2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 85–101. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Description of regional blow-up in a porous-medium equation ^{∗}

Carmen Cortázar, Manuel del Pino, & Manuel Elgueta

Abstract

We describe the (finite-time) blow-up phenomenon for a non-negative solution of a porous medium equation of the form

u_{t} = Δu^{m} + u^{m}

in the entire space. Here m > 1 and the initial condition is assumed compactly supported. Blow-up takes place exactly inside a finite number of balls with same radii and exhibiting the same self-similar profile.

1 Introduction

This paper deals with the description of the blow-up phenomenon in the porousmedium equation in ℝ^{N}, N ≥ 1,

\begin{matrix}u_{t} = Δu^{m} + u^{m} \\ u(x, 0) = u_{0}(x) \end{matrix}  (1.1)

where m > 1 and u_{0}(x) is a compactly supported, not identically zero nonnegative function whose regularity will be specified later. This gives rise to the interesting phenomenon of regional blow-up, meaning this blow-up taking place only in a compact set with nonempty interior.

The purpose of this note is to describe the blow-up in (1.1) in the following sense: we show that for any initial condition the solution u develops (exactly) a finite number of similar spherical hot spots: more precisely, there is a finite number of disjoint balls with common radii R^{∗} outside which the solution remains uniformly bounded, while inside each of them it develops a common self-similar radially symmetric profile (T̄ − t)^{−1/(m−1)}w_{∗}(r), where r is the distance to the center of these balls and w_{∗} is a strictly positive function.

∗Mathematics Subject Classifications: 35B40, 35B45, 35J40.

Key words: Multiple-bump, pattern formation, mathematical biology,

singular perturbation.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published October 21, 2002.

The presence of regional blow-up in this equation was first observed and studied in the case N = 1 in [11]. The elliptic problem found when searching by separation of variables a solution of the form

u(x, t) = (T̄ − t)^{−1/(m−1)}θ(x)

has been studied for radial symmetry in [2, 3, 4, 15].

The result contained in this paper is a sequel of the work [5] where the following partial result was established: Let T̄ > 0 be the time at which blowup occurs. Let t_{n} be any sequence t_{n} ↑ T̄. Then there is a subsequence of t_{n} which we still denote t_{n}, and a nontrivial compactly supported solution w(x) of the elliptic equation

Δw^{m} + w^{m} − \frac{1}{m − 1}w = 0,  (1.2)

such that (T̄ − t_{n})^{1/(m−1)}u(x, t_{n}) → w(x) uniformly.

On the other hand, it was established in [3] that the support of any finite energy solution w of (1.2) consists of a finite number of disjoint balls of the same radii and that the solution is radially symmetric inside each of them. This radially symmetric solution w_{∗}(|x|) turns out to be unique, as established in [4]. Thus w can be written as

w(x) = ∑_{j=1}^{k} w_{∗}(|x − x_{j}|),  (1.3)

where, if R^{∗} is the radius of the support of w_{∗}, |x_{i} − x_{j}| ≥ 2R^{∗} for i ≠ j.

Let BU(u_{0}) be the set of blow-up points of u, namely the set of points x for which there are sequences x_{n} → x and t_{n} → T̄ such that u(x_{n}, t_{n}) → +∞. It was also shown in [5] that this set is compact and it is precisely constituted by the union of the supports of all possible limiting w’s. The important point unsolved in [5] was whether there is an actual unique blow-up profile, rather than oscillation between different limiting configurations. The question turns out to be rather subtle, and we answer it affirmatively in the following result.

