ASYMPTOTIC BEHAVIOUR FOR A NONLOCAL DIFFUSION EQUATION ON A LATTICE LIVIU I. IGNAT AND JULIO D. ROSSI Abstract. In this paper we study the asymptotic behaviour as t → ∞ of solutions to a P nonlocal diffusion problem on a lattice, namely, u0n (t) = j∈Zd Jn−j uj (t) − un (t) with t ≥ 0 P and n ∈ Zd . We assume that J is nonnegative and verifies n∈Zd Jn = 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.

1. Introduction In this paper our main concern is the study of the asymptotic behaviour of the following nonlocal equation on a lattice ( un 0 (t) = (J ∗ u)n (t) − un (t), t ≥ 0, n ∈ Zd , (1.1) un (0) = ϕn , n ∈ Zd , where by (J ∗ u) we denote the discrete convolution, X (J ∗ u)n = Jn−j uj . j∈Zd

Trough the paper we assume that the kernel J is nonnegative and satisfies, X Jn = 1. (1.2) n∈Zd

Equation (1.1), is called nonlocal diffusion equation. Continuous analogous to (1.1), like ut (x, t) = J ∗ u(x, t) − u(x, t), have been recently widely used to model diffusion processes, see, for example, [2], [3], [5], [6], [8], [9], [10], [16] and [17]. In particular, let us mention that these equations are also used in models of neuronal activity, see [7], [11], [13] and [14]. Also there is a discrete counterpart for nonlocal models, see [1], [3] and references therein. In all these models the asymptotic behaviour of the solution (see [4]) is relevant, both from its pure mathematical and its applied point of view. Concerning (1.1), as stated in [9] (see also [3]), if ui (t) is thought of as the density of a single population at the point i at time t, and Ji−j is thought of as the probability distribution of jumping from location i to location j, then (J ∗ u)(t) is the rate at which individuals are arriving to position i from all other places and −ui (t) is the rate at which they are leaving location i to travel to all other sites. This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies equation (1.1). To study the asymptotic beahviour of solutions to (1.1) let us introduce the discrete Laplacian given by d X (∆d u)n = (un+ek − 2un + un−ek ), k=1 1

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L. I. IGNAT AND J.D. ROSSI

where {ek }dk=1 is the canonical basis on Rd . Note that this is a local diffusion operator. Our first result says that the asymptotic behaviour as t → ∞ of solutions to (1.1) is the same as the one for the evolution equation associated to a fractional power of the discrete Laplacian. Theorem 1.1. Let u be a solution of equation (1.1) with ϕ ∈ l1 (Zd ). If there exist positive constants α and A such that b = 1 − A|ξ|α + o(|ξ|α ), J(ξ)

as ξ → 0,

then the asymptotic behaviour of u(t) is given by lim td/α ku(t) − v(t)kl∞ (Zd ) = 0,

t→∞

where v is solution of v 0 = −A(−∆d )α/2 v with initial datum vn (0) = ϕn , n ∈ Zd . In view of this result, we analyze the asymptotic profile of the solutions to v 0 = −A(−∆d )α/2 v. Theorem 1.2. Let us consider ϕ ∈ l1 (Zd ). Then the solution to ( v 0 (t) = −A(−∆d )α/2 v, t > 0, v(0) = ϕ, satisfies

¯ ¯ ´ ³X ¯ ¯ lim sup ¯td/α v([jt1/α ], t) − ϕn GA (j)¯ = 0,

t→∞ j∈Zd

n∈Zd

where GA is defined by

Z α

A

eixξ e−A|ξ| dξ,

G (x) = Rd

and [·] is the floor function. 2. Proofs of the results In our analysis, we make use of the semidiscrete Fourier transform (SDFT) (we refer to [12] and [15] for the mains properties of the SDTF). For any v ∈ l2 (Zd ) we define its SDFT by: X vb(ξ) = e−iξ·j vj , ξ ∈ [−π, π]d . j∈Zd

b = 1. In view of property (1.2), Jb belongs to L∞ ([−π, π]) and J(0) Proof of Theorem 1.1. Applying the SDFT to the solutions of equation (1.1) we get b u(t, ξ) − u u b 0 (t, ξ) = J(ξ)b b(t, ξ),

ξ ∈ [−π, π]d , t > 0.

