ON NONLINEAR NONLOCAL DIFFUSION EQUATIONS
ANDREW DICKERSON, EMMANUEL ESTRADA, TUCKER HARTLAND, WILLIAM
JAMIESON, LAUREN LEWANDOWSKI, PETRONELA RADU, AND RAVI SHANKAR
Abstract. This is a study of a class of nonlocal nonlinear diffusion equations
(NNDEs). We present several new qualitative results for nonlocal Dirichlet
problems. It is shown that solutions with positive initial data remain positive
through time, even for nonlinear problems; in addition, we prove that solutions
to these equations obey a strong maximum principle. A striking result shows
that nonlocal solutions must have some irregularity at the boundary; otherwise,
we have ill-posedness of the initial value problem. In addition, for several
general classes of NNDEs and nonlocal reaction-diffusion equations, we obtain
explicit differential inequalities that bound above and below the solutions’
energy decay. Explicit solutions to these inequalities (hence, explicit rates of
decay) are found for some special cases. Finally, we present a proof of the
nonlocal Poincaré inequality with an explicit constant.
1. Introduction
Our work concerns the following nonlocal nonlinear reaction diffusion equation
for function u:
ut = D[kGu] + R.
(1.1)
Here, D represents a “nonlocal divergence” and G a “nonlocal gradient”; these
are nonlocal operators that suitably extend the classical ∇· (div) and ∇ (grad)
operators. Equation (1.1) represents a nonlocal extension of the classical reactiondiffusion equation:
ut = ∇ · (k∇u) + R.
(1.2)
In both equations (1.1) and (1.2), functions k and R are respectively called the
“diffusion coefficient” and the “reactivity”.
The integro-differential equation (1.1) contains, as a special case, the nonlocal
linear diffusion equation
ut = µ ∗ u − u
(1.3)
found in [4], where the nonlocal operators allow us to write D(Gu) = µ ∗ u − u with
µ ∈ L1 denoting a probability density. Equation (1.1) also includes other nonlinear
nonlocal versions of diffusion equations, such as nonlocal versions of the porous
medium equations, the nonlocal p-Laplace equation [4, Chapter 6], and the fast
diffusion equation [4, Chapter 5]. For example, the latter has the following form:
(γ(u))t = µ ∗ u − u.
2010 Mathematics Subject Classification. Primary: 45P05 Secondary: 35L35,
Key words and phrases. nonlocal diffusion, nonlinear diffusion, peridynamics, partial integral
differential equations, decay estimates.
1
2
A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
Assuming γ is invertible, a change of variables shows that we recover (1.1) for an
appropriate choice of k.
There are many natural phenomena that are well modeled by nonlocal processes.
Some applications of nonlocal models include image processing [14] and biological
systems, including population models [5] and swarming systems [23]. See also [4]
for applications in nonlocal diffusion and the references therein for other types of
applications.
The success of using nonlocal operators to model these diverse phenomena stems
from the lack of spatial derivatives in the associated integro-differential equations.
By exchanging spatial differential operators for integral operators, we allow for
more general and realistic solutions that require less regularity, such as functions
with discontinuities or singularities. Thus, physical phenomena that exhibit singular solutions and/or nonlocality (dynamic fracture exhibits both features - see
the peridynamic formulation introduced in [27]) can not be modeled by classical
equations, but can be successfully modeled through integral equations.
Our results shed light on the qualitative behaviors of classes of nonlinear nonlocal diffusion equations. These results extend and complement others previously
obtained in the nonlocal literature. Wellposedness has been studied in [4] for linear
and p-Laplacian equations, and we also mention results for reaction-diffusion equations [28] and anomalous diffusion equations [9]. Decay rate estimates have been
obtained in [19] for local porous medium, fast diffusion, and p-Laplacian equations,
in [4, 12, 10, 18, 16] for nonlocal linear and p-Laplacian equations, and in [4] for
more general reaction-diffusion equations. Nonlinear blowup in reaction-diffusion
equations has been established for the local case in [21]; in [25] for a linear diffusion
equation with a reaction term; in [11] for nonlocal p-Laplacian equations; and in
[29] for inhomogeneous problems.
The main contributions of this paper are:
• Positivity of solutions: We show that solutions to nonlocal diffusion
equations with nonnegative initial data will remain nonnegative for all future times t. This comparison principle is well known for linear nonlocal
diffusion [4].
• Strong maximum principle: We show that the nonlinear equation (1.1)
admits a strong maximum principle similar to those of classical diffusion
(see e.g. [26] pg 34), linear diffusion [6, 13], and partial integro-differential
equations [20].
• Irregularity at the boundary: We show that (1.1) admits only the
trivial solution for all (t, x) ∈ [0, T ]× Ω̄ if we impose C([0, T ]× Ω̄) regularity
on the solution u. Unless the initial condition u0 is trivial, the problem
becomes then ill-posed when searching for regular solutions. This striking
result shows a clear dichotomy between weak solutions and nonlocal ones,
in the sense that weak solutions with sufficient regularity satisfy a problem
in the classical sense, whereas nonlocal solutions may not be assumed to be
regular, in particular across the boundary.
• Decay rates: We obtain explicit energy decay rates for several classes
of diffusion coefficients k and choices for the reactivity R. We also derive
differential inequalities for energy decay lower bounds (in general, these
bounds are suboptimal). When R = 0, the solution to the inequality is a
NONLOCAL DIFFUSION EQUATIONS
3
decaying exponential. For nonzero R, we use this lower bound to identify
equations whose solutions exhibit nonlinear blowup.
In § 2, we give an overview of nonlocal diffusion equations and the nonlocal
framework we use in this paper. Sections 3, 4, and 5 include proofs of the comparison principle, strong maximum principle, and irregularity result, respectively.
Finally, in § 6, we discuss energy decay rate estimates.
2. Nonlocal nonlinear diffusion
After discussing the nonlinear nonlocal diffusion problem to solve, we discuss
how this problem connects with the operator equation presented in (1.1). We define
these nonlocal operators and use them to derive our nonlocal nonlinear diffusion
equation.