Theorem 1.1 Let u(x, t) be the solution of (1.1), where u_{0}(x) is compactly supported, continuous and such that u_{0}^{m} ∈ H^{1}(ℝ^{N}). Let T̄ > 0 be the blow-up time of this solution. Then there are points x_{1}, … x_{k} ∈ ℝ^{N} such that

lim_{t→T̄}(T̄ − t)^{1/(m−1)}u(x, t) = ∑_{j=1}^{k} w_{∗}(|x − x_{j}|)

uniformly. Here w_{∗}(|x|) is the unique compactly supported, radially symmetric solution of (1.2). If R^{∗} is the radius of its support then we also have |x_{i} − x_{j}| ≥ 2R^{∗} for i ≠ j. Moreover u(x, t) remains uniformly bounded up to its blow-up time on compact subsets of ℝ^{N} \ ⋃_{j=1}^{k} B̄(x_{j}, R^{∗}). In other words,

BU(u_{0}) = ⋃_{j=1}^{k} B̄(x_{j}, R^{∗}).

Next we describe the proof of the above results. Let us introduce the change of variables

v(x, t) = (T̄ − τ)^{1/(m−1)}u(x, τ)|_{τ=T̄(1−e^{−t})}.  (1.4)

It is readily checked that v is globally defined in time and satisfies the equation

v_{t} = Δv^{m} + v^{m} − \frac{1}{m − 1}v  (1.5)
v(x, 0) = T̄^{1/(m−1)}u_{0}(x).  (1.6)

From [5, Proposition 4.1], v is a bounded function and given a sequence t_{n} → +∞ there is a subsequence, which we denote in the same way, and a nontrivial, compactly supported solution of (1.2) so that

v(x, t_{n})^{m} → w(x)^{m} as n → ∞,

both in uniform and H^{1}-senses. Thus our task in establishing Theorem 1.1 is precisely to prove that the limit w(x) is actually the same along every sequence t_{n} → +∞. Here w(x) has the form (1.3), and for the sake of simplicity we restrict ourselves in what follows of this paper only to the case k = 1. The proof for k > 1 is similar if |x_{i} −x_{j}| > 2R^{∗} for all i ≠ j, only to the expense of some extra notation. Instead, if |x_{i}−x_{j}| = 2R^{∗} for some i ≠ j, the situation is substantially more involved. The complete proof is contained in the forthcoming work [7]. It should be mentioned that conditions for one-ball blow-up (i.e. k = 1) are provided in [5]. For instance, if the support of the initial condition is contained in a ball of radius less than R^{∗} this is the case. The one-ball blow up can be shown to be stable, with basically the same proof below. This means that if an initial condition u_{0} leads to blow-up with k = 1 in Theorem 1.1, then this is the case for all sufficiently close initial data, where the limiting ball is also close to that associated to u_{0}. Instead, the two-ball blow-up is not stable as the following example shows. Let us fix points x_{1} and x_{2} with |x_{1} − x_{2}| > 2R^{∗}. Then the function

u(x, t) = (T̄_{1} − t)^{−1/(m−1)}w_{∗}(|x − x_{1}|) + (T̄_{2} − t)^{−1/(m−1)}w_{∗}(|x − x_{2}|)

solves equation (1.1) for 0 < t < min{T̄_{1}, T̄_{2}}. If T̄_{1} = T̄_{2}, then two-ball blow-up takes place, which however disappears as soon as T̄_{1} and T̄_{2} differ, no matter how close they are. This example suggests that one-ball blow-up may actually hold for “generic” initial data.

The main feature of equation (1.5) is the presence of a Lyapunov functional for it, namely

J(z) = \frac{1}{2} ∫ (|∇z^{m}|^{2} − z^{2m})dx + \frac{m}{m^{2} − 1} ∫ z^{m+1}dx.  (1.7)

In fact we have that the mapping t ↦ J(v(·, t)) is decreasing on t > 0 and

lim_{t→+∞} J(v(·, t)) = J(w).

Here and in what follows the integral symbol without limits specified means integration on the whole ℝ^{N}. The presence of this functional implies that limit points of the trajectory must be steady states. The problem of uniqueness of asymptotic limits in nonlinear heat equations under the presence of a Lyapunov functional has been analyzed in a number of works. A general result due to L. Simon [24] shows the uniqueness of the limit for uniformly parabolic equations in a uniform real analytic setting on a compact manifold. Analiticity cannot be lifted in general in this result, at least in the non-autonomous setting, as shown in [22]. Needless to say, the compactly supported setting we deal with makes our situation highly non uniformly analytic.

