A NON-LOCAL DIFFUSION EQUATION WHOSE SOLUTIONS DEVELOP A FREE BOUNDARY.

CARMEN CORTAZAR, MANUEL ELGUETA, AND JULIO D. ROSSI

Abstract. Let J : ℝ → ℝ be a nonnegative, smooth compactly supported function such that ∫_{ℝ} J(r)dr = 1. We consider the nonlocal diffusion problem

u_{t}(x, t) = ∫_{ℝ} J (\frac{x − y}{u(y, t)}) dy − u(x, t) in ℝ × [0, ∞)

with a nonnegative initial condition. Under suitable hypotheses we prove existence, uniqueness, as well as the validity of a comparison principle for solutions of this problem. Moreover we show that if u(·, 0) is bounded and compactly supported, then u(·, t) is compactly supported for all positive times t. This implies the existence of a free boundary, analog to the corresponding one for the porous media equation, for this model.

1. Introduction

Let J : ℝ → ℝ be a nonnegative, smooth function with ∫_{ℝ} J(r)dr = 1. Assume also that J is supported in [−1, 1], is strictly increasing in [−1, 0] and strictly decreasing in [0, 1].

Equations of the form

(1.1) u_{t}(x, t) = J ∗ u − u(x, t) = ∫_{ℝ} J(x − y)u(y, t)dy − u(x, t),

and variations of it, have been recently widely used to model diffusion processes, see [2], [4], [5], [6], [8]. As stated in [5] if u(x, t) is thought of as a density at the point x at time t and J(x − y) is thought of as the probability distribution of jumping from location y to location x, then (J ∗ u)(x, t) is the rate at which individuals are arriving to position x from all other places and −u(x, t) = − ∫_{ℝ} J(y − x)u(x, t)dy is the rate

Key words and phrases. Nonlocal diffusion, free boundaries. Supported by Universidad de Buenos Aires under grant TX048, by ANPCyT

PICT No. 03-00000-00137, by CONICET (Argentina) and by FONDECYT (Chile) project number. 2000 Mathematics Subject Classification 35K57, 35B40.

at which they are leaving location x to travel to all other sites. This consideration, in the absence of external sources, leads immediately to the fact that the density u satisfies equation (1.1).

Equation (1.1), so called nonlocal diffusion equation, shares many properties with the classical heat equation

u_{t} = Δu

such as: bounded stationary solutions are constant, a maximum principle holds for both of them and, even if J is compactly supported, perturbations propagate with infinite speed. By this we understand that if u is a nonnegative non trivial solution, then u(x, t) > 0 for all x ∈ ℝ and all t > 0 no matter whether the non trivial initial condition u(x, 0) vanishes in some region.

Another classical equation that has been used to model diffusion is the well known porous medium equation,

u_{t} = Δu^{m}

with m > 1. This equation also shares several properties with the heat equation but there is a fundamental difference, in this case if the initial data u(·, 0) is compactly supported, then u(·, t) has compact support for all t > 0. In such a case, if the support of the initial condition is a finite interval, one can define the right and left free boundaries of the solution by

s_{+}(t) = sup{x / u(x, t) > 0}

and

s_{−}(t) = inf{x / u(x, t) > 0}

respectively. Properties and the behavior of the free boundary for the porous medium equation have been largely studied over the past years. See for example [1], [7] and the corresponding bibliography. It is worth mentioning that this phenomena also arises in the context of the Stefan problem, see [3] and the references therein.

The purpose of this note is to present a simple nonlocal model for diffusion whose solutions, with compactly supported bounded initial data, develop a free boundary. To do this we propose a model where the diffusion at a point depends on the density. The simplest situation we can think of is when the probability distribution of jumping from location y to location x is given by

J (\frac{x − y}{u(y, t)}) \frac{1}{u(y, t)}

when u(y, t) > 0 and 0 otherwise. In this case the rate at which individuals are arriving to position x from all other places is

∫_{ℝ} J (\frac{x − y}{u(y, t)}) dy

and the rate at which they are leaving location x to travel to all other sites is

−u(x, t) = − ∫_{ℝ} J (\frac{y − x}{u(x, t)}) dy.