2.1. The initial-boundary value problem. Let Ω and Γ be bounded open subsets of Rn with Γ a collar surrounding Ω. We write (1.1) explicitly as a nonlocal
nonlinear diffusion equation:
Z
ut (t, x) =
k[u(t, y) − u(t, x)]µ dy + R(t, x, u(t, x)), (t, x) ∈ (0, T ) × Ω. (2.1)
Ω∪Γ
Above we have denoted k = k(t, x, y, u(t, x), u(t, y)) and µ(x, y) = µ(x − y). We
solve (2.1) for u : [0, T ) × (Ω ∪ Γ) → R subject to the Dirichlet boundary condition
u(t, x) = g(t, x),
(t, x) ∈ (0, T ) × Γ
(2.2)
x ∈ Ω ∪ Γ.
(2.3)
and with initial condition
u(0, x) = u0 (x),
We assume that k ≥ 0 is symmetric and µ to be radial so they satisfy:
k(t, x, y, u, v) = k(t, y, x, v, u),
(2.4)
µ(z) = µ(|z|).
For brevity, we will often omit the arguments of k and µ in our computations
whenever the context will make it clear what they are. In addition, the reaction
term R will be taken to be zero unless otherwise stated.
We now state the function spaces to which µ, k, and u belong. In all cases, we
assume that µ(|z|) ∈ L1 (Ω ∪ Γ) and that k is:
(k1) locally bounded in its first three arguments, i.e.
k(·, ·, ·, v, u) ∈ L∞ (R, Rn , Rn ) for every v, u ∈ R;
(k2) continuous in its last two arguments.
Additionally, we will impose some growth requirement properties of k. Thus, we
define the exponent sk as follows:
(
)
sk := inf
s≥0:
lim sup
|k(t, x, y, u, v)|/(|u| + |v|)s < ∞
∪ {∞}.
(2.5)
min(|u|,|v|)→∞
For example, if k(t, x, y, u, v) = (u + v)s for some s ≥ 0, then sk = s. On the other
hand, if k(t, x, y, u, v) = exp((u − v)2 ), then s = ∞.
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A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
For any p, q ≥ 1 satisfying 1/p + 1/q = 1 and p, q ≥ 1, we let the appropriate
function spaces be as follows:
µ ∈ L1 (Rn ) ∩ Lq (Rn ),
(2.6)
u ∈ Lp(1+sk ) (Ω ∪ Γ).
This choice ensures that the integral in (2.1) is well-defined.
2.2. Nonlocal operators and spaces. Following [7], we define the set of nonlocal
operators used in (1.1). Using such a nonlocal framework is advantageous, since
it allows us to draw powerful analogies with classical calculus; for example, this
framework gives a natural extension of integration by parts.
Let α : Rn → Rn be an anti-symmetric function in the sense that α(−z) = −α(z).
For u : Ω ∪ Γ → R and f : (Ω ∪ Γ)2 → Rn , we define:
(i) Nonlocal (two-point) gradient
Gu(x, y) := (u(y) − u(x)) α(y − x),
x, y ∈ Ω ∪ Γ,
(ii) Nonlocal Divergence
Z
Df (x) :=
(f (x, y) + f (y, x)) · α(x, y) dy,
x ∈ Ω.
(2.7)
(2.8)
Ω∪Γ
Composing these two operators with a (symmetric) two-point function k : (Ω ∪
Γ)2 → R gives a generalization of ∇ · (k∇u):
(iii) Nonlocal Composition
Z
D[kGu](x) = 2
k(x, y)(u(y) − u(x))|α(x − y)|2 dy
Ω∪Γ
Z
(2.9)
=
k(x, y)(u(y) − u(x))µ(x − y) dy, x ∈ Ω,
Ω∪Γ
where we defined
µ(z) := 2|α(z)|2 ,
z ∈ Rn .
(2.10)
We call µ the “kernel” of the nonlocal integral operator D(kG). We take it to be
integrable and radially symmetric (cf. (2.4) and (2.6)).
The definition (2.10) demonstrates the consistency between the operator equation (1.1) and the integro-differential equation (2.1). We interchange between each
formulation whenever it is convenient.
We also see that this formulation is consistent with other formulations of nonlocal
diffusion equations. Consider
R the nonlocal linear diffusion equation (1.3). Let k = 1
in (2.1) and suppose that Rn µ(z) dz = 1. Then (2.9) becomes:
Z
DGu(x) := Lu(x) =
u(y)µ(x − y) dy − u(x) = µ ∗ u(x) − u(x), x ∈ Ω.
Ω∪Γ
This recovers the right hand side of (1.3). We call L the “nonlocal Laplacian”, since
it generalizes the classical Laplace operator ∆ = ∇2 .
We take µ to be supported in the δ-ball Bδ (0) for some δ > 0. In peridynamics, δ
is usually called the “horizon” of the kernel µ. Given this definition of a “localized
kernel”, we define our domain Ω ∪ Γ as follows. We let Ω be a bounded, smooth,
connected open subset of Rn with positive measure |Ω|, and we let Γ be the set of
those points in Rn not in Ω of a distance less than δ from Ω. In other words, Γ is an
NONLOCAL DIFFUSION EQUATIONS
5
open “collar” of width δ that surrounds the “body” Ω. We do not treat problems
on Rn or kernels µ with infinite horizon δ in this paper.
2.3. Derivation of (2.1). We motivate (2.1) with parallel derivations of the classical (1.2) and nonlocal (1.1) diffusion equations. We do this by combining the
general local conservation law with a modified Fick’s law for the diffusion flux.
The local and nonlocal conservation laws for a quantity u : [0, T ) × Rn → R are
as follows:
ut = −∇ · q,
(2.11)
ut = −DQ.
Here, q : [0, T ) × Rn and Q : [0, T ) × (Rn )2 are respectively the local and nonlocal
fluxes.
These conservation laws (2.11) state that the change of u within a small region of
space must be related to the flux (i.e. flow) q of u through this region’s boundary.
To make this an equation that determines u, we must assume a constitutive law for
the fluxes q and Q. We use the classical and nonlocal Fick laws of diffusion:
q(t, x) = −kL (t, x, u(t, x))∇u(t, x),
(2.12)
Q(t, x, y) = −kN (t, x, y, u(t, x), u(t, x))Gu(t, x, y).
Here, kL and kN are respectively the local and nonlocal diffusion coefficients. They
are nonnegative functions that depend on the physical medium in consideration.
