonnegative, continuous function such that \(\int _{{\mathbb {R}}^N}K(y,x)\,dy=1\) for all \(x\in {\mathbb {R}}^N\). Nonlocal dispersal equation of the form
and variations of it have been widely used to model diffusion process [1, 3]. As stated in [9, 11], if u(x, t) is thought as a density at position x at time t and the probability distribution that individuals jump from y to x is given by K(x, y), then \(\int _{{\mathbb {R}}^N}K(x,y)u(y,t)\,dy\) denotes the rate at which individuals are arriving to position x from all other places and \(u(x,t)=\int _{{\mathbb {R}}^N}K(y,x)u(x,t)\,dy\) is the rate at which they are leaving position x to all other places. This consideration, in the absence of external sources, leads immediately to that u(x, t) satisfies (1.1). For recent references on nonlocal dispersal equations, see [2,3,4, 17, 19] and references therein.
It is known from [1, 9] that nonlocal dispersal equation shares many properties with the classical heat equation. Moreover, Cortazar et al. [6] proved that a suitable rescaled nonlocal equation with convolution kernel function can approximate the classical heat equation with Dirichlet boundary condition. We refer to [5, 13,14,15,16, 18] for the recent study of nonlocal rescaled problems. In the present paper, we study nonlocal dispersal problem with non-homogeneous kernel functions. We then analyze the approximation solutions when the dispersal kernel is rescaled. To do this, let us first consider the nonlocal dispersal equation
where \(\Omega \) is a bounded smooth domain of \({\mathbb {R}}^N\), \(J: {\mathbb {R}}^N\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a nonnegative dispersal kernel such that \(\int _{{\mathbb {R}}^N}J(x,y)dy=1\) for any \(x\in {\mathbb {R}}^N\). The function g(x, t) is defined for \(x\in {\mathbb {R}}^{N}\backslash \Omega ,\) \(t>0\) and \(u_{0}(x)\) is defined for \(x\in \Omega \). In (1.2), the values of u(x, t) are prescribed outside \(\Omega \), which is analogous to the Dirichlet boundary condition for heat equation [1, 4]. Throughout this paper, we make the following assumptions.
-
(A1) \(J: {\mathbb {R}}^N\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is nonnegative, smooth, \(J(0,0)>0\), \(J(x,y)=J(|x|,|y|)\) for \(x,y\in {\mathbb {R}}^N\). Moreover, \(\int _{{\mathbb {R}}^N}J(x,y)dy=1\) and \(\int _{{\mathbb {R}}^N}J(x,y)|y|^2dy<\infty \) for any \(x\in {\mathbb {R}}^N\).
-
(A2) The function g(x, t) and \(u_0(x)\) are smooth functions.
It follows from the assumption (A1) that the dispersal kernel function J may not be a symmetric function. However, we shall prove that the rescaled nonlocal dispersal equation of (1.2) is analogous to heat equation
Note that the regularity of solution u(x, t) to (1.3) is related to the properties of \(\Omega \), \(u_0(x)\) and g(x, t), see [7, 8]. So in this paper, we assume that u(x, t) is the unique solution to (1.3) and
for some \(0<\alpha <1\). Take \(\varepsilon >0\), we consider the rescaled kernel function of J(x) as follows
here d(x) is given by
By (A1), we know that d(x) is positive and finite for \(x\in \Omega \). We then consider the nonlocal dispersal equation
Existence and uniqueness of solutions to (1.5) will be established in Sect. 2. We show that there exists a unique solution \(u_\varepsilon (x,t)\) to (1.5) such that
Now, we are ready to state the main result.
Theorem 1.1
Assume that \(u\in C^{2+\alpha ,1+\alpha /2}({\overline{\Omega }}\times [0,T])\) is the solution of (1.3) and \(u_{\varepsilon }(x,t)\) is the solution of (1.5), respectively. Then, there exists \(C=C(T)\) such that
From Theorem 1.1, we can see that the nonlocal dispersal Eq. (1.5) is similar to the Dirichlet heat equation (1.3). It follows from the classical works of Ignat and Rossi [10] that the asymmetric nonlocal dispersal equation may be similar to the convection–diffusion equation. However, our result shows that the nonlocal dispersal Eq. (1.5) may also be similar to the diffusion equation without convection.
In the second part of this paper, let us consider the nonlocal dispersal equation
For \(\varepsilon >0\), we use the rescaled kernel function
and study the rescaled nonlocal dispersal equation
here, d(x) is given in (1.4). Existence and uniqueness of solutions to (1.6) and (1.7) will be established in Sect. 2. We show that there exists a unique solution \(u^\varepsilon (x,t)\) to (1.5) such that
In order to get a simple statement, we assume further that J satisfies the following condition.
-
(A3) There exists \(c>0\) such that \(J(x,y)=0\) for \(|x|>c\) and \(|y|>c\).