As before this consideration, in the absence of external sources, leads immediately to the fact that the density u has to satisfy

u_{t}(x, t) = ∫_{ℝ} J (\frac{x − y}{u(y, t)}) dy − u(x, t).

As for the initial data, although we are mostly interested in functions u(·, 0) ∈ L^{1}(ℝ)∩L^{∞}(ℝ) it is more convenient, for technical reasons that will become clear later, to consider a slightly more general set of initial conditions. So in this paper we will deal with the problem

(1.2)  \begin{matrix}u_{t}(x, t) = ∫_{ℝ} J (\frac{x − y}{u(y, t)}) dy − u(x, t) in ℝ × [0, ∞). \\ u(x, 0) = c + w_{0}(x) on ℝ, \end{matrix}

where c ≥ 0, w_{0} ∈ L^{1}(ℝ) and w_{0} ≥ 0.

Most of the results contained in this note can be obtained in several dimensions without many changes in the elementary arguments but, we have chosen to treat the one dimensional case for the sake of simplicity of the exposition.

We will address in this paper the questions of existence, uniqueness, comparison principles and some basic facts about the free boundary for solutions of problem (1.2). Several further questions, such as the decay rate of solutions, the speed at which the free boundary moves, the existence of the so called waiting times for the free boundary and many others, are left open. Also one can consider equations involving a source term and to study, for example, the blow up phenomena. We hope such questions can be answered by us or by someone else in the near future.

2. Existence and uniqueness.

The existence and uniqueness result will be a consequence of Banach’s fixed point theorem and it is convenient to give some preliminaries before giving its proof.

Fix t_{0} > 0 and consider the Banach space C([0, t_{0}]; L^{1}) with the norm

|∥w∥| = max_{0≤t≤t_{0}} ∥w(·, t)∥_{L^{1}}.

Let

X_{t_{0}} = {w ∈ C([0, t_{0}]; L^{1}) / w ≥ 0}

which is a closed subset of C([0, t_{0}]; L^{1}).

We will obtain the solution in the form u(x, t) = w(x, t) + c where w is a fixed point of the operator T_{w_{0}} : X_{t_{0}} → X_{t_{0}} defined by

\begin{aligned}T_{w_{0}}(w)(x, t) = & ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} J (\frac{x − y}{w(y, s) + c}) dy ds \\ & +e^{−t}w_{0}(x) − c(1 − e^{−t}). \end{aligned}

The following lemma is the main ingredient of our proof.

Lemma 2.1. Let w_{0}, z_{0} non negative functions such that w_{0}, z_{0} ∈ L^{1}(ℝ) and w, z ∈ X_{t_{0}}, then

|||T_{w_{0}}(w) − T_{z_{0}}(z)||| ≤ (1 − e^{−t_{0}})|||w − z||| + ||w_{0} − z_{0}||_{L^{1}(ℝ)}.

Proof: We have

∫_{ℝ} |T_{w_{0}}(w)(x, t) − T_{z_{0}}(z)(x, t)| dx

≤ ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} |∫_{ℝ} (J (\frac{x − y}{w(y, s) + c}) − J (\frac{x − y}{z(y, s) + c})) dy| dx ds

+e^{−t} ∫_{ℝ} |w_{0} − z_{0}|(y) dy.

Now set

A^{+}(s) = {y / w(y, s) ≥ z(y, s)}

and

A^{−}(s) = {y / w(y, s) < z(y, s)}.

We have now

∫_{ℝ} |∫_{ℝ} (J (\frac{x − y}{w(y, s) + c}) − J (\frac{x − y}{z(y, s) + c})) dy| dx

≤ ∫_{ℝ} ∫_{A^{+}(s)} (J (\frac{x − y}{w(y, s) + c}) − J (\frac{x − y}{z(y, s) + c})) dy dx

+ ∫_{ℝ} ∫_{A^{−}(s)} (J (\frac{x − y}{z(y, s) + c}) − J (\frac{x − y}{w(y, s) + c})) dy dx.

