REFINED ASYMPTOTIC EXPANSIONS FOR NONLOCAL DIFFUSION EQUATIONS LIVIU I. IGNAT AND JULIO D. ROSSI Abstract. We study the asymptotic behavior for solutions to nonlocal diffusion models of the form ut = J ∗ u − u in the whole Rd with an initial condition u(x, 0) = u0 (x). Under suitable hypotheses on J (involving its Fourier transform) and u0 , it is proved an expansion of the form
X (−1)|α| Z
u0 (x)xα dx ∂ α Kt 6 Ct−A ,
u(u) − α! Lq (Rd ) |α|6k
where Kt is the regular part of the fundamental solution and the exponent A depends on J, q, k and the dimension d. Moreover, we can obtain bounds for the difference between the terms in this expansion and the corresponding ones for the expansion of the evolution given by fractional powers of the Laplacian, s vt (x, t) = −(−∆) 2 v(x, t).
1. Introduction In this paper we study the asymptotic behavior as t → ∞ of solutions to the nonlocal evolution equation ut (x, t) = J ∗ u − u(x, t), t > 0, x ∈ Rd , (1.1) u(x, 0) = u0 (x), x ∈ Rd , R where J : Rd → R verifies Rd J(x)dx = 1. Equations like (1.1) and variations of it, have been recently widely used to model diffusion processes, for example, in biology, dislocations dynamics, etc. See, for example, [3], [4], [6], [7], [11], [12], [8], [14] and [15]. As stated in [11], if u(x, t) is thought of as the density of a single population at the point x at time t, and J(x − y) is thought of as the probability distribution of jumping from location y R to location x, then (J ∗ u)(x, t) = RN J(y − x)u(y, t) dy is the rate at which individuals are arriving to position x from all other places and R −u(x, t) = − RN J(y −x)u(x, t) dy is the rate at which they are leaving Key words and phrases. Nonlocal diffusion, asymptotic behavior, fractional Laplacian. 2000 Mathematics Subject Classification. 35B40, 45A05, 45M05. 1
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L. I. IGNAT AND J.D. ROSSI
location x 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). Equation (1.1), is called nonlocal diffusion equation since the diffusion of the density u at a point x and time t does not only depend on u(x, t), but on all the values of u in a neighborhood of x through the convolution term J ∗ u. When J is nonnegative and compactly supported, this equation shares many properties with the classical heat equation, ut = cuxx , such as: bounded stationary solutions are constant, a maximum principle holds for both of them and perturbations propagate with infinite speed, see [11]. However, there is no regularizing effect in general. For instance, if J is rapidly decaying (or compactly supported) the singularity of the source solution, that is a solution of (1.1) with initial condition a delta measure, u0 = δ0 , remains with an exponential decay. In fact, this fundamental solution can be decomposed as w(x, t) = e−t δ0 +Kt (x) where Kt (x) is smooth, see Lemma 2.1. In this way we see that there is no regularizing effect since the solution u of (1.1) can be written as u(t) = w(t) ∗ u0 = e−t u0 + Kt ∗ u0 with Kt smooth, which means that u(·, t) is as regular as u0 is. For the heat equation a precise asymptotic expansion in terms of the fundamental solution and its derivatives was found in [9]. In fact, if Gt denotes the fundamental solution of the heat equation, namely, 2 Gt (x) = (4πt)−d/2 e−|x| /(4t) , under adequate assumptions on the initial condition, we have,
X (−1)|α| Z
α α (1.2) u0 (x)x ∂ Gt 6 Ct−A
u(x, t) − q d α! L (R ) Rd |α|6k
with A = ( d2 )( (k+1) + (1 − 1q )). As pointed out by the authors in [9], d the same asymptotic expansion can be done in a more general setting, s dealing with the equations ut = −(−∆) 2 u, s > 0. Now we need to introduce some notation. We will say that f ∼ g as ξ ∼ 0 if |f (ξ) − g(ξ)| = o(g(ξ)) when ξ → 0 and f . g if there exists a constant c independent of the relevant quantities such that f 6 cg. In the sequel we denote by Jb the Fourier transform of J. Our main objective here is to study if an expansion analogous to (1.2) holds for the non-local problem (1.1). Concerning the first term, b − 1 ∼ −|ξ|s as ξ ∼ 0, then the in [5] it is proved that if J verifies J(ξ) asymptotic behavior of the solution to (1.1), u(x, t), is given by d