Related uniqueness results in parabolic problems, nondegenerate and degenerate, are contained in the works [1, 6, 8, 9, 10, 13, 17, 16, 20]. In [6], a re-normalization method based on L. Simon’s ideas, used in classifying singularities in an elliptic problem in [18] was adapted to a semilinear heat equation. The general framework of this method is what we will use here. Alternative methods for degenerate equations of porous-medium type, in one and higher dimensions, have been devised in [9], [10]. Those techniques do not apply to the nonlinearity of equation (1.6), in particular those in [10], based on analyticity, because of the presence of compactly supported steady states. This is explicitly commented in [10] and posed as an open question.

2 Proof of the main results

For functions v_{1} and v_{2} defined on ℝ^{N} we consider the “distance” between them defined as

d(u_{1}, u_{2}) = ( ∫ (u_{1}(x)^{m} − u_{2}(x)^{m})(u_{1}(x) − u_{2}(x))dx)^{1/2}.

We observe that there is a constant D > 0 for which

d(u_{1}, u_{3}) ≤ D( d(u_{1}, u_{2}) + d(u_{2}, u_{3}) ).

The theorem stated in the previous section (in the case k = 1) will be a direct consequence of the following result,

Proposition 2.1 There exist positive numbers T and C such that if v is a solution of equation (1.6) defined on 0 ≤ t < +∞, such that for some sequence t_{n} → +∞, setting

v_{n}(x, t) ≡ v(x, t_{n} + t), w_{n}(x) = w_{∗}(|x − x_{n}|),

one has

η_{n} ≡ sup_{t∈[0,T]} d(v_{n}(t), w_{n}) → 0 as n → ∞.  (2.1)

Then there exists a point x̄_{n} with |x̄_{n} − x_{n}| ≤ Cη_{n} and

sup_{t∈[0,T]} d(v_{n}(t), w̄_{n}) ≤ \frac{η_{n}}{2}

for all n sufficiently large, and where w̄_{n}(x) = w_{∗}(|x − x̄_{n}|).

As a consequence, we obtain the validity of the following fact: There exist positive numbers T, δ, C and t^{∗}with the following property:

Let v(x, t) be a solution of (1.6), defined in 0 < t < ∞ as in the statement of the theorem. Consider a point x_{1} and set w(x) = w_{∗}(|x − x_{1}|). Assume that t_{0} > t^{∗} is such that

η = sup_{t_{0}≤t≤t_{0}+T} d(v(t), w) < δ.

Then there exists a points x̄_{1} with |x̄_{1} − x_{1}| ≤ Cη such that

sup_{t_{0}+T ≤t≤t_{0}+2T} d(v(t), w̄) ≤ \frac{η}{2} ,

where w̄(x) = w_{∗}(|x − x̄_{1}|).

Let us see how Theorem 1.1 (for k = 1) follows from this assertion. Let ε be given and let us write w as w(x) = w_{∗}(|x − x_{1}|). Let δ_{0} < δ, with δ the number predicted by Proposition 2.1. Assume that for some t_{0} > t^{∗} we have η_{1} ≡ sup_{t_{0}≤t≤t_{0}+T} d(v(t), w) ≤ δ_{0}, where T and t^{∗} are the numbers given by Proposition 2.1. We find then that there is a point x_{2} with |x_{1} − x_{2}| ≤ Cη_{1} such that η_{2} ≡ sup_{t_{0}+T ≤t≤t_{0}+2T} d(v(t), w_{2}) ≤ \frac{η_{1}}{2} . where w_{2}(x) = w_{∗}(|x − x_{2}|). Since η_{2} ≤ η_{1}/2 < δ, we can apply again Proposition 2.1 to find a point x_{3} with now |x_{3} − x_{2}| ≤ C \frac{η_{1}}{2} such that η_{3} = sup_{t∈[t_{0}+2T,t_{0}+3T]} d(v(t), w_{3}) ≤ \frac{η_{1}}{4} where w_{3}(x) = w_{∗}(|x − x_{3}|). Iterating this procedure we find a sequence x_{j}, j = 1, 2, … such that |x_{j+1} − x_{j}| ≤ C \frac{η}{2^{j}} and