We require that kN (t, x, x, u, u) = kL (t, x, u) for consistency between the local and
nonlocal frameworks.
Combining (2.11) and (2.12) yields (1.1) and (1.2). Using the operator definition
(2.9), we obtain (2.1). We see that this nonlocal formulation allows for drawing
strong analogies with the physically accepted classical framework.
2.4. Nonlocal Operations. The nonlocal operators G and D satisfy some useful
identities analogous to those of their classical counterparts.
Proposition 2.1 (Nonlocal integration by parts). We have
Z
Z
Z
u(x)Df (x) dx = −
Gu(x, y) · f (x, y) dx dy.
Ω∪Γ
Ω∪Γ
(2.13)
Ω∪Γ
We frequently use the following identity. It allows us to rewrite the equation
(2.1) for u in terms of one for kukL2 (Ω) .
Corollary 2.2. Suppose u vanishes on Γ. Then for a two-point function f , the
following identity holds:
Z
Z
Z
u(x)D[f Gu](x) dx = −
f (x, y)|Gu(x, y)|2 dx dy.
(2.14)
Ω∪Γ?
Ω∪Γ
Ω∪Γ
Poincaré’s inequality, in addition to being essential for proving the wellposedness of linear steady-state diffusion [8], is critical for obtaining energy decay rate
estimates on bounded domains. We present the nonlocal Poincaré inequality of
[1, 2, 3, 22] in the following form and give a proof in the Appendix.
Theorem 2.3 (Nonlocal Poincaré’s inequality). Let u : Ω ∪ Γ → R vanish on
Γ. Then, for the kernel α : Rn → Rn of (2.7) and (2.8), we have the following
estimate:
Ckuk2L2 (Ω) ≤ kG(u)k2L2 ((Ω∪Γ)2 ) ,
(2.15)
6
A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
where C > 0 depends on α, Ω ∪ Γ, and δ.
The proof is a modification to the one in [15] used to derive an explicit form for
a (local) Poincaré constant. We use this approach to find an explicit constant for
the nonlocal inequality. An explicit (even if not necessarily optimal) constant is
relevant for comparing numerical simulations to theoretical results, for example.
3. Comparison principle
The main result of this section shows that if the initial data u0 is non-negative,
then the solution u remains non-negative in time. This result was proven in [4]
for linear equations (i.e. k(.) = 1).; here we extend it to nonlinear equations with
non-negative conductivities k(t, x, y, u(x), u(y)).
We understand that a solution u to the problem (2.1) solves the following
(Volterra-Fredholm) integral equation for x in Ω and t > 0:
Z tZ
u(t, x) = u0 (x) +
k(s, x, y, u(s, x), u(s, y))[u(s, y) − u(s, x)]µ dy ds (3.1)
0
Rn
Of course, u solves (3.1) subject to the boundary condition in (2.1). Note that, by
virtue of solving (3.1), we know that u − u0 is in C 1 [0, T ) ∩ C(Ω).
Theorem 3.1 (Positivity Preservation). Assume u in C 1 [0, T ) solves (2.1) for
non-negative conductivity k. If u0 ≥ 0 a.e. on Ω ∪ Γ, and if g ≥ 0 a.e. on (0, t) × Γ,
then u ≥ 0 a.e. on [0, T ) × (Ω ∪ Γ).
Proof. Suppose this is false. Since u0 ≥ 0 a.e., continuity in time shows that there
is a smallest time t0 ≥ 0 such that:
(i) u(t0 , x) ≥ 0 a.e. on Ω,
(ii) there is a set U ⊂ Ω of positive measure such that ut (t0 , x) < 0 and u(t0 , x) =
0 everywhere on U .
By virtue of u being in C 1 [0, T ), (2.1) holds at t = 0 (hence at t0 ≥ 0). Let x0
be in U . Since (2.1) holds at x0 , and since u(t0 , x0 ) = 0, we have:
Z
Z
ut (t0 , x0 ) =
µ(|x − y|)k[u(t0 , y) − u(t0 , x0 )] dy =
µ(|x − y|)ku(t0 , y) dy.
Rn
Rn
Since µ and k are non-negative, and since u(t0 , y) ≥ 0 a.e. on Ω ∪ Γ by (i), we
see that ut (t0 , x0 ) ≥ 0, a contradiction to (ii).
This result is a natural generalization of Theorem 3.1. We use it later to prove
irregularity at the boundary (cf. Theorem 5.1).
Corollary 3.2. Suppose that u ∈ C 1 [0, T ) solves (2.1) for non-negative conductivity k, and suppose that g+ (t) = supx∈Γ g(t, x) and g− (t) = inf x∈Γ g(t, x) exist. If
u0 ≥ g+ (0) (≤ g− (0)) a.e. in Ω, then u(t, .) ≥ g− (t) (resp. ≤ g+ (t)) a.e. in Ω ∪ Γ
for each t ≥ 0.
Proof. Apply Theorem 3.1 to u − g− (resp. g+ − u).
4. Strong Maximum Principle
We show that solutions u to (2.1) that are in C 1 ([0, T ), L∞ (Ω∪Γ)) obey a strong
maximum principle entirely analogous to that of the classical heat equation. To do
NONLOCAL DIFFUSION EQUATIONS
7
this, we need some preliminary results. We suppose that u satisfies a constant-intime Dirichlet condition:
u(t, x) ≡ g(x) in (0, T ) × Γ,
(4.1)
where g is in L∞ (Γ). We assume that this boundary condition is compatible with
the initial condition u0 in (2.3).
We show that the L∞ (Ω∪Γ) norm of solutions to (2.1) decays with time. Explicit
decay estimates were obtained in [17] for the nonlocal Cauchy problem involving
linear diffusion and nonlinear convection.
Lemma 4.1 (L∞ decay). Suppose u in C 1 ([0, T ), L∞ (Ω ∪ Γ)) solves (2.1) subject
to (2.3) and (4.1). Then dku(., t)k∞ /dt ≤ 0 for each t > 0, where k.k∞ denotes
the essential supremum.
Proof. For some t > 0, there exists a nonempty set X contained in Ω ∪ Γ such that
ess lim supy→x0 |u(t, y)| ≥ |u(t, x)| a.e. in Ω ∪ Γ for each x0 in X. There are three
cases to consider.
(i) If X ⊂ Γ, then it is clear from (4.1) that dku(t, )k∞ /dt = 0.