lim t s max |u(x, t) − v(x, t)| = 0,
t→+∞
x
ASYMPTOTICS FOR NONLOCAL DIFFUSION
3
s
where v is the solution of vt (x, t) = −(−∆) 2 v(x, t) with initial condition v(x, 0) = u0 (x). As a consequence, the decay rate is given by d ku(·, t)kL∞ (Rd ) 6 C t− s and the asymptotic profile is as follows,
Z
d 1 s u0 G (y) lim t s u(yt s , t) −
∞ d = 0,
t→+∞ d L (R )
R
cs (ξ) = e−|ξ|s . where Gs (y) satisfies G Here we find a complete expansion for u(x, t), a solution to (1.1), in terms of the derivatives of the regular part of the fundamental solution, Kt . As we have mentioned, the fundamental solution w(x, t) of problem (1.1) satisfies w(x, t) = e−t δ0 (x) + Kt (x), where the function Kt (the regular part of the fundamental solution) is given by b ct (ξ) = e−t (etJ(ξ) K − 1). In contrast with the previous analysis done in [5] where the long time behavior is studied in the L∞ (Rd )-norm, here we also consider Lq (Rd ) norms. We focus in the case 2 6 q 6 ∞ where we use HausdorffYoung’s inequality and Plancherel’s identity as main tools. The case 1 ≤ q < 2 will be treated elsewhere. Theorem 1.1. Let be s and m positive such that b − 1 ∼ −|ξ|s , (1.3) J(ξ) ξ∼0 and 1 , |ξ| → ∞. |ξ|m Then for any 2 6 q 6 ∞ and k + 1 < m − d there exists a constant C = C(q, k)k|x|k+1 u0 kL1 (Rd ) such that
X (−1)|α| Z
α α (1.5) u(x, t) − u (x)x ∂ K 6 Ct−A
0 t q d α! L (R ) Rd
(1.4)
b |J(ξ)| .
|α|6k
for all u0 ∈ L1 (Rd , 1 + |x|k+1 ). Here A =
(k+1) s
+ ds (1 − 1q ).
Remark 1.1. The condition k + 1 < m − d guarantees that all the partial derivatives ∂ α Kt of order |α| = k + 1 make sense. In addition if Jb decays at infinity faster than any polynomial, c(m) b (1.6) ∀ m > 0, ∃ c(m) such that |J(ξ)| 6 , |ξ| → ∞, |ξ|m then the expansion (1.5) holds for all k.
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L. I. IGNAT AND J.D. ROSSI
b − 1 ∼ −|ξ|2 as Note that, when J has an expansion of the form J(ξ) ξ ∼ 0 (this happens for example if J is compactly supported), then the decay rate in L∞ (Rd ) of the solutions to the non-local problem (1.1) and d the heat equation coincide (in both cases they decay as t− 2 ). Moreover, the first order term also coincide (in both cases it is a Gaussian). See [5] and Theorem 1.1. Our next aim is to study if the higher order terms of the asymptotic expansion that we have found in Theorem 1.1 have some relation with the corresponding ones for the heat equation. Our next results say that the difference between them is of lower order. Again we have to deal with 2 6 q 6 ∞. Theorem 1.2. Let J as in Theorem 1.1 and assume in addition that there exists r > 0 such that b − (1 − |ξ|s ) ∼ B|ξ|s+r , J(ξ)
(1.7)
ξ ∼ 0,
for some real number B. Then for any 2 6 q 6 ∞ and |α| 6 m − d there exists a positive constant C = C(q, d, s, r) such that the following holds d
1
k∂ α Kt − ∂ α Gst kLq (Rd ) 6 Ct− s (1− q ) t−
(1.8)
|α|+r s
,
cst (ξ) = exp(−t|ξ|s ). where Gst is defined by its Fourier transform G Note that these results do not imply that the asymptotic expansion P (−1)|α| R u0 (x)xα ∂ α Kt coincides with the expansion that holds |α|6k α! R P |α| s for the equation ut = −(−∆) 2 u: |α|6k (−1) u0 (x)xα ∂ α Gst . They α! only say that the corresponding terms agree up to a better order. When J is compactly supported or rapidly decaying at infinity, then s = 2 and we obtain an expansion analogous to the one that holds for the heat equation. Finally, we present a result that gives the first two terms in the asymptotic expansion with very weak assumptions on J. Theorem 1.3. Let u0 ∈ L1 (Rd ) with ub0 ∈ L1 (Rd ) and s < l be two b − (1 − |ξ|s ) ∼ B|ξ|l , when ξ ∼ 0, for positive numbers such that J(ξ) some real number B. Then for any 2 6 q 6 ∞ (1.9)
d
1
lim t s (1− q )+
t→∞
l−s s
l
ku(t) − v(t) − Bt[(−∆) 2 v](t)kLq (Rd ) → 0, s
where v is the solution to vt = −(−∆) 2 v with v(x, 0) = u0 (x).