sup_{t∈[t_{0}+jT,t_{0}+(j+1)T]} d(v(t), w_{j}) ≤ \frac{η_{1}}{2^{j}}

with w_{j}(x) = w_{∗}(|x − x_{j}|).

Now let t be any number greater than t_{0}. Then t ∈ (t_{0} +jT, t_{0} +(j +1)T] for some j. Using that w_{∗} is Hölder continuous we see that d(w, w_{j}) ≤ C|x_{j} − x_{1}|^{a} for some a, C > 0. Moreover, |x_{j} − x_{1}| ≤ C ∑_{l=1}^{∞} η_{1}2^{−l} = Cη_{1}, hence

d(v(t), w) ≤ A{(η_{1})^{2} + C^{a}η_{1}^{a}}.

We conclude that there is a δ > 0 such that if η_{1} < δ, then d(v(t), w) < ε for all t > t_{0}. Finally, from [5, Proposition 4.1], there is a sequence s_{k} → ∞ such that v(x, s_{k} + τ) → w(x) for some nontrivial solution of (1.2), uniformly in x and for τ in bounded intervals, in particular for τ ∈ [0, T]. We recall that the space support of v is contained inside a ball independent of the time variable. It follows that, given ε > 0, there is indeed a number t_{0} > 0 such that η_{1} < δ. Since ε is arbitrary, we have actually established that w is the unique limit point of the trajectory v(·, t), and the proof of the theorem is complete. □

The remaining of this paper will be devoted to the proof of Proposition 2.1.

3 Preliminaries

Let v_{n} be a sequence as in the statement of Proposition 2.1. Then we have that v_{n}(x, t)^{m} − w_{n}(x)^{m} → 0 in L^{∞} and H^{1}-senses in ℝ^{N}, uniformly locally in time t ∈ [0, ∞), as it follows from the results in [5]. For T > 0 fixed, which we will choose later, we use in what follows the following notation.

ϕ_{n}(x, t) ≡ \frac{v_{n}(x, t) − w_{n}(x)}{η_{n}}.  (3.1)

Then ϕ_{n} satisfies the equation

\frac{∂ϕ_{n}}{∂t} = mΔ(w̃_{n}^{m−1}ϕ_{n}) + mw̃_{n}^{m−1}ϕ_{n} − \frac{1}{m − 1}ϕ_{n}  (3.2)

where

w̃_{n}(x, t)^{m−1} ≡ ∫_{0}^{1}(w_{n}(x) + t(v_{n}(x, t) − w_{n}(x)))^{m−1}dt.  (3.3)

Let us observe that, by definition of the number η_{n} we have that

(m ∫ w_{n}(x, t)^{m−1}ϕ_{n}(x, t)^{2}dx)^{1/2} ≤ 1

for all t ∈ [0, T]. In the limit w_{n} must converges (up to subsequences) to a steady state w of equation (1.2). Then w(x) = w_{∗}(|x−x_{1}|) for certain x_{1} ∈ ℝ^{N}. In what follows we shall assume, with no loss of generality that x_{1} = 0

These facts suggest that on interior sets of the support B of the limit w(x) of the w_{n} we should see convergence in certain sense of ϕ_{n} to a solution of the degenerate parabolic equation

ϕ_{t} = mΔ(w^{m−1}ϕ) + mw^{m−1}ϕ − \frac{1}{m − 1}ϕ.  (3.4)

By a weak solution of equation (3.4), we understand a function ϕ which is smooth in the interior of B = B × (0, ∞), and satisfies (3.4) there, such that moreover

∫_{0}^{s} ∫_{B}(m|∇(w^{m−1}ϕ)|^{2}m(w^{m−1}ϕ)^{2} + \frac{1}{m − 1}ϕ^{2})dxdt < +∞ for all s > 0.