Solving this ODE we find that (2.1)

b

u b(t, ξ) = et(J(ξ)−1) ϕ(ξ), b

ξ ∈ [−π, π]d , t > 0.

In the same way, v, the solution to v 0 = −A(−∆d )α/2 v satisfies (2.2)

α (ξ)

vb(t, ξ) = e−Atp

ϕ(ξ), b

ξ ∈ [−π, π]d , t > 0,

A NONLOCAL PROBLEM ON A LATTICE

3

where d ³ X ¡ ξk ¢´1/2 p(ξ) = 4 sin2 . 2 k=1

Using the Fourier representation of u and v given by (2.1) and (2.2) we find that Z ku(t) − v(t)kl∞ (Zd ) ≤ |b u(ξ, t) − vb(ξ, t)|dξ [−π,π]d Z b − 1)) − exp(−Atpα (ξ))||ϕ(ξ)|dξ. = | exp(t(J(ξ) b [−π,π]d

By our hypothesis there exists a positive R < π such that b |J(ξ)| ≤1−

|ξ|α , |ξ| ≤ R. 2

Once R has been fixed, there exists δ > 0 such that b |J(ξ)| ≤1−δ

for all ξ ∈ ΩR = {ξ ∈ [−π, π]d , |ξ| > R}.

Hence, it is easy to see that Z Z α b b t(J(ξ)−1) −Atpα (ξ) |e −e ||ϕ(ξ)|dξ b ≤ kϕk b L∞ ([−π,π]d ) (et(|J(ξ)|−1) + e−Atp (ξ) )dξ ξ∈ΩR ξ∈ΩR Z ≤ kϕk b L∞ ([−π,π]d ) (e−tδ + exp(−At inf pα (ξ))dξ. ξ∈ΩR

ξ∈ΩR

Tacking into account that the right hand side in the last inequality is exponentially small, it remains to analyze the term Z I(t) = |b u(ξ, t) − vb(ξ, t)|dξ. |ξ|≤R

Let us choose a function r(t) → 0 such that r(t)t1/α → ∞ as t → ∞. The remaining term I(t) satisfies: Z α b I(t) = |et(J(ξ)−1) − e−Atp (ξ) ||ϕ(ξ)|dξ b ≤ I1 (t) + I2 (t) |ξ|≤R

where

Z

and

b

||ϕ(ξ)|dξ b

|ξ|≤r(t)

Z I2 (t) =

α (ξ)

|et(J(ξ)−1) − e−Atp

I1 (t) =

b

α (ξ)

|et(J(ξ)−1) − e−Atp

||ϕ(ξ)|dξ. b

r(t)≤|ξ|≤R

Using that, for some positive constant c, the following holds c|ξ| ≤ p(ξ) ≤ |ξ|

for all ξ ∈ [−π, π]d ,

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L. I. IGNAT AND J.D. ROSSI

the term I2 (t) can be estimated as follows:

Z α b (e−Atp (ξ) + et(|J(ξ)|−1) )dξ td/α I2 (t) ≤ td/α kϕk b L∞ ([−π,π]d ) r(t)≤|ξ|≤R Z α α (e−Atp (ξ) + e−t|ξ| /2 )dξ ≤ td/α kϕkL1 (Zd ) r(t)≤|ξ|≤R Z α e−Bt|ξ| dξ ≤ td/α kϕkL1 (Zd ) r(t)≤|ξ|≤R Z α = kϕkL1 (Zd ) e−B|ξ| dξ t1/α r(t)|ξ|≤t1/α

≤ kϕkL1 (Zd ) td/α e−Btr

α (t)

→ 0.

To estimate I1 (t) we first observe that there exists a function h(ξ) with h(ξ) → 0 as |ξ| → 0 and such that b − 1 − A|ξ|α | ≤ |ξ|h(ξ) |J(ξ) for all ξ in a sufficiently small ball centered at the origin. Thus for all such ξ b − 1 − Apα (ξ)| ≤ |ξ|α h(ξ) + ||ξ|α − pα (ξ)| . |ξ|α h(ξ) + |ξ|3α . |J(ξ) In view of this property we get