ASYMPTOTICS FOR NONLOCAL DIFFUSION
Moreover (1.10) Z
d + l −1 1 1
s s s s lim t u(yt , t) − v(yt , t) − Bh(y) t→∞
5
u0
d
R
= 0,
L∞ (Rd )
s where h is given by b h(ξ) = e−|ξ| |ξ|l .
Let us point out that the asymptotic expansion given by (1.5) involves Kt (and its derivatives) which is not explicit. On the other hand, the two-term asymptotic expansion (1.9) involves Gst , a well known explicit kernel (v is just the convolution of Gst and u0 ). However, our ideas and methods allow us to find only two terms in the latter expansion. 2. Proofs of the results 2.1. Preliminaries. First, let us obtain a representation of the solution using Fourier variables. A proof of existence and uniqueness of solutions using the Fourier transform (see [13]) is given in [5]. We repeat the main arguments here for the sake of completeness. Theorem 2.1. Let u0 ∈ L1 (Rd ) such that ub0 ∈ L1 (Rd ). There exists a unique solution u ∈ C 0 ([0, ∞); L1 (Rd )) of (1.1), and it is given by u b(ξ, t) = e(J(ξ)−1)t ub0 (ξ). b
Proof. We have Z J(x − y)u(y, t) dy − u(x, t).
ut (x, t) = J ∗ u − u(x, t) = Rd
b − 1). Applying the Fourier transform we obtain u bt (ξ, t) = u b(ξ, t)(J(ξ) b b (J(ξ)−1)t 1 d (J(ξ)−1)t Hence, u b(ξ, t) = e ub0 (ξ). Since ub0 ∈ L (R ) and e is continuous and bounded, the result follows by taking the inverse of the Fourier transform. Now we prove a lemma concerning the fundamental solution of (1.1). Lemma 2.1. Let J ∈ S(Rd ), the space of rapidly decreasing functions. The fundamental solution of (1.1), that is the solution of (1.1) with initial condition u0 = δ0 , can be decomposed as (2.11)
w(x, t) = e−t δ0 (x) + Kt (x),
where the function Kt is smooth and given by b ct (ξ) = e−t (etJ(ξ) K − 1).
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L. I. IGNAT AND J.D. ROSSI
Moreover, if u is a solution of (1.1) it can be written as Z u(x, t) = (w ∗ u0 )(x, t) = w(x − z, t)u0 (z) dz. Rd
b − 1). Proof. By the previous result we have w bt (ξ, t) = w(ξ, b t)(J(ξ) b Hence, as the initial datum verifies ub0 = δ0 = 1, we get w(ξ, b t) = e(J(ξ)−1)t = e−t + e−t (eJ(ξ)t − 1). b
b
The first part of the lemma follows applying the inverse Fourier transform in S(Rd ). To finish the proof we just observe that w ∗ u0 is a solution of (1.1) (just use Fubini’s theorem) with (w ∗ u0 )(x, 0) = u0 (x). b →0 Remark 2.1. The above proof together with the fact that J(ξ) (since J ∈ L1 (Rd )) shows that if Jb ∈ L1 (Rd ) then the same decomposition (2.11) holds and the result also applies. To prove our result we need some estimates on the kernel Kt . 