(3.5)

Lemma 3.1 There is a subsequence of ϕ_{n} which we denote in the same way, for which ϕ_{n}(x, t) → ϕ(x, t) (up to a subsequence) in the uniform C^{1}-sense over compact subsets of B. Moreover, ϕ is a weak solution of (3.4).

Before proving Lemma 3.1 we introduce some notation. Let us write w_{n} = w_{∗}(|x − x_{n}|) and consider the functions

ψ_{n} = \frac{v_{n}^{m} − w_{n}^{m}}{η_{n}} .  (3.6)
G_{n} = \frac{1}{η_{n}^{2}} \{\frac{v_{n}^{m+1}}{m + 1} − \frac{w_{n}^{m+1}}{m + 1} − w_{n}^{m}(v_{n} − w_{n})\}.  (3.7)
H_{n} = \frac{(m − 1)(v_{n}^{m+1} − w_{n}^{m+1}) + (m + 1)(v_{n}w_{n}^{m} − v_{n}^{m}w_{n})}{η_{n}^{2}}.  (3.8)

It is easily checked the existence of constants C_{1} and C_{2}, depending only on m such that the following inequalities hold:

C_{1}G_{n} ≤ \frac{(v_{n}^{m} − w_{n}^{m})(v_{n} − w_{n})}{η_{n}^{2}} ≤ C_{2}G_{n}.  (3.9)

We have the validity of the following relation:

∫ \frac{∂}{∂t}G_{n} ≤ \frac{1}{m^{2} − 1} ∫ H_{n}.  (3.10)

In fact, let J be the Lyapunov functional for (1.5) given by (1.7). Let us set I = 2(J(v_{n}) − J(w_{n})). Then I ≥ 0. After integrating by parts and using the equations satisfied by v_{n} and w_{n} we get

I = − ∫ v_{nt}(v_{n}^{m}−w_{n}^{m})−\frac{1}{m − 1} ∫ (v_{n}+w_{n})(v_{n}^{m}−w_{n}^{m})+\frac{2m}{m^{2} − 1} ∫ (v_{n}^{m+1}−w_{n}^{m+1}).

Now, we have that η_{n}^{2} \frac{∂G_{n}}{∂t} = v_{nt}(v_{n}^{m} − w_{n}^{m}), so

I = −η_{n}^{2} ∫ \frac{∂G_{n}}{∂t} + \frac{1}{m − 1} ∫ (v_{n}w_{n}^{m} − v_{n}^{m}w_{n}) + \frac{1}{m + 1} ∫ (v_{n}^{m+1} − w_{n}^{m+1}).

Using the definition of H_{n}, inequality (3.10) follows.

Let us observe the following consequence of (3.10): since ∫ G_{n}(x, 0) ≤ C, and by definition |H_{n}| ≤ CG_{n}, then there exist constants a, b > 0 such that for all n, t,

∫ G_{n}(x, t) ≤ be^{at}.  (3.11)

Proof of Lemma 3.1. Let us now analyze the convergence of ϕ_{n}. We note that ψ_{n} defined by (3.6) satisfies the equation

(ϕ_{n})_{t} = Δψ_{n} + ψ_{n} − \frac{1}{m − 1}ϕ_{n}.  (3.12)

Integrating (3.12) against ψ_{n}, recalling that \frac{∂}{∂t}G_{n} = ψ_{n} \frac{∂ϕ_{n}}{∂t}, we obtain

\frac{∂}{∂t} ∫ G_{n}dx = ∫ \frac{∂ϕ_{n}}{∂t} ψ_{n}dx = − ∫ |∇ψ_{n}|^{2}dx + ∫ ψ_{n}^{2} − \frac{1}{m − 1} ∫ ϕ_{n}ψ_{n}. (3.13)

Note that given t > 0, there exists C(t) > 0 such that

∫_{0}^{t} ∫ (|∇ψ_{n}(·, s)|^{2} + |ψ_{n}(·, s)|^{2})ds ≤ C(t)  (3.14)

for all n. In fact, let us recall that

∫ G_{n}(s, x)dx ≤ be^{as}.