Z α α b I1 (t) ≤ td/α kϕk b L∞ ([−π,π]d ) e−Atp (ξ) |et(J(ξ)−1−Atp (ξ)) − 1|dξ |ξ|≤r(t) Z α ≤ td/α kϕkL1 (Zd ) e−Atp (ξ) t|ξ|α (h(ξ) + |ξ|3α )dξ |ξ|≤r(t) Z α ≤ td/α kϕkL1 (Zd ) e−Bt|ξ| (t|ξ|α h(ξ) + t|ξ|4α )dξ. |ξ|≤r(t)

The last term in the right hand side verifies Z Z α td/α e−Bt|ξ| t|ξ|4α dξ ≤ t−3

α

e−B|η| |η|4α dη → 0.

|η|≤r(t)t1/α

|ξ|≤r(t)

Hence we have to analyze the first one. In this case, by the same change of variables, we get Z Z α α td/α e−Bt|ξ| t|ξ|α h(ξ) = |η|α e−Bt|η| h(ηt−1/α ). |η|
|ξ|≤r(t)

Applying Lebesgue convergence theorem we obtain that also this term converges to zero as t → ∞. This ends the proof. ¤ Now we prove our second result, Theorem 1.2, that describes the asymptotic profile of solutions to v 0 = −A(−∆d )α/2 v. Proof of Theorem 1.2. Using the Fourier representation of v we have Z α v(j, t) = e−Atp (ξ) eijξ ϕ(ξ) b dξ, j ∈ Zd , t > 0. [−π,π]d

A NONLOCAL PROBLEM ON A LATTICE

Thus

Z α (ξ)

td/α v([jt1/α ], t) = td/α Z

e−Atp

ei[jt

1/α ]ξ

ϕ(ξ) b dξ

[−π,π]d α (ξt−1/α )

e−Atp

= [−πt1/α ,πt1/α ]d

and

³ [jt1/α ] ´ exp iξ 1/α ϕ(ξt b −1/α ) dξ t

Z α

GA (j) =

e−A|ξ| eiξj dξ. Rd

Denoting A I(j, t) = td/α v([jt1/α ], t) − ϕ(0)G b (j)

we obtain

Z ¯ ¯Z α ¯ ¯ −Atpα (ξt−1/α ) ijξ |I(j, t)| ≤ ¯ e e − ϕ(0) b e−A|ξ| eiξj dξ ¯ [−πt1/α ,πt1/α ]d Rd Z ¯ ¯ α −1/α ) ¯eijξ − eiξ[jt1/α ]t−1/α ¯|ϕ(ξt + e−Atp (ξt b −1/α )|dξ [−πt1/α ,πt1/α ]d

= I1 (j, t) + I2 (j, t). Therefore we have to get bounds for I1 (j, t) and I2 (j, t). Step I. Estimates for I2 (t). For I2 (t) we have the rough estimate Z ¯ ¡ jt1/α ξ − [jt1/α ]ξ ¢¯¯ α −1/α ) ¯ |I2 (j, t)| ≤ kϕk b L∞ ([−π,π]d ) e−Atp (ξt ¯dξ ¯ sin 2t1/α [−πt1/α ,πt1/α ]d Z ¯ 1/α ξ − [jt1/α ]ξ ¯ α −1/α ) ¯ jt ¯ ≤ kϕkL1 (Zd ) e−Atp (ξt ¯ ¯dξ 1/α 2t [−πt1/α ,πt1/α ]d Z α −1/α ) |ξ| dξ ≤ kϕkL1 (Zd ) e−Atp (ξt 1/α 1/α d t1/α [−πt ,πt ] Z α . t−1/α kϕkL1 (Zd ) e−c(α)|ξ| |ξ|dξ → 0. [−πt1/α ,πt1/α ]d

Step II. Estimates for I1 (t). Observe that I1 satisfies: Z α −1/α ) α I1 (j, t) ≤ |e−Atp (ξt − e−A|ξ| ||ϕ(ξt b −1/α )|dξ 1/α 1/α d [−πt ,πt ] Z α + e−A|ξ| |ϕ(ξt b −1/α ) − ϕ(0)|dξ b [−πt1/α ,πt1/α ]d Z α +|ϕ(0)| b e−A|ξ| dξ 1/α ,πt1/α ]d ξ ∈[−πt /

= I3 (t) + I4 (t) + I5 (t). In the case of the last integral, easily follows that |ξ| & t1/α . Thus Z α I5 (t) . e−A|ξ| dξ → 0. |ξ|&t1/α