2.2. Estimates on Kt . In this subsection we obtain the long time behavior of the kernel Kt and its derivatives. The behavior of Lq (Rd )-norms with 2 6 q 6 ∞ follows by HausdorffYoung’s inequality in the case q = ∞ and Plancherel’s identity for q = 2. Lemma 2.2. Let 2 6 q 6 ∞ and J satisfying (1.3) and (1.4). Then for all indexes α such that |α| < m − d there exists a constant c(q, α) such that |α| d 1 k∂ α Kt kLq (Rd ) 6 c(q, α) t− s (1− q )− s holds for sufficiently large t. Moreover, if J satisfies (1.6) then the same result holds with no restriction on α. Proof of Lemma 2.2. We consider the cases q = 2 and q = ∞. The other cases follow by interpolation. We denote by e.s. the exponentially small terms. First, let us consider the case q = ∞. Using the definition of Kt , b c Kt (ξ) = e−t (etJ(ξ) − 1), we get, for any x ∈ Rd , Z b α −t |∂ Kt (x)| 6 e |ξ||α| |etJ(ξ) − 1| dξ. Rd
Using that |ey − 1| 6 2|y| for |y| small, say |y| 6 c0 , we obtain that 2t b b |etJ(ξ) − 1| 6 2t|J(ξ)| 6 m |ξ|
ASYMPTOTICS FOR NONLOCAL DIFFUSION
7
1
for all |ξ| ≥ h(t) = (c0 t) m . Then Z Z b −t |α| tJ(ξ) −t e |ξ| |e − 1| dξ . te |ξ|≥h(t)
|ξ|≥h(t)
|ξ||α| dξ 6 te−t c(m − |α|) m |ξ|
provided that |α| < m − d. Is easy to see that if (1.6) holds no restriction on the indexes α has to be assumed. It remains to estimate Z b −t e |ξ||α| |etJ(ξ) − 1|dξ. |ξ|6h(t)
We observe that the term e−t we concentrate on I(t) = e
−t
R |ξ|6h(t)
Z
|ξ||α| dξ is exponentially small, so |etJ(ξ) ||ξ||α| dξ. b
|ξ|6h(t)
Now, let us choose R > 0 such that |ξ|s for all |ξ| 6 R. 2 Once R is fixed, there exists δ > 0 with b |J(ξ)| 61−
(2.12)
b |J(ξ)| 6 1 − δ for all |ξ| ≥ R.
(2.13) Then −t
Z
|α|
−t
Z
|I(t)| 6 e |e ||ξ| dξ + e |etJ(ξ) ||ξ||α| dξ |ξ|6R R6|ξ|6h(t) Z Z b . et(|J(ξ)|−1) |ξ||α| dξ + e−tδ |ξ||α| dξ |ξ|6R R6|ξ|6h(t) Z t|ξ|s . e− 2 |ξ||α| + e.s. |ξ|6R Z |α| |η|s |α| − s − ds − 2 |α| − s − ds = t e |η| + e.s. . t . 1 b tJ(ξ)
b
|η|6Rt s
Now, for q = 2, by Plancherel’s identity we have Z b α 2 −2t k∂ Kt kL2 (Rd ) ≤ e |etJ(ξ) − 1|2 |ξ|2|α| dξ. Rd
Putting out the exponentially small terms, it remains to estimate Z b |et(J(ξ)−1) |2 |ξ|2|α| dξ, |ξ|6R
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L. I. IGNAT AND J.D. ROSSI
where R is given by (2.12). The behavior of Jb near zero gives Z Z 2|α| d s b t(J(ξ)−1) 2 2|α| |e | |ξ| dξ . e−t|ξ| |ξ|2|α| dξ . t− s − s , |ξ|6R
|ξ|6R
which finishes the proof.
Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. Following [9] we obtain that the initial condition u0 ∈ L1 (Rd , 1 + |x|k+1 ) has the following decomposition X X (−1)|α| Z α u0 x dx Dα δ0 + D α Fα u0 = α! |α|=k+1
|α|6k
where kFα kL1 (Rd ) 6 ku0 kL1 (Rd , |x|k+1 ) for all multi-indexes α with |α| = k + 1. In view of (2.11) the solution u of (1.1) satisfies u(x, t) = e−t u0 (x) + (Kt ∗ u0 )(x). The first term being exponentially small it suffices to analyze the long time behavior of Kt ∗ u0 . Using the above decomposition and Lemma 2.2 we get
X (−1)|α| Z
u0 (x)xα dx ∂ α Kt 6
Kt ∗ u0 − α! Lq (Rd ) |α|6k X 6 k∂ α Kt ∗ Fα kLq (Rd ) |α|=k+1
X
6
k∂ α Kt kLq (Rd ) kFα kL1 (Rd )
|α|=k+1 d
1
. t− s (1− q ) t−
(k+1) s
This ends the proof.
ku0 kL1 (Rd , |x|k+1 ) .