We shall prove that the rescaled nonlocal dispersal Eq. (1.7) will approximate the convection–diffusion equation
where \(q(x)=(q_1(x),q_2(x),\cdots ,q_N(x))\) and \(q_i(x)\) is given by
for \(i=1,2,\cdots ,N\) and \(y=(y_1,y_2,\cdots ,y_N)\).
Theorem 1.2
Assume that \((A1)-(A3)\) hold. Let \(u\in C^{2+\alpha ,1+\alpha /2}({\overline{\Omega }}\times [0,T])\) be the solution of (1.8) and \(u^{\varepsilon }(x,t)\) be the solution of (1.7), respectively. Then, we have
By Theorem 1.2, we know that the nonlocal Dirichlet Eq. (1.7) is similar to the classical convection–diffusion equation. Our main results reveal that nonlocal dispersal equation with non-homogeneous kernel function may also be similar to convection–diffusion equation.
The rest of the paper is organized as follows. In Sect. 2, we prove existence and uniqueness of solutions to our nonlocal models. The main results are proved in Sect. 3.
2 Existence and Uniqueness
In this section, we establish the existence and uniqueness of solutions to our main models. Here by a solution of (1.2), it is understood in an integral sense.
Definition 2.1
A solution of (1.2) is a function \(u\in C([0,\infty ); L^1(\Omega ))\) such that
and
The solution of (1.6) can be defined by a similar way. Since the argument for (1.6) is analogous, we first study the model (1.2). Existence and uniqueness will be obtained by Banach’s fixed point theorem. Fix \(t_0>0\) and consider the space
We then know that \(X _{t_0}\) is a Banach space with norm
Define \({\mathcal {T}}: X _{t_0}\rightarrow X _{t_0}\) by
where it is assumed that
Lemma 2.2
Assume that \(u_0,v_0\in L^1(\Omega )\). Then, there exists \(C>0\) depending only on J and \(\Omega \) such that
for \(u,v\in X _{t_0}\).
Proof
Note that
we obtain that
for some \(C>0\). \(\square \)
Theorem 2.3
For every \(u_0\in L^1(\Omega )\), there exists a unique solution u(x, t) to (1.2) and
Proof
By the definition of \({\mathcal {T}}_{u_0}\), we know that \({\mathcal {T}}_{u_0}\) maps \(X _{t_0}\) into \(X _{t_0}\). Then, it follows from Lemma 2.2 that \({\mathcal {T}}_{u_0}\) is a contraction map if we choose \(t_0\) small enough such that \(Ct_0<1\). By Banach’s fixed point theorem, we get the existence and uniqueness of solutions in the interval \([0,t_0]\). Then, we take \(u(x,t_0)\in L^1(\Omega )\) as initial value and we can obtain a solution up to \([0,2t_0]\). Iterating this procedure, we get a solution u(x, t) such that
But for any \(\delta \ne 0\) and \(t>0\), we have
It follows from (2.2) and Lebesgue theorem that
Hence, we know that (2.1) holds. \(\square \)
Analogously, we obtain the following result.
Theorem 2.4
For every \(u_0\in L^1(\Omega )\), there exists a unique solution u(x, t) to (1.6) and
At the end of this section, we give the comparison principle. The sub-super solutions are defined as follows.
Definition 2.5
A function \(u\in C((0,T); L^1(\Omega ))\) is a super-solution to (1.2) if
The sub-solution is defined analogously by reversing the inequalities.
We have the following results on sub-super solutions. One can see [6] for a similar proof.
Theorem 2.6
Assume that u(x, t), v(x, t) are a pair of super-sub solutions to (1.2). Then, \(u(x,t)\ge v(x,t)\) for \((x,t)\in \Omega \times (0,\infty )\).
Theorem 2.7
Assume that u(x, t), v(x, t) are a pair of super-sub solutions to (1.6). Then, \(u(x,t)\ge v(x,t)\) for \((x,t)\in \Omega \times (0,\infty )\).
3 Proof of Main Results
In this section, we shall prove the main results of this paper. We first consider the case of nonlocal dispersal Eq. (1.2).
Proof of Theorem 1.1
In (1.3), the functions g(x, t) and \(u_0(x)\) are smooth, and we then can extend u(x, t) to the whole space, see [8, 12]. Let \({\widetilde{v}}(x,t)\) be a \(C^{2+\alpha ,1+\alpha /2}\) extension of u(x, t) to \({\mathbb {R}}^{N}\times [0,T]\) and consider the operator
We can see that \({\widetilde{v}}(x,t)\) satisfies
where
Since \(G(x,t)={\widetilde{v}}(x,t)-g(x,t)\) is smooth and \(G(x,t)=0\) for \(x\in \partial \Omega \), we can find \(M_{1}>0\), such that
for x satisfying \(\text {dist}(x,\partial \Omega )\le \varepsilon \).