5

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L. I. IGNAT AND J.D. ROSSI

For I4 we have the following estimate: Z α I4 (t) = e−A|ξ| |ϕ(ξt b −1/α ) − ϕ(0)|χ b [−πt1/α ,πt1/α ]d dξ Rd

and the Lebesgue dominated convergence theorem guarantees that I4 (t) → 0 as t → ∞. Using that p(ξ) satisfies c|ξ| ≤ p(ξ) ≤ |ξ| for some positive c and the mean value theorem we get: Z α −1/α ) e−Atp (ξt |tpα (ξt−1/α ) − |ξ|α | dξ I3 (t) ≤ kϕk b L∞ ([−π,π]d ) 1/α 1/α d [−πt ,πt ] Z α . kϕkL1 (Zd ) e−c|ξ| |tpα (ξt−1/α ) − |ξ|α | dξ. [−πt1/α ,πt1/α ]d

Applying again the dominated convergence theorem we obtain that I3 (t) → 0 as t → ∞. The proof is now complete. ¤ Acknowledgements. L. I. Ignat partially supported by the grants MTM2005-00714 and PROFIT CIT-370200-2005-10 of the Spanish MEC, SIMUMAT of CAM and CEEX-M3-C312677 of the Romanian MEC. J. D. Rossi partially supported by SIMUMAT (Spain), UBA X066, CONICET and ANPCyT PICT 05009 (Argentina). References [1] P. Bates and A. Chmaj. A discrete convolution model for phase transitions. Arch. Rat. Mech. Anal., 150, 281–305, (1999). [2] P. Bates, P. Fife, X. Ren and X. Wang. Travelling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal., 138, 105–136, (1997). [3] C. Carrillo and P. Fife. Spatial effects in discrete generation population models. J. Math. Biol., 50(2), 161–188, (2005). [4] E. Chasseigne, M. Chaves and J. D. Rossi. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl., 86, 271–291, (2006). [5] X Chen. Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differential Equations, 2, 125–160, (1997). [6] C. Cortazar, M. Elgueta and J. D. Rossi. A non-local diffusion equation whose solutions develop a free boundary. Annales Henri Poincar´e, 6(2), 269–281, (2005). [7] R. Curtu and B. Ermentrout. Pattern formation in a network of excitatory and inhibitory cells with adaptation. SIAM J. Appl. Dyn. Syst. 3(3), 191–231, (2004). [8] F. Da Lio, N. Forcadel and R. Monneau. Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics. to appear in J. Eur. Math. Soc. [9] P. Fife. Some nonclassical trends in parabolic and parabolic-like evolutions. Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003. [10] P. Fife and X. Wang. A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions. Adv. Differential Equations, 3(1), 85–110, (1998). [11] S. E. Folias and P. C. Bressloff. Breathers in Two-Dimensional Neural Media. Phys. Rev. Letters, 95, 208107, 1–4, (2005). [12] T. W. K¨ orner. Fourier analysis. Cambridge University Press, Cambridge, 1988. [13] D. J. Pinto and B. G. Ermentrout. Spatially structured activity in synaptically coupled neuronal networks. I. Traveling fronts and pulses. SIAM J. Appl. Math. 62(1), 206–225, (2001). [14] D. J. Pinto and B. G. Ermentrout. Spatially structured activity in synaptically coupled neuronal networks. II. Lateral inhibition and standing pulses. SIAM J. Appl. Math. 62(1), 226–243, (2001). [15] L.N. Trefethen, Spectral methods in MATLAB, Software, Environments and Tools, Society for Industrial and Applied Mathematics, 2000.

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[16] X. Wang. Metaestability and stability of patterns in a convolution model for phase transitions. J. Differential Equations, 183, 434–461, (2002). [17] L. Zhang. Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks. J. Differential Equations, 197(1), 162–196, (2004). L. I. Ignat ´ ticas, Departamento de Matema ´ noma de Madrid, U. Auto 28049 Madrid, Spain and Institute of Mathematics of the Romanian Academy, P.O.Box 1-764, RO-014700 Bucharest, Romania. E-mail address: [email protected] Web page: http://www.uam.es/liviu.ignat J. D. Rossi ´ tica, FCEyN UBA (1428) Depto. Matema Buenos Aires, Argentina. E-mail address: [email protected] Web page: http://mate.dm.uba.ar/∼jrossi