2.3. Asymptotics for the higher order terms. In this subsection we prove Theorem 1.2. Proof of Theorem 1.2. Recall that we have defined Gst by its Fourier cst = exp(−t|ξ|s ). transform G We consider the case q = ∞, the case q = 2 can be handled similarly and the rest of the cases, 2 < q < ∞, follow again by interpolation. Writing each of the two terms in Fourier variables we obtain Z s b α α s k∂ Kt − ∂ Gt kL∞ (Rd ) 6 |ξ||α| e−t (etJ(ξ) − 1) − e−t|ξ| dξ. Rd
ASYMPTOTICS FOR NONLOCAL DIFFUSION
Let us choose a positive R such that b − 1 + |ξ|s 6 C|ξ|r+s , J(ξ)
9
for |ξ| 6 R,
satisfying (2.13) for some δ > 0. For |ξ| ≥ R all the terms are exponentially small as t → ∞. Thus the behavior of the difference ∂ α Kt −∂ α Gt is given by the following integral: Z b s I(t) = |ξ||α| et(J(ξ)−1) − e−t|ξ| dξ. |ξ|6R
In view of the elementary inequality |ey − 1| 6 c(R)|y| for all |y| 6 R we obtain that Z s s b |ξ||α| e−t|ξ| et(J(ξ)−1+|ξ| ) − 1 dξ I(t) = Z |ξ|6R s b − 1 + |ξ|s ) dξ . |ξ||α| e−t|ξ| t(J(ξ) |ξ|6R Z s .t |ξ||α| e−t|ξ| |ξ|s+r dξ |ξ|6R |α| − ds − rs − s
.t
.
This finishes the proof.
2.4. A different approach. In this final subsection we obtain the first two terms in the asymptotic expansion of the solution under less restrictive hypotheses on J. Proof of Theorem 1.3. The method that we use here is just to estimate l the difference ku(t)−v(t)−Bt(−∆) 2 v(t)kLq (Rd ) using Fourier variables. As before, it is enough to consider the cases q = 2 and q = ∞. We analyze the case q = ∞, the case q = 2 follows in the same manner by applying Plancherel’s identity. By Hausdorff-Young’s inequality we get l
ku(t) − v(t) − tB(−∆) 2 v(t)kL∞ (Rd ) Z \2l 6 b(t, ξ) − vb(t, ξ) − tB (−∆) v(t, ξ) dξ u ZRd b t(J(ξ)−1) −t|ξ|s l = −e (1 + tB|ξ| )||ub0 (ξ) dξ. e Rd
As before, let us choose R > 0 such that |ξ|s b |J(ξ)| 61− , |ξ| 6 R. 2 Then there exists δ > 0 such that b |J(ξ)| 6 1 − δ, |ξ| ≥ R.
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L. I. IGNAT AND J.D. ROSSI
Hence
Z
|et(J(ξ)−1) ||ub0 (ξ)|dξ 6 e−δt kub0 kL1 (Rd ) b
|ξ|≥R
and Z
b t(J(ξ)−1)
1 t− l 6|ξ|6R
|e
Z
s
||ub0 (ξ)|dξ 6 kub0 kL∞ (Rd ) e−t|ξ| /2 −1 t l 6|ξ|6R Z s d −|ξ|s /2 − ds −t1− l /4 . t− s e dξ . t e . 1 1 |ξ|≥t s − l
Also Z 1 |ξ|≥t− l
−t|ξ|s
Z
l
(1 + tB|ξ| )|ub0 (ξ)|dξ . kub0 kL∞ (Rd ) Z −|η|s 1− ds − sl e .t |η|l dξ 1−1 |η|≥t s l Z s s 1− ds − sl −t1− l /2 .t e e−|η| /2 |η|l dξ. 1 1
e
1 |ξ|≥t− l
s
e−t|ξ| t|ξ|l dξ
|η|≥t s − l
Therefore, we have to analyze Z s b I(t) = |et(J(ξ)−1) − e−t|ξ| (1 + tB|ξ|l )||ub0 (ξ)| dξ. 1 |ξ|6t− l
b = 1 − |ξ|s + B|ξ|l + |ξ|l f (ξ) where f (ξ) → 0 as |ξ| → 0. We write J(ξ) Thus I(t) 6 I1 (t) + I2 (t) where Z s l l e−t|ξ| |eBt|ξ| +t|ξ| f (ξ) − (1 + Bt|ξ|l + t|ξ|l f (ξ))||ub0 (ξ)| dξ I1 (t) = 1 |ξ|6t− l
and
Z I2 (t) =
s
1 |ξ|6t− l
e−t|ξ| t|ξ|l f (ξ)|ub0 (ξ)| dξ.