Since the integrands are non negative we can apply Fubini’s theorem to get

\begin{matrix}∫_{ℝ} ∫_{A^{+}(s)} (J (\frac{x − y}{w(y, s) + c}) − J (\frac{x − y}{z(y, s) + c})) dy dx \\ = ∫_{A^{+}(s)}(w(y, s) − z(y, s))dy \end{matrix}

and similarly for the integral over A^{−}(s). Therefore we obtain

∫_{ℝ} |∫_{ℝ} (J (\frac{x − y}{w(y, s) + c}) − J (\frac{x − y}{z(y, s) + c})) dy| dx

≤ ∫_{ℝ} |w(y, s) − z(y, s)| dy.

Hence we get

∥|T_{w_{0}}(w) − T_{z_{0}}(z)|∥ ≤ (1 − e^{−t_{0}})|∥w − z∥| + ||w_{0} − z_{0}||_{L^{1}(ℝ)}

as desired. □

We can state now the main result of this section.

Theorem 2.1. For every nonnegative w_{0} ∈ L^{1} and every constant c ≥ 0, there exists a unique solution u, such that (u−c) ∈ C([0, ∞); L^{1}), of (1.2). Moreover, the solution verifies u(x, t) ≥ c and preserves the total mass above c, that is

(2.1)  ∫_{ℝ}(u(y, t) − c) dy = ∫_{ℝ} w_{0}(y) dy.

Proof: We check first that T maps X_{t_{0}} into X_{t_{0}}. Since w ≥ 0 we have

J (\frac{x − y}{w(y, s) + c}) ≥ J (\frac{x − y}{c} )

and hence

(2.2)  \begin{aligned}T (w)(x, t) ≥ & ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} J (\frac{x − y}{c} ) dy ds \\ & +e^{−t}w_{0}(x) − c(1 − e^{−t}) = e^{−t}w_{0}(x) ≥ 0. \end{aligned}

Taking z_{0}, z ≡ 0 in Lemma 2.1 we get that T (w) ∈ C([0, t_{0}]; L^{1}). Now taking z_{0} ≡ w_{0} in Lemma 2.1 we get that T_{w_{0}} is a strict con-

traction in X_{t_{0}} and the existence and uniqueness part of the theorem follows from Banach’s fixed point theorem.

We finally prove that if u = w + c is the solution, then the integral in x of w is preserved. Since

0 = ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} J (\frac{x − y}{c} ) dy ds − c(1 − e^{−t}),

we can write

w(x, t) = ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ}(J (\frac{x − y}{w(y, s) + c})−J (\frac{x − y}{c} )) dy ds+e^{−t}w_{0}(x).

The integrand in the above formula is non negative so we can integrate in x and apply Fubini’s theorem to obtain

(2.3) ∫_{ℝ} w(x, t)dx = ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} w(y, s) dy ds + e^{−t} ∫_{ℝ} w_{0}(x)dx

from where it follows that

\frac{d}{dt} ∫_{ℝ} w(x, t)dx = 0

and the theorem is proved. □

We will need in what follows the following lemma which is a direct corollary of the proof of Theorem 2.1 and is a first version of the comparison principle of Section 3 below.

Lemma 2.2. With the above notation if 0 ≤ w(x, 0) ≤ M for all x ∈ ℝ, then w(x, t) ≤ M for all (x, t) ∈ ℝ × [0, ∞).

Proof: Under the given hypotheses one has that if w(x, t) ≤ M, then

T_{w_{0}}(w)(x, t) =

∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} J (\frac{x − y}{w(y, s) + c}) dy ds + e^{−t}w_{0}(x) − c(1 − e^{−t})

≤ ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} J (\frac{x − y}{M + c}) dy ds + e^{−t}M − c(1 − e^{−t}) = M.

The lemma follows by the uniqueness of the fixed point for T_{w_{0}}. □

Lemma 2.1, Theorem 2.1, Lemma 2.2 and their proofs have several immediate consequences that we state as a series of remarks for the sake of future references.