(ii) On the other hand, if there is some x0 ∈ X ∩ Ω, then (2.1) holds at x0 :
Z
ut (x0 , t) =
kµ(x0 − y)[u(t, y) − u(x0 , t)] dy.
Bδ (x0 )
If u(t, x0 ) ≥ u(t, y) a.e. in Ω ∪ Γ, then ut (t, x0 ) ≤ 0. Similarly, if u(t, x0 ) ≤
u(t, y) a.e. in Ω∪Γ, then ut (t, x0 ) ≥ 0. Either way, we find that ∂t |u(t, x0 )| ≤
0, which gives the result.
(iii) Suppose that X ⊂ ∂Γ. If X ⊂ ∂Γ \ ∂Ω, then ess lim supy→x0 |g(y)| is
constant in time, and the first case holds. On the other hand, if X ⊂ ∂Ω∩∂Γ
with x0 ∈ X, then we can approximate x0 arbitrarily well by a sequence
of points xn in Ω such that |u(t, xn )| % |u(t, x0 )|. We pass to the limit in
(2.1) for x = xn and recover the second case.
The next lemma is trivial.
Lemma 4.2 (Transformation to homogeneous Dirichlet conditions). If u solves
(2.1) subject to (2.3) and (4.1), and if g is a constant, then v := u − g solves (2.1)
with k̄(x, y, v1 , v2 ) := k(x, y, v1 + g, v2 + g) in place of k in (2.1), v0 := u0 − g
in place of u0 in (2.3), and 0 in place of g in (4.1), where k̄ satisfies the same
properties (non-negativity, symmetry) as k.
The next inequality is useful for obtaining lower bounds.
Lemma 4.3 (Reverse Poincaré’s Inequality). Let u : Ω ∪ Γ → R vanish on Γ. Then
kGuk2L2 ((Ω∪Γ)2 ) ≤ C 0 kuk2 , where C 0 = 4kµkL1 .
Proof. Since u|Γ = 0, we can rewrite the norm of G as follows:
Z
Z
2
kGukL2 ((Ω∪Γ)2 ) =
|[u(y) − u(x)]α(x − y)|2 dy dx.
Ω∪Γ
Ω∪Γ
8
A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
Using the elementary inequality (a + b)2 ≤ 2a2 + 2b2 , we find that
Z
Z
2
kGukL2 ((Ω∪Γ)2 ) ≤ 2
[u(y)2 + u(x)2 ]|α(x − y)|2 dy dx
Ω∪Γ Ω∪Γ
Z
Z
=4
u(x)2
|α(x − y)|2 dy dx
Ω∪Γ
Ω∪Γ
= 2kµkL1 kuk2L2 (Ω∪Γ) ,
where we used that µ = 2|α|2 is symmetric and supported on Bδ (0).
The following result applies for constant boundary conditions and gives (in general, poor) lower bounds on the solution’s energy decay.
Lemma 4.4 (Decay Rate Lower Bound). Suppose that u in C 1 ((0, T ), L∞ (Ω ∪ Γ))
solves (2.1) subject to (2.3) and (4.1), with g constant. Then the following estimate
holds:
Z t
0
ku(t, ) − gkL2 (Ω) ≥ ku(0, ) − gkL2 (Ω) exp −C
σ(s) ds ,
(4.2)
0
0
where σ(t) = sup k(τ, x, y, φ, θ), and C is as in Lemma 4.3. The supremum is
taken over τ ∈ (0, t) (x, y) ∈ (Ω ∪ Γ)2 , and (φ, θ) ∈ [range (u0 − g)]2 .
Proof. Using v := u − g as in Lemma 4.2, we multiply the Lemma 4.2 equivalent
of (2.1) by v(t, x), integrate over Ω, and apply the identity (2.14). The following
equality holds:
Z
Z
d
2
kvkL2 (Ω) = −2
k̄[v(t, y) − v(t, x)]2 |α(x − y)|2 dy dx
dt
Ω∪Γ Ω∪Γ
Z
Z
≥ −2σ(t)
[v(t, y) − v(t, x)]2 |α(x − y)|2 dy dx
Ω∪Γ
Ω∪Γ
= −2σkGvk2L2 ((Ω∪Γ)2 ) ,
where σ is given above; note that we used Lemma 4.1 to deduce the inequality
sup(x,y,τ )∈(Ω∪Γ)2 ×(0,t) k(t, x, y, v(x, τ ), v(y, τ )) ≤ σ(t) for each t. Applying Lemma
4.3, we obtain:
d
kvk2L2 (Ω) ≥ −2σC 0 kvk2L2 (Ω) .
dt
The result follows from Gronwall’s inequality.
We can now prove the Strong Maximum Principle. Some of these ideas are used
later in proving irregularity at the boundary (cf. Theorem 5.1).
Theorem 4.1 (Strong Maximum Principle). Suppose that u in C 1 ((0, T ), L∞ (Ω ∪
Γ)) uniquely solves (2.1) subject to (2.3) and (4.1). Suppose that k satisfies condition (i) in Theorem 5.1. Let Θ = [0, T ] × Ω. If u achieves an essentially global
extremum in Θ \ ∂Θ, then g must be a constant, and u ≡ g on Ω ∪ Γ.
Proof. Assume that there is some (t0 , x0 ) in Θ\∂Θ such that u(t0 , x0 ) is an essential
extremum. Then ut (t0 , x0 ) = 0; since (2.1) holds at x0 , this implies that:
Z
0=
k[u(t, y) − u(t, x0 )]µ(y − x0 ) dy.
Bδ (x0 )
NONLOCAL DIFFUSION EQUATIONS
9
Since the integrand is one-signed almost everywhere, we find that u(t0 , .)|Bδ (x0 ) ≡
u(t0 , x0 ) a.e. Choosing a new x0 in this ball and repeating this argument (a finite
number of times less than |Ω|/|Bδ |) shows that u(t0 , .)|Ω ≡ u(t0 , x0 ). In fact, since
(2.1) holds at points x sufficiently close to ∂Ω, such that the integration range
extends into Γ, this implies that u(t0 , x0 ) = g(x) for almost every x in Γ, so that g
is constant almost everywhere. We see that u(t0 , .) ≡ g.