For I1 we have Z
s
e−t|ξ| (t|ξ|l + t|ξ|l |f (ξ)|)2 dξ I1 (t) 6 kub0 kL∞ (Rd ) −1 l |ξ|6t Z d 2l s . e−t|ξ| t2 |ξ|2l dξ . t− s +2− s 1 |ξ|6t− l
and then
d
l
l
t s + s −1 I1 (t) . t1− s → 0, It remains to prove that d
l
t s + s −1 I2 (t) → 0,
t → ∞.
t → ∞.
ASYMPTOTICS FOR NONLOCAL DIFFUSION
Making a change of variable we obtain Z d l −1+ s I (t) 6 ku ts b0 kL∞ (Rd ) 2 1
s
1 |ξ|6t s − l
11
1
e−|ξ| |ξ|l f (ξt− s ) dξ.
The integrand is dominated by kf kL∞ (Rd ) |ξ|l exp(−|ξ|s ), which belongs 1 to L1 (Rd ). Hence, as f (ξ/t s ) → 0 when t → ∞, this shows that d
l
t s + s −1 I2 (t) → 0, and finishes the proof of (1.9). Thanks to (1.9), the proof of (1.10) is reduced to show that
Z
d+ l
l 1
s s [(−∆) 2 v](yt s , t) − h(y) lim t u 0
∞ d = 0. t→∞ d R
L (R )
d
For any y ∈ R by making a change of variables we obtain Z l 1 l 1 d s + e−|ξ| |ξ|l eiyξ ub0 (ξ/t s ). I(y, t) = t s s [(−∆) 2 v](yt s , t) = Rd
Thus, using the dominated convergence theorem we obtain
Z Z
1 s
I(y, t) − h(y) e−|ξ| |ξ|l |ub0 (ξ/t s ) − ub0 (0)| dξ → 0 u0 6
d d ∞ d R
L (R )
R
as t → ∞.
Acknowledgements. L. I. Ignat partially supported by the grant CEx06-M3-102, the reintegration grant RP-3, contract 4-01/10/2007 of CNCSIS Romania and by the grant MTM2005-00714 of the Spanish MEC, the DOMINO Project CIT-370200-2005-10 in the PROFIT program (Spain). J. D. Rossi partially supported by Fundaci´on Antorchas, CONICET and ANPCyT PICT 05009 (Argentina). References [1] S. Barza, V. Burenkov, J. Peˇcari´c and L.-E. Persson. Sharp multidimensional multiplicative inequalities for weighted Lp spaces with homogeneous weights. Math. Inequal. Appl., 1(1), 53–67, (1998). [2] P. Bates and A. Chmaj. A discrete convolution model for phase transitions. Arch. Rat. Mech. Anal., 150, 281–305, (1999). [3] 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). [4] C. Carrillo and P. Fife. Spatial effects in discrete generation population models. J. Math. Biol., 50(2), 161–188, (2005). [5] E. Chasseigne, M. Chaves and J. D. Rossi. Asymptotic behaviour for nonlocal diffusion equations. J. Math. Pures Appl., 86, (2006), 271–291. [6] X Chen. Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differential Equations, 2, 125–160, (1997).
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[7] 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). [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. Preprint. [9] J. Duoandikoetxea and E. Zuazua. Moments, masses de Dirac et decomposition de fonctions. (Moments, Dirac deltas and expansion of functions). C. R. Acad. Sci. Paris Ser. I Math., 315(6), 693-698, (1992). [10] M. Escobedo and E. Zuazua. Large time behavior for convection-diffusion equations in RN . J. Funct. Anal., 100(1), 119-161, (1991). [11] P. Fife. Some nonclassical trends in parabolic and parabolic-like evolutions. Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003. [12] 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). [13] T. W. K¨ orner. Fourier analysis. Cambridge University Press, Cambridge, 1988. [14] X. Wang. Metaestability and stability of patterns in a convolution model for phase transitions. J. Differential Equations, 183, 434–461, (2002). [15] 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 Institute of Mathematics of the Romanian Academy, P.O.Box 1-764, RO-014700 Bucharest, Romania. E-mail address:
[email protected] Julio D. Rossi IMDEA Matematicas, C-IX, Campus UAM, Madrid, Spain. ´ tica, FCEyN UBA (1428) On leave from Departamento de Matema Buenos Aires, Argentina. E-mail address:
[email protected] Web page: http://mate.dm.uba.ar/∼jrossi