Remark 2.1. Solutions of (1.2) depend continuously on the initial condition in the following sense. If u and v are solutions of (1.2), then

max_{0≤t≤t_{0}} ∥u(·, t) − v(·, t)∥_{L^{1}(ℝ)} ≤ e^{t_{0}}||u(·, 0) − v(·, 0)||_{L^{1}(ℝ)}

for all t_{0} ≥ 0.

Remark 2.2. The function u is a solution of (1.2) if and only if

u(x, t) = ∫_{0}^{t} e^{−(t−s)} ∫_{ℝ} J (\frac{x − y}{u(y, s)}) dy ds + e^{−t}u(x, 0).

Remark 2.3. From the previous remark and Lemma 2.2 we get that if c > 0 and u(·, 0) ∈ C^{k}(ℝ) with 0 ≤ k ≤ ∞, then u(·, t) ∈ C^{k}(ℝ) for all t ≥ 0. Moreover if w_{0} is a compactly supported C^{1} function, then there exists a constant K depending on c, J and w_{0} such that

|u_{t}(x, t)| , |\frac{∂u}{∂x}(x, t)| ≤ K.

Remark 2.4. A consequence of Remark 2.3 and of (2.1) is that if c > 0 and w_{0} is a compactly supported C^{1} function, then

lim_{|x|→∞} u(x, t) = c uniformly on compact intervals [0, T].

Remark 2.5. It follows from inequality (2.2) that

w(x, t) ≥ e^{−t}w(x, 0).

In particular, in the case that u(·, 0) ∈ L^{1}(ℝ), the support of u(·, t) does not shrink as time increases. By this we understand that if u(x_{0}, t_{0}) > 0, then u(x_{0}, t) > 0 for all t ≥ t_{0}.

3. Comparison Principle.

Comparison principles like the one below have proven to be a very useful tool in studying diffusion problems.

Theorem 3.2. Let u and v be continuous solutions of (1.2). If

u(x, 0) ≤ v(x, 0) for all x ∈ ℝ,

then

(3.1)  u(x, t) ≤ v(x, t) for all (x, t) ∈ ℝ × [0, ∞).

Proof: We assume first that

u(x, 0) = c + w(x, 0) and v(x, 0) = d + z(x, 0)

with 0 < c < d and u(x, 0) < v(x, 0). Moreover we assume for a moment that w(x, 0) and z(x, 0) are compactly supported C^{1} functions. In this case there exists δ > 0 such that u(x, 0) + δ < v(x, 0). Assume, for a contradiction that the conclusion does not hold. In view of Remark 2.4 we have that there exists a time t_{0} > 0 and a point x_{0} ∈ ℝ such that u(x_{0}, t_{0}) = v(x_{0}, t_{0}) and u(x, t) ≤ v(x, t) for all (x, t) ∈ ℝ × [0, t_{0}].

Let us consider the set B = {x ∈ ℝ / u(x, t_{0}) = v(x, t_{0})}. Clearly B is non empty and closed.

Let x_{1} ∈ B. We have then

0 ≤ (u − v)_{t}(x_{1}, t_{0}) = ∫_{ℝ} (J (\frac{x_{1} − y}{u(y, t_{0})}) − J (\frac{x_{1} − y}{v(y, t_{0})})) dy ≤ 0

which implies

u(y, t_{0}) = v(y, t_{0}) for all y ∈ (x_{1} − c, x_{1} + c).

Hence B is open. It follows that B = ℝ which is the desired contradiction since (u(·, t_{0}) − c) ∈ L^{1}(ℝ).

We now get rid of the extra hypothesis that w(x, 0) and z(x, 0) are compactly supported C^{1} functions. In order to do this let w_{n}(x, 0) and z_{n}(x, 0) be sequences of compactly supported C^{1} functions such that w_{n}(x, 0) → w(x, 0) and z_{n}(x, 0) → z(x, 0) in L^{1}(ℝ) as n → ∞ and, moreover, u_{n}(x, 0) = c + w_{n}(x, 0) < v_{n}(x, 0) = d + z_{n}(x, 0). Let u_{n} and v_{n} be the solutions with initial data u_{n}(x, 0) and v_{n}(x, 0) respectively. By the previous argument one has u_{n} ≤ v_{n} an the result follows by letting n → ∞ in view of Remark 2.1.