Consider the function w : [t0 , T ] × (Ω ∪ Γ) → R defined by w ≡ g. Observe that
this function satisfies (2.1) identically, the boundary condition (4.1), and an “initial
condition” (at t = t0 ) w(t, x0 ) = u(t, x0 ) = g. Thus, w solves our problem (for u)
on Ω ∪ Γ for all t ≥ t0 . Since the solution u to our problem is unique, we conclude
that u(t, x) = w(t, x) ≡ g on Ω ∪ Γ for t ≥ t0 .
Observe that ku − gkL2 (Ω) = 0 at t = t0 . But by Lemma 4.4, we have that
ku(t0 , ) − gkL2 (Ω) ≥ C 00 ku(0, ) − gkL2 (Ω) for some C 00 > 0. Combining these two
equations shows that, in fact, u(0, x) − g ≡ 0 a.e. in order for our extremum point
(t0 , x0 ) to exist. Since u(, ) ≡ g solves our problem and such a solution is unique,
we can only conclude that u ≡ g on Ω ∪ Γ.
5. Irregularity at the boundary
An interesting feature of the nonlocal problem (2.1) is that we cannot impose too
much regularity on solutions u without forcing them to be trivial. The conclusion
is similar to that of a maximum principle, but the hypotheses are much different.
Moreover, unless the initial data is trivial, the problem (2.1) is ill-posed in this
regularity. This is quite a contrast with classical diffusion problems, for which
solutions naturally admit strong regularity.
We prove that if u is in C 1 ([0, T ), C(Ω)) (namely, continuous on the closure of Ω),
then there are many cases for which it must be identically constant. In paricular,
if the initial data u0 is non-negative, and if u is zero on Γ, then either u0 ≡ 0,
or there is no such solution u. This situation does not arise if we allow u to be
discontinuous at ∂Ω.
Theorem 5.1. Consider the problem (2.1) for a solution u in C 1 ([0, T ), C(Ω)).
Suppose that the following hypotheses are true:
(i) Either of these two conditions: (a) k > 0 (e.g. k = 1 + |k̄|), or (b) k ≥ 0,
and k(t, x, y, u, v) = 0 implies u = ±v (e.g. k(t, x, y, u, v) = |u ∓ v|p ).
(ii) Let g± be as in Corollary 3.2, and suppose that u0 ≥ g+ (0) (≤ g− (0)) in Ω.
(iii) ∂t g− (t) ≤ 0 (resp. ∂t g+ (t) ≥ 0) for each t in (0, T ).
(iv) ess lim inf y→x g(t, y) ≡ g− (t) (resp. ess lim supy→x g(t, y) ≡ g+ (t)) for each
x in ∂Ω and each t in (0, T ). I.e. g is minimized (resp. maximized) on the boundary
of Ω.
Then, in fact, u0 (x), g(t, x), u(t, x) ≡ g− (0) on their respective domains.
Proof. Let us consider the case of u0 ≥ g+ (0) in (ii). By the continuity of u in Ω
and its differentiability in [0, T ), (2.1) holds at each x in ∂Ω and at each t ≥ 0:
Z
ut (t, x) = ∂t g− (t) =
µk[u(t, y) − u(t, x)] dy
Bδ (x)
Z
=
µk[u(t, y) − g− (t)] dy.
Bδ (x)
10
A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
By (ii) and Corollary 3.2, we have that u(t, y) ≥ g− (t). Now, (iii) shows that
ut (t, x) ≤ 0, but (ii) and (iv) show that the integrand is non-negative; thus, the
integrand is zero. By (i) and the continuity of u in Ω, this implies that u(t, y) ≡
g− (t) on Bδ (x) ∩ Ω and that u(t, y) ≡ ±g− (t) a.e. on Bδ (x) ∩ Γ. If u(t, y) ≡ −g− (t)
a.e. in Bδ (x) ∩ Γ, then since g(t, y) = u(t, y) in Γ, (iv) implies that g(t, y) ≡ 0 a.e.
in Bδ (x) ∩ Γ. In either case, we must have u(t, y) ≡ g− (t) for almost each y in
Bδ (x).
This holds for each x in ∂Ω. So, in fact, we have that g(t, x) ≡ g− (t) a.e. in
Γ and for each t ≥ 0. Since ut (t, x) = ∂t g− (t) = 0 at each t ≥ 0 from above, we
conclude that g(t, x) = g− (t) ≡ g− (0) in Γ.
Repeating this argument at some x0 in Bδ (x) ∩ Ω shows that u(t, y) ≡ g− (0)
in Bδ (x0 ) as well. Continuing a number of times less than |Ω|/|Bδ |, we find that
u(t, y) ≡ g− (0) on Ω ∪ Γ. Since this applies to u at t = 0, we see that u0 (y) ≡ g− (0)
in Ω.
The argument for the case of u0 ≤ g− (0) proceeds identically.
We specialize this rather general result to a simpler case.
Corollary 5.2. Suppose that u in C 1 ([0, T ), C(Ω)) solves the linear nonlocal diffusion equation:
Z
ut = µ ∗ u − u =
µ(x − y)[u(t, y) − u(t, x)] dy, (t, x) ∈ (0, T ) × Γ
Bδ (x)
subject to homogeneous boundary conditions:
u(t, x) = 0,
(t, x) ∈ (0, T ) × Γ.
If the initial data is non-negative, namely
u(0, x) = u0 (x) ≥ 0,
x ∈ Ω,
then, in fact, u0 (x) ≡ 0 on Ω, and u ≡ 0 on Ω ∪ Γ.
Remark 5.1. The overly restrictive condition of u being in C 1 ([0, T ), C(Ω)) was
actually (incorrectly) assumed in [4].
6. Decay Rates
We showed in Lemma 4.4 that the energy decay of solutions u to the nonlocal
nonlinear diffusion equation (2.1) can be bounded from below by an exponential.
We now investigate decay estimates from above. For special classes of diffusion
coefficients k and reactivities R in (2.1), we are able to obtain an explicit differential
inequality for the energy kukL2 (Ω) . We solve this inequality for several practically
interesting special cases.
6.1. Classes of equations that admit estimation. Let us consider the following
class of diffusion coefficient:
k(t, x, y, u, v) =
N
X
1
|Ki (t, y, v) − Ki (t, x, u)|2 ,
|v − u|2 i=1
ki,− (t, x)|fi (u)| ≤ |Ki (t, x, u)| ≤ ki,+ (t, x)|fi (u)|,
1 ≤ i ≤ N.