Since the function v is bounded we see that |ψ_{n}| ≤ C|ϕ_{n}|. Hence, using (3.9), we get that ψ_{n} ϕ_{n} < CG_{n} and ψ_{n}^{2} ≤ CG_{n}. Now integrating relation (3.13) in time, between s = 0 and s = t and using relation (3.11) we get (3.14).

As a consequence of the last result, the sequence ψ_{n} can be assumed, after passing to a subsequence, weakly convergent in L^{2}((0, S); H^{1}(ℝ^{N})) for each S > 0. Let ψ(x, s) be this limit. Recall that we are assuming w(x) = w_{∗}(|x|) , so that the support of w is the ball B = B(0, R^{∗}). Then if we define ϕ = w^{−(m−1)}ψ on B = B × (0, ∞) then

∫_{0}^{S} ∫_{B} |∇(w^{m−1}ϕ)|^{2} + |w^{m−1}ϕ|^{2} < +∞

for each S > 0. We recall that ϕ_{n} satisfies the equation

(ϕ_{n})_{t} = Δ(a_{n}ϕ_{n}) + a_{n}ϕ_{n} − ϕ_{n}  (3.15)

where \frac{a_{n}=(v_{n}^{m}−w^{m})/(}{v_{n}−w)}. Then a_{n} → mw^{m−1} uniformly. Hence over compacts of B the coefficient a_{n} is uniformly positive and bounded. The standard theory for quasilinear nondegenerate parabolic equations, see [19], gives that this convergence is also uniform in the C^{1}-sense over compacts of B, so that ∇a_{n} is also bounded there. Again the theory for nondegenerate parabolic equations in [19] provides uniform estimates for C^{1,α} norms over compacts of B, from where C^{1} convergence follows. The fact that ϕ is a weak solution of (3.4) is now easily checked. □

Let us consider the eigenvalue problem

mΔ(w^{m−1}ϕ) + mw^{m−1}ϕ − \frac{1}{m − 1}ϕ + λϕ = 0.  (3.16)

with ϕ such that w^{m−1}ϕ ∈ H^{1}(B) and w^{\frac{m−1}{2}} ϕ ∈ L^{2}(B). Here w(x) = w_{∗}(|x|) and B = B(0, R^{∗}).

Concerning this problem, we have the validity of the following result.

Lemma 3.2 There is only a finite number of negative eigenvalues (repeated according to multiplicity)

λ_{1} ≤ λ_{2} ≤ … ≤ λ_{k} < 0,

with associated eigenfunctions ϕ_{1}, …, ϕ_{k}, normalized so that ∫_{B} ϕ_{i}ϕ_{j}w^{m−1} = δ_{ij}. Besides, the only eigenfunctions of (3.16) for λ = 0 are linear combinations of the functions \frac{∂w}{∂_{x_{i}}}.

Let N be the finite-dimensional vector space spanned by all eigenfunctions in H associated to non-positive eigenvalues. Then there exists a number λ^{∗} > 0 such that

λ^{∗} ∫_{B} w^{m−1}|ϕ̃|^{2} ≤ ∫_{B} m( |∇(w^{m−1}ϕ̃)|^{2} + m|w^{m−1}ϕ̃|^{2} − \frac{w^{m−1}}{m − 1}ϕ̃^{2})dx

for all ϕ̃ ∈ H with ∫ w^{m−1}ϕ̃ζ = 0, for all ζ ∈ N.