In order to prove the theorem in the general case pick strictly decreasing sequences a_{n} and b_{n} such that 0 < a_{n} < b_{n} and b_{n} → 0 as n → ∞. Let u_{n} and v_{n} be the solutions with initial conditions u_{n}(x, 0) = u(x, 0) + a_{n} and v_{n}(x, 0) = v(x, 0) + b_{n} respectively. According to the previous argument one has u_{n} ≤ v_{n}. Moreover u_{n+1} ≤ u_{n} and v_{n+1} ≤ v_{n}. By Remark 2.2, after an application of the monotone convergence theorem, it follows that u_{n}(x, t) → u(x, t) and v_{n}(x, t) → v(x, t) as n → ∞ and the theorem is proved. □

An immediate consequence of the comparison principle and Remark 2.4 is the following corollary that extends Remark 2.4 to the case c = 0.

Corollary 3.1. If c = 0 and w_{0} is a compactly supported C^{1} function, then

lim_{|x|→∞} u(x, t) = 0 uniformly on compact intervals [0, T].

4. The free boundary.

In this section we will prove that solutions of (1.2) with compactly supported continuous initial data do have a free boundary in the sense that

s_{+}(t) = sup{x / u(x, t) > 0} < +∞

and

s_{−}(t) = inf{x / u(x, t) > 0} > −∞

for all t ≥ 0. It follows from Remark 2.5 that s_{+} and s_{−} are non decreasing and non increasing functions respectively. Moreover we will also prove in this section that the supports of u(·, t) eventually fill at least half a ray of the space, in particular either lim_{t→∞} s_{+}(t) = ∞ or lim_{t→∞} s_{−}(t) = −∞. In the case that J is even, that is the case of an isotropic media, the supports eventually cover the whole of ℝ.

The following theorem implies the existence of free boundaries.

Theorem 4.3. If u(·, 0) is compactly supported and bounded then u(·, t) is also compactly supported for all t ≥ 0.

Proof: Due to the scaling invariance of the equation, namely if u(x, t) is a solution then for any λ > 0 the function v_{λ}(x, t) = λu(\frac{x}{λ}, t) is also a solution, we can restrict ourselves to initial data supported in [−1, 1] and such that sup_{x∈ℝ}u(x, 0) ≤ 1.

We note first that

u_{t}(x, t) ≤ ∫_{ℝ} J (\frac{x − y}{u(y, t)}) dy. (4.1)

Therefore, since 0 ≤ u ≤ 1, we get by (4.1) that

u(x, t) ≤ \frac{1}{2} for all t ≤ \frac{1}{2} and all x such that |x| ≥ 1.

Now if |x| ≥ 2 and t ≤ \frac{1}{2} we have that |x − y| ≤ u(y, t) implies that |y| ≥ 1 and hence u(y, t) ≤ \frac{1}{2}. Therefore, again by (4.1), we have

u(x, t) ≤ \frac{1}{4} for all t ≤ \frac{1}{2} and all x such that |x| ≥ 2.

We look now at the case |x| ≥ 2 + \frac{1}{2} and t ≤ \frac{1}{2}. In this case |x − y| ≤ u(y, t) implies that |y| ≥ 2 and hence u(y, t) ≤ \frac{1}{4}. Again by (4.1), we have

u(x, t) ≤ \frac{1}{8} for all t ≤ \frac{1}{2} and all x such that |x| ≥ 2 + \frac{1}{2}.

Repeating this procedure we obtain by induction that for any integer n ≥ 1 one has

u(x, t) ≤ \frac{1}{2^{n+2}} for all t ≤ \frac{1}{2} and all x such that |x| ≥ 2 + ∑_{k=1}^{n} \frac{1}{2^{k}} .

It follows that the support of u(·, t) is contained in the interval [−3, 3] for all t ≤ \frac{1}{2} as we wanted to prove. □

In order to prove our next result we need a preliminary lemma.