(6.1)
(6.2)
NONLOCAL DIFFUSION EQUATIONS
11
We also consider a reaction term R in (2.1) that is given as follows:
N
R(t, x, u) = R0 (t, x)u/2 −
1 X
Ri (t, x, u),
2u i=1
2
2
ri,−
(t, x)fi (u)2 ≤ Ri (t, x, u) ≤ ri,+
fi (u)2 ,
1 ≤ i ≤ N.
(6.3)
(6.4)
Here, if they are not identically zero, we have ki,± ≥ ki,±,0 > 0 and |ri,± | ≥
ri,±,0 > 0; R0 is locally bounded. Each function fi (u) is locally Lipschitz, has
convex absolute value, vanishes at u = 0, and satisfies the following estimate for
each u:
fi (u)2 ≥ |fi (u2 )|.
(6.5)
The simplest examples of functions that obey this estimate include those for which
equality is identically obtained (e.g. f (u) = |u|p or sgn(u)|u|p for p ≥ 1), but
there are also other examples, such as any globally Lipschitz convex function that
satisfies L ≤ |f (v) − f (u)|/|v − u| ≤ L2 − 2 for some L ≥ 1 and ≥ 0 (e.g.
f (u) = u(3+2 arctan(u)/π)). We consider explicit special cases in later subsections.
These classes of coefficients k and reactivities R allow us to apply the nonlocal Poincaré inequality (Theorem 2.15) to obtain an explicit differential inequality
governing L2 norm decay. We illustrate this as follows.
Theorem 6.1 (L2 decay). Let u be a solution of (2.1) with coefficient k and reactivity R of the forms (6.1) and (6.3) subject to a homogeneous boundary condition
(i.e. g = 0 in (2.2)). If R0+ (t) := ess supx∈Ω R0 (t, x) is sufficiently small, then
kukL2 (Ω) → 0 as t → ∞.
Proof. We multiply (2.1) by u(t, x), integrate over Ω, and apply (2.14). By the
estimate (6.4), we have:
Z
Z
N
X
d
kuk2L2 (Ω) = −
2
|α[Ki (t, y, u(t, y)) − Ki (t, x, u(t, x))]|2 dy dx
dt
Ω∪Γ
Ω∪Γ
i=1
Z
Z
−
Ri (t, x, u(t, x)) dx +
R0 (t, x)u(t, x)2 dx
Ω
≤−
N X
Ω
2kGKi k2L2 (Ω∪Γ)2 + kri,− (t, x)fi (u)k2L2 (Ω) + R0+ (t)kuk2L2 (Ω) .
i=1
Applying to this equation the nonlocal Poincaré inequality, the estimate (6.2), that
the functions hi,− (t) := ess inf x∈Ω hi,− (t, x) and ri,− (t) := ess inf x∈Ω |ri,− (t, x)|
12
A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
exist, and Jensen’s inequality combined with 6.5, we obtain:
N X
d
kuk2L2 (Ω) ≤ −
2CkKi k2L2 (Ω∪Γ) + kri,− (t, x)fi (u)k2L2 (Ω) + R0+ (t)kuk2L2 (Ω)
dt
i=1
≤−
N
X
2Ckki,− (t, x)fi (u)k22 + kri,− (t, x)fi (u)k22 + R0+ (t)kuk22
i=1
≤ −|Ω|
N
X
2
2
2Cki,−
(t) + ri,−
(t)
≤ −|Ω|
|fi (u(t, x)2 )| d(x/|Ω|) + R0+ (t)kuk22
Ω
i=1
N
X
Z
2
2Cki,−
+
2
ri,−
Z
2
fi 1
u(t, x) dx + R0+ kuk22 .
|Ω|
Ω
i=1
This gives the following differential inequality for kuk2L2 (Ω) :
N
X
d
2
2
kuk2L2 (Ω) ≤ −|Ω|
2Cki,−
(t) + ri,−
(t) fi |Ω|−1 kuk2L2 (Ω) + R0+ (t)kuk2L2 (Ω) .
dt
i=1
(6.6)
There are two cases to consider. If none of the fi satisfies |fi (u)| ≥ fi,0 |u| for some
fi,0 > 0 and each u, then we impose that R0+ (t) ≤ 0. On the other hand, if each fi
for i : 1 ≤ i ≤ m and some m ≤ N satisfies this, then we only need assume that:
R0+ (t) <
m
X
2
2
(2Cki,−
(t) + ri,−
(t))fi,0 ,
t > 0.
(6.7)
i=1
Now, by virtue of the convexity of each |fi |, the fact that they each vanish at
zero, and the estimate (6.5), we see that each fi (s) = 0 in (6.6) if and only if s = 0.
It follows from basic ODE analysis and the condition (6.7) that kuk2L2 (Ω) tends
towards its unique equilibrium point (i.e. ku(, t)kL2 (Ω) → 0 as t → ∞).
Let us consider some interesting special cases of the diffusion coefficient (6.1) for
which we can explicitly solve the differential inequality for kukL2 (Ω) .
6.2. Porous Medium Equation. The classical porous medium equation (PME)
is of the form
ut = ∆um = ∇ · (mum−1 ∇u), m > 1.
(6.8)
This equation models isentropic gas through a porous medium or groundwater
infiltration.
The nonlocal extension of (6.8) is obtained from (2.1) by using the following
conductivity:
β+1
2
v
− uβ+1
k(u, v) =
, β = (m − 1)/2 > 0,
(6.9)
v−u
which corresponds to N = 1, f (u) = uβ+1 , and k± (t, x) = 1 in (6.1) and R0 =
r± = 0 in (6.3).
The differential inequality (6.6) takes the form:
h
iβ+1
d
kuk2L2 (Ω) ≤ −2C|Ω|−β kuk2L2 (Ω)
,
dt
NONLOCAL DIFFUSION EQUATIONS
13
so Gronwall’s inequality gives the following estimate:
ku(t, )k2L2 (Ω) ≤ h
|Ω|ku(0, )k2L2 (Ω)
|Ω|β + 2βCtku(0, )k2β
L2 |Ω|
i1/β .