The existence and uniqueness of solution \(u_{\varepsilon }(x,t)\) to (1.5) are followed by Theorem 2.3. Define \(\omega _{\varepsilon }(x,t)={\widetilde{v}}(x,t)-u_{\varepsilon }(x,t)\), then we get
Note that \({\widetilde{v}}\in C^{2+\alpha ,1+\alpha /2}({\overline{\Omega }}\times [0,T])\), we claim that there exists \(M_{2}>0\) such that
In fact, we know that
But \({{\widetilde{v}}}\in C^{2+\alpha ,1+\alpha /2}({\overline{\Omega }}\times [0,T])\), a simple argument from Taylor’s theorem shows that
By the assumption (A1), we have
for \(i=1,2\cdots ,N\) and
for \(i,j=1,2\cdots ,N\) and \(i\ne j\). Accordingly,
Hence,
This gives that (3.3) holds.
Now, denote
For \(x\in \Omega \), we have
In view of (3.1–3.2), by choosing \(M_2\) large, we obtain
for \(x\in {\mathbb {R}}^{N}\backslash \Omega \) such that \(dist(x,\partial \Omega )\le \varepsilon \) and \(t\in [0,T]\). Moreover, it is clear that
Thanks to (3.4–3.5), we have \({\overline{w}}(x,t)\) is the super-solution of (3.1). This yields
By a similar way, we can show that
is a sub-solution and
Hence,
and we end the proof. \(\square \)
In order to prove Theorem 1.2, we consider the elliptic equation
where \(\varepsilon >0\), the coefficients
and
for \(i,j=1,2\cdots ,N\). We know from [8, 12] that (3.6) admits a unique solution
We then have the following results.
Lemma 3.1
Assume that \((A1)-(A3)\) hold. Let v(x, t) and \(v^\varepsilon (x,t)\) be the solutions of (1.8) and (3.6), respectively. Then, we have
Proof
Since
for \(i,j=1,2,\cdots ,N\) and
for \(i\ne j\), we get
On the other hand, we have
and so
for \(i=1,2,\cdots ,N\). Hence, we know that (3.7) holds. \(\square \)
Lemma 3.2
Assume that \((A1)-(A3)\) hold. Let \(v^\varepsilon (x,t)\) be the solution of (3.6) and \(u^{\varepsilon }(x,t)\) be the solution of (1.7), respectively. Then, we have
Proof
Let \({{\widetilde{v}}}^\varepsilon (x,t)\in {\mathbb {R}}^N\times [0,T]\) be the extension of \(v^\varepsilon (x,t)\) to \({\mathbb {R}}^{N}\times [0,T]\), where \(v^\varepsilon (x,t)\) is the unique solution to (3.6). Define the operator
Then, \({{\widetilde{v}}}^\varepsilon (x,t)\) satisfies
where
Besides, since \(G(x,t)={{\widetilde{v}}}^\varepsilon (x,t)-g(x,t)\) is smooth and \(G(x,t)=0 \) if \(x\in \partial \Omega \), then there exists \(M_1>0\) such that
for x such that \(\text {dist}(x,\partial \Omega )\le \varepsilon \).
Denote \(w^\varepsilon (x,t)={{\widetilde{v}}}^\varepsilon (x,t)-u^\varepsilon (x,t)\), then we have
We claim that for \(\varepsilon >0\) is small, there exists \(M_2>0\) such that
In fact, we always have
Set \(z=(x-y)/\varepsilon \), then we get
this also shows that (3.8) holds,
Denote
For \(x\in \Omega \), by the claim above, we have
We can take \(K_2\) large enough such that
for \(x\in {\mathbb {R}}^{N}\backslash \Omega \) satisfying \(dist(x,\partial \Omega )\le \varepsilon \) and \(t\in [0,T]\). Moreover, we have
Thanks to (3.9–3.10), we use the comparison principle to obtain
Similarly, we can show that
Hence,
We end the proof. \(\square \)
At last, let \(v\in C^{2+\alpha ,1+\alpha /2}({\overline{\Omega }}\times [0,T])\) be the solution of (1.8). Then, we have
It follows from Lemmas 3.1–3.2 that
and we end the proof of Theorem 1.2.
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Acknowledgements
J.W. Sun would like to thank Professor Wenxian Shen for useful discussions. This work was supported by NSF of China (11401277,11731005), NSF of Gansu Province (21JR7RA535,21JR7RA537) and Fundamental Research Funds for the Central Universities (lzujbky-2021-52).
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Communicated by Syakila Ahmad.
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Du, Y., Sun, JW. Approximation Solutions of Some Nonlocal Dispersal Problems. Bull. Malays. Math. Sci. Soc. 46, 8 (2023). https://doi.org/10.1007/s40840-022-01403-z
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DOI: https://doi.org/10.1007/s40840-022-01403-z