Lemma 4.3. If u(x, 0) is continuous and not constant, then the function

M(t) = max_{x∈ℝ}u(x, t)

is strictly decreasing.

Proof: It is clear, by comparison with a constant, that M(t) decreases as t increases. Moreover by Remark 2.5 one has M(t) > c for all t ≥ 0. Fix t_{0} ≥ 0 and let t_{1} > t_{0}. Let us consider the set

C = {x / u(x, t_{1}) = M(t_{0})}.

The set C is clearly closed. Since u(x, t) ≤ M(t_{0}) for all t ≥ t_{0} we have that at any point x_{0} ∈ C one must have

0 ≤ u_{t}(x_{0}, t_{1}) = ∫_{ℝ} J (\frac{x_{0} − y}{u(y, t_{1})}) dy − u(x_{0}, t_{1}) ≤ 0.

This implies that u(x, t_{1}) = M(t_{0}) for all x in a neighborhood of x_{0} and hence C is open. Consequently either C = ℝ or C is empty. It is clear that C ≠ ℝ, so C = ∅ and the lemma is proved. □

We are now in a position to prove that at least one of the free boundaries go to infinity.

Theorem 4.4. Let u be the solution of problem (1.2) with c = 0 and w_{0} ≠ 0. Then either

lim_{t→∞} s_{+}(t) = ∞  or  lim_{t→∞} s_{−}(t) = −∞

and the supports of u(·, t) eventually cover an infinite half ray of ℝ. If J is an even function the supports eventually cover the whole of ℝ.

Proof: By comparison, and the invariance under translations of the equation, it is enough to prove the theorem under the assumptions that w_{0} ∈ C^{1}, its support is the interval [−A, A] and it is symmetric with respect to the origin.

We claim first that the support of u(·, t) is not uniformly bounded. Assume for a contradiction that there exists L > 0 such that u(x, t) = 0 for all x such that |x| ≥ L and all t ≥ 0. Since ∫_{ℝ} u(x, t)dx = ∫_{ℝ} u(x, 0)dx > 0 there exists C > 0 such that

lim_{t→∞} M(t) = C.

Let v(x, 0) be a smooth function supported in [−L−1, L+1] such that 0 ≤ v(x, 0) ≤ C and v(x, 0) ≡ C if x ∈ [−L, L]. Let us denote by v(x, t) the solution of (1.2) with this initial condition. By Lemma 4.3 we have that

max_{x∈ℝ}v(x, 1) < C.

Now for any integer n > 0 let v_{n}(x, 0) be a smooth compactly function supported in [−L − 2, L + 2] such that 0 ≤ v(x, 0) ≤ C + \frac{1}{n} and v_{n}(x, 0) ≡ C + \frac{1}{n} if x ∈ [−L, L]. Assume further that

v_{n+1}(x, 0) ≤ v_{n}(x, 0)

and denote by v_{n}(x, t) the solution of (1.2) with initial condition v_{n}(x, 0). By comparison it follows that

v_{n+1}(x, t) ≤ v_{n}(x, t).

Using Remark 2.2 and the monotone convergence theorem one has

v_{n}(x, 1) → v(x, 1) in [−L − 2, L + 2] as t → ∞.

Moreover, being the limit continuous the convergence is uniform by Dini’s theorem. Consequently there exists n_{0} such that

max_{x∈ℝ}v_{n_{0}}(x, 1) < C.

On the other hand there exists t_{0} such that

u(x, t_{0}) ≤ v_{n_{0}}(x, 0).

This implies, by comparison, that

max_{x∈ℝ}u(x, t_{0} + 1) < C

a contradiction that proves the claim.

We are ready now to prove the statement of the theorem.

We claim that if there exists x_{0} ≥ A such that u(x_{0}, t) = 0 for all t ≥ 0, then

u(x, t) = 0 for all (x, t) ∈ [x_{0}, ∞) × [0, ∞).

Indeed, let d > 0 and we will prove that

(4.2) u(x, t) ≤ d for all x ≥ A and all t ≥ 0.