For large times t, this tends towards |Ω|/(2βCt)1/β , so the decay rate is rational.
The PME (6.8) is often called a “slow diffusion equation”, which seems to be an
apt moniker given this decay rate. Indeed, the linear case corresponding to m = 0
has an exponential decay rate.
6.3. Reaction Diffusion Equation. We consider a Newell-Whitehead-type reaction diffusion equation of the form:
1
ut = ∆u + u(a − u2p ),
2
a, p > 0.
(6.10)
For a = p = 1, this equation describes Rayleigh-Benard convection [24].
We obtain the nonlocal extension of (6.10) from (2.1) by setting k(t, x, y, u, v) = 1
and R(t, x, u) = u(a − u2p )/2. Note that this corresponds to N = 2, f1 (u) = u,
f2 (u) = up+1 , k1,± = 1, and k2,± = 0 in (6.1)-(6.2) and corresponds to R0 = a,
r1,± = 0, and r2,± = 1 in (6.3)-(6.4).
Substituting this information into the differential inequality (6.6), we obtain:
d
kuk2L2 (Ω) ≤ (a − 2C)kuk2L2 (Ω) − |Ω|−p kuk2p+2
L2 (Ω) .
dt
Since p > 0, we impose via (6.7) that a ≤ 2C(Ω, δ, n) so that kukL2 (Ω) → 0 as
t → ∞.
The solution to this differential inequality is:
"
#1/p
(2C − a)e(a−2C)pt
2
kukL2 (Ω) ≤
.
|Ω|−p (1 − e(a−2C)pt ) + (2C − a)ku(0, )k−2p
L2 (Ω)
As t → ∞, this tends towards an exponential. However, this function decays faster
than the exponential does for small times t. This is due to the nonlinearity.
Note also that, as a % 2C, this upper bound tends toward ku(0, )k22 /[1 +
1/p
ptku(0, )k2p
, so the decay rate becomes rational, as in the PME case.
2 ]
Another interesting feature is that the value of ku(0, )kL2 (Ω) does not get “lost”
as t → ∞ for this class of equation. Indeed, for the PME, the asymptotic energy
profile becomes independent of its initial state as t → ∞. Thus suggests that some
sort of “inverse scattering method” may be possible for this equation.
6.4. Non-constant boundary conditions. For linear nonlocal reaction-diffusion,
we show that it is possible to get explicit estimates for decay rates when the boundary condition (2.2) is not identically zero or constant. This case corresponds to
k = 1 and R = R0 u in (2.1).
Proposition 6.2. Suppose that u in L2 (Ω ∪ Γ) solves the following linear nonlocal
reaction-diffusion equation:
ut = DGu + R0 u,
14
A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
subject to (2.2) and (2.3), with µ in L2 (Rn ). Suppose that both R0 and kg(t, )kL2 (Γ)
are locally bounded functions. Then the following estimate holds:
Z t
2
Φ+ (t)
2
eΦ+ (t)−Φ+ (s) kg(t, )k2L2 (Γ) ds, (6.11)
ku(t, )kL2 (Ω) ≤ e
ku(0, )kL2 (Ω) + M
0
where Φ+ is a continuous function, and M > 0 is a constant.
Proof. Let ḡ : Ω ∪ Γ → R be an extension of g that is zero in Ω. Then v := u − ḡ
satisfies the following initial-boundary volume problem:
vt = DGv + R0 v + DGḡ,
v(t, x) = 0,
(t, x) ∈ (0, T ) × Γ,
v(0, x) = v0 (x) := u0 (x) − ḡ(0, x),
x ∈ Ω ∪ Γ.
As before, we multiply the first equation by v, integrate over Ω, and apply the
identity (2.14). The following inequality holds:
Z
1 d
kvk2L2 (Ω) ≤ (R0+ (t) − C)kvk2L2 (Ω) +
v(t, x)DGḡ(t, x) dx,
2 dt
Ω
where R0+ (t) := ess supx∈Ω R0 (t, x). Applying Young’s inequality to the last term
and putting L := DG, we obtain an inequality for kuk2L2 (Ω) :
1 d
1
1
+
2
kvkL2 (Ω) ≤ R0 (t) − C +
kvk2L2 (Ω) + kLḡ(t, )k2L2 (Ω) .
2 dt
2
2
From [8], the nonlocal Laplacian L is a bounded linear operator from L2 to L2 . So,
there exists a constant M > 0 such that kLḡk2L2 (Ω) ≤ M kgk2L2 (Γ) . Since v ≡ u in
Ω, we obtain the following inequality:
d
kuk2L2 (Ω) ≤ 2R0+ (t) − 2C + 1 kuk2L2 (Ω) + M kg(t, )k2L2 (Γ) .
dt
Rt
Putting Φ+ (t) := (1−2C)t+2 0 R0+ (s) ds, the result (6.11) follows from Gronwall’s
inequality.
We can also find an explicit lower bound.
Proposition 6.3. In Proposition 6.2, the following lower bound holds:
√ Z t Φ (t)−Φ (s)
Φ− (t)
−
2
2
e −
kGg(t, , )kL2 (Γ2 ) ds,
ku(t, )kL (Ω) ≥ e
ku(0, )kL (Ω) − C 0
0
where Φ− is continuous, and C 0 is as in Lemma 4.3.
Proof. Using a similar procedure to that of Lemma 4.4, we have the following
inequality:
Z
d
v(t, x)DGḡ(t, x) dx,
kvkL2 (Ω) kvkL2 (Ω) ≥ (R0− (t) − C 0 )kvk2L2 Ω) +
dt
Ω
where R0− (t) := ess inf x∈Ω R0 (t, x).
To the last term, we apply integration by parts (2.14), Hölder’s inequality, and
Lemma 4.3. Dividing by kvk, we arrive at the following inequality:
√
d
kvkL2 (Ω) ≥ (R0− (t) − C 0 )kvkL2 (Ω) − C 0 kGḡ(t, , )kL2 (Ω∪Γ)2 .
dt
We note that v ≡ u in Ω and Rthat ḡ ≡ g in Γ and zero otherwise. The result follows
t
by putting Φ− (t) := −C 0 t + 0 R0− (s) ds.