Since u(x_{0}, t) ≡ 0 one has

u(x, t) ≤ |x − x_{0}| for all x ∈ ℝ and all t ≥ 0.

Moreover u(x, 0) = 0 for all x ≥ x_{0}. So if (4.2) does not hold, using Corollary 3.1, there exists a point x_{1} ∈ ℝ with x_{1} ≥ x_{0} + d and a time t_{1} > 0 such that u(x_{1}, t_{1}) = d and u(x, t) ≤ d for all (x, t) ∈ ℝ × [0, t_{1}]. As in the proof of Theorem 3.2 we consider the set

B = {x ≥ x_{0} + d / u(x, t_{1}) = d}

which is clearly closed. Also at a point x_{2} ∈ B one has

0 ≤ (d − u)_{t}(x_{2}, t_{1}) = ∫_{ℝ} (J (\frac{x_{2} − y}{d} ) − J (\frac{x_{2} − y}{u(y, t_{0})})) dy ≤ 0

which implies

u(y, t_{0}) = d for all y ∈ (x_{2} − d, x_{2} + d).

It follows that B is open and hence B = [x_{0}, ∞) which is a contradiction that proves (4.2). Since d > 0 was chosen arbitrarily the claim follows. An analog of the above claim holds for points −x_{1} < −A such that u(−x_{1}, t) = 0 for all t ≥ 0. Such a points x_{0} and x_{1} can not exist simultaneously because this contradicts the fact that the supports of u(·, t) are not uniformly bounded. This, plus the fact that if J and u(·, 0) are even functions then u(·, t) is even for all t ≥ 0, proves the theorem. □

Finally we give an example of a non symmetric function J such that the supports of solutions u(·, t), with compactly supported bounded initial data, do not eventually cover the whole of ℝ.

We will show that for a special choice of J the function

u(x) = x_{+} = \{ \begin{matrix}0 & if & x ≤ 0 \\ x & if & x ≥ 0 \end{matrix}

satisfies

(4.3)  0 = ∫_{ℝ} J (\frac{x − y}{u(y)} ) dy − u(x).

It is immediate that if x ≤ 0, then

∫_{ℝ} J (\frac{x − y}{u(y)} ) dy = 0

and hence (4.3) is satisfied.

As for the case x > 0 we have that \frac{|x−y|}{y_{+}} ≤ 1 implies 0 < x ≤ 2y and hence

∫_{ℝ} J (\frac{x − y}{u(y)} ) dy − u(x)

= ∫ ∞_{\frac{x}{2}} J (\frac{x}{y} − 1) dy − x

= x (∫_{−1}^{1} J(r)\frac{dr}{(1 + r)^{2}} − 1) .

Now we choose J such that, in addition to the hypotheses already made, satisfies

∫_{−1}^{1} J(r)\frac{dr}{(1 + r)^{2}} = 1

and (4.3) also holds.

The desired example follows now by a comparison argument, like the one of the proof of Theorem 3.2, using the function x_{+}, or a translation of it, as a barrier.

References

[1] D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, A. Fasano and M. Primicerio eds. Lecture Notes in Math. 1224, Springer Verlag, (1986).

[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] Cannon, J. R.; Primicerio, Mario. A Stefan problem involving the appearance of a phase. SIAM J. Math. Anal. 4 (1973), 141–148.

[4] X Chen. Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differential Equations, 2, 125-160, (1997).

[5] P. Fife. Some nonclassical trends in parabolic and parabolic-like evolutions. Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003.

[6] C. Lederman and N. Wolanski. A free boundary problem from nonlocal combustion. Preprint

[7] J. L. Vazquez. An introduction to the mathematical theory of the porous medium equation. In ”Shape optimization and free boundaries” (M.C.Delfour eds.), Dordrecht, Boston and Leiden. 347-389, 1992.

[8] X. Wang. Metaestability and stability of patterns in a convolution model for phase transitions. Preprint.

Departamento de Matemática, Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile.

E-mail address: ccortaza@mat.puc.cl, melgueta@mat.puc.cl,
jrossi@dm.uba.ar