NONLOCAL DIFFUSION EQUATIONS
15
6.5. Lower bounds and blowup. We extend the lower-bound differential inequality in Lemma 4.4 to some nonlocal reaction-diffusion equations (i.e. R 6= 0
in (2.1)). One interesting consequence of this result is that we can easily spot
equations for which solutions will blow up.
We consider a reactivity R of the form:
N
R(t, x, u) = R0 (t, x)u +
1X
Ri (t, x, u),
u i=1
(6.12)
where each Ri is given as in (6.4), the exception being that we now require r±,i > 0.
Note that the only difference of this reactivity with the previous case is the plus
sign, which corresponds to an influx (rather than an outflux). Of course, we are
now allowing for a completely general diffusion coefficient k.
We say that a function f is “superlinear” if lim inf |u|→∞ |f (u)/up | = ∞ for some
p > 1.
Proposition 6.4. Let u solve (2.1) subject to (2.2) and (2.3). Suppose that k ≥ 0,
g ≡ 0, and R is given in (6.12). If any of the fi is superlinear, and if σ in Lemma
4.4 is sufficiently small, then u blows up in finite time.
Proof. Following the same procedures as in Lemma 4.4 and Theorem 6.1, we have
the following differential inequality for kuk2L2 (Ω) :
N
X
1 d
2
kuk2L2 (Ω) ≥ (R0− (t) − C 0 σ(t))kuk2L2 (Ω) + |Ω|
ri,+
(t) fi |Ω|−1 ku)k2L2 (Ω) ,
2 dt
i=1
(6.13)
where R0− (t) := ess inf x∈Ω R0 (t, x).
Suppose for some m : 0 ≤ m < N that |fi (u)| ≥ fi,0 |u| for some constants
fi,0 ≥ 0 and for each u; we assume that fN is a superlinear function. If, for each
t > 0, σ satisfies the following condition:
m
X
2
C 0 σ(t) ≤ R0− (t) +
ri,+
(t)fi,0 ,
(6.14)
i=1
then we see that the following inequality holds:
1 d
2
kuk2L2 (Ω) ≥ |Ω|rN,+
(t) fN |Ω|−1 ku)k2L2 (Ω) .
2 dt
By the convexity of |fN | and the fact that it vanishes at zero, we see that, unless
ku(0, )k = 0, this lower bound must be an increasing function of t. Since fN is
superlinear, it follows from basic ODE theory that kukL2 (Ω) → ∞ as t % t0 for
some t0 > 0.
Remark 6.1. Since σ is a measure of the size of the diffusion coefficient k, this result
essentially says that blowup occurs if the effects of nonlocal diffusion are not strong
enough to counteract the nonlinear effects of the reactivity R.
Remark 6.2. When the nonlinear terms of R(t, x, u) in (6.12) are zero, we can
obtain explicit lower bounds for kukL2 (Ω) similar to in Lemma 4.4. Examples of
equations that fall under this framework include the p-Laplacian equation, the
filtration equation, and reaction-diffusion equations with linear reactivity. In each
case, the lower bound is exponential.
16
A. DICKERSON, E. ESTRADA, T. HARTLAND, W. JAMIESON, L. LEWANDOWSKI, P. RADU, AND R. SHANKAR
Acknowledgments
This work was funded by the NSF DMS Award 1263132.
7. Appendix
Proof of Poincaré’s inequality. We suppose only that α satisfies the following condition:
Z
0<
y · α(y) dy := A < ∞.
(7.1)
Rn
As shown in [7], this integral must be equal to n in order for nonlocal first derivatives
to converge to classical derivatives, so this assumption is natural (and also needed
in [22] in a different form).
Extend u to Rn by setting it to be zero on Rn \ Ω. Then the following identity
holds for any x0 ∈ Rn :
Z Z
α(y − x) · (x0 − y)u2 (y) dy dx = 0.
(7.2)
Rn
Rn
Indeed, letting x → x + y in the integral, we see that the integral over x of the
anti-symmetric function α(x)
vanishes. This first step is analogous to that in [15],
R
for which we have that Rn ∇ · ((x0 − y)u2 (y)) dy = 0 for each u with compact
support.
Rewriting (7.2), letting y → x + y, and using (7.1), we obtain:
Z
Z
Z
u2 (y)
(y − x0 ) · α(y − x) dx dy = A
u2 (y) dy
Rn
Rn
Rn
Z Z
(7.3)
=
α(y) · (x0 − y − x)(u(x + y) − u(x))(u(x) + u(x + y)) dy dx.
Rn
Rn
Since supp α = Bδ , and supp u = Ω, we can rewrite the integral over (x, y) ∈
Rn × Rn as one over (x, y) ∈ Ω ∪ Γ × Bδ .
Using Hölder’s inequality, (7.3) becomes:
1/2
1/2
Akuk2L2 (Ω) ≤ B1 B2 ,
(7.4)
where
Z
Z
B1 :=
Ω∪Γ
Z
Z
B2 :=
Ω∪Γ
[α(y) · (x0 − y − x)(u(x + y) − u(x))]2 dy dx,
Bδ
[u(x) + u(x + y)]2 dy dx = 4|Bδ |kuk2L2 (Ω) .
Bδ
Given the following estimate
|α(y) · (x0 − x − y)| ≤ |α(y)||x0 − x − y| ≤ |α(y)|(|x0 − x| + |y|),
we see that, for any x0 ∈ Rn , (7.4) becomes:
Z
1/2
Z
1/2
2
2
AkukL2 (Ω) ≤ 2|Bδ |
|Gu(x, y)| (|x0 − x| + |y|) dx dy
.
Ω∪Γ
(7.5)
Bδ
Letting r := inf x0 ∈Ω∪Γ supx∈Ω∪Γ |x0 − x| be the “radius” of Ω ∪ Γ and |y| ≤ δ
be that of Bδ , we substitute these estimates into (7.5) and obtain (2.15), where the
NONLOCAL DIFFUSION EQUATIONS
17
constant C(α, Ω ∪ Γ, δ) is given by:
C=
A
.
2|Bδ |1/2 (r + δ)
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E-mail address: [email protected]
E-mail address: [email protected]
E-mail address: [email protected]
E-mail address: [email protected]
E-mail address: [email protected]
E-mail address: [email protected]
E-mail address: [email protected]
University of Nebraska-Lincoln, Department of Mathematics, Lincoln, NE 68588