The proof of the above lemma, which can be found in [7] is based on standard compactness arguments in Sobolev spaces. The fact that w^{m−1} vanishes quadratically on the boundary of its support plays a role here.

We can state now the following lemma.

Lemma 3.3 Let ϕ(x, t) be the function found in Lemma 3.1, then

ϕ(x, t) = ∑_{j=1}^{k} D_{j}e^{−λ_{j}t}ϕ_{i}(x) + ∑_{i=1}^{N} C_{i} \frac{∂w(x)}{∂x_{i}} + θ(x, t),  (3.17)

where θ(x, t) converges to zero as t → +∞, exponentially uniformly inside compact sets of the set A

Proof: Let us consider the expansion in B,

ϕ(x, 0) = ∑_{i=1}^{k} D_{i}ϕ_{i} + ∑_{i=1}^{N} C_{i} \frac{∂w}{∂x_{i}} + θ(x)

where

∫_{B} θϕ_{i}w^{m−1} = ∫_{B} θ \frac{∂w}{∂x_{j}} w^{m−1} = 0  (3.18)

for all i, j. Now let us consider the function

ϕ̃(x, t) = ϕ(x, t) − ∑_{i=1}^{k} D_{i}e^{−λ_{i}t}ϕ_{i} − ∑_{i=1}^{N} C_{i} \frac{∂w}{∂x_{i}} .

Let us observe that

∑ C_{i}^{2} + ∑ D_{i}^{2} ≤ ∫_{B} w^{m−1}ϕ^{2}(x, 0)dx ≤ 1.

Clearly ϕ̃(x, t) satisfies equation (3.4). We claim that

∫_{B} ϕ̃(·, s)ϕ_{i}w^{m−1} = ∫_{B} ϕ̃(·, s)\frac{∂w}{∂x_{j}} w^{m−1} = 0

for all s > 0. In fact, if we set for instance φ(s) = ∫_{B} ϕ̃(·, s)ϕ_{i}w^{m−1}, then φ′(s) = −λ_{i}φ(s). Since φ(0) = 0, the claim follows. Now, let us set η(s) = ∫_{B} w^{m−1}ϕ̃(·, s)^{2}. Then

η′(s) = −2 ∫_{B} m(|∇(w^{m−1}ϕ̃)|^{2} − m(w^{m−1}ϕ̃)^{2} + \frac{w^{m−1}}{m − 1}ϕ̃^{2})dx.

Since ϕ̃ satisfies the above orthogonality relations, it follows from Lemma 3.2 that

η′(s) ≤ −2λ^{∗}η(s),

with λ^{∗} > 0, and hence

η(s) ≤ η(0)e^{−2λ^{∗}s}.

Also, η(0) ≤ 1. Finally, linear parabolic regularity implies that exponential decay at this rate for ϕ̃ also holds uniformly on compact subsets of B. □

4 Analysis near the boundary

In last section we have found the validity of convergence of ϕ_{n} to ϕ essentially in the interior of the support of the limiting w. Here we will show estimates which provide control of ϕ_{n} near the boundary of the support of w. Our main purpose is to show that the contribution of the region near the boundary on the integral of G_{n} is basically negligible.

Lemma 4.1 Let ε > 0 be given. Then there exist numbers 0 < r_{0} < R^{∗} and s^{∗} > 0 such that for each given s̄ ≥ s^{∗} and all n sufficiently large we have

sup_{s∈[s^{∗},s̄]} ∫_{{|x|>r_{0}}} G_{n}(s, x)dx < ε.

The proof is divided into two steps. Step 1: We make the following claim:

There exist numbers A and c, depending only on m, with the following property: Given ϵ > 0 and 0 < r_{0} < R^{∗}, with r_{0} sufficiently close to R^{∗} and any t > 0, we have that for all n sufficiently large,

∫_{|x|≥r_{1}} G_{n}(x, t)dx ≤

A[e^{−ct} + ϵ + w∗^{\frac{3m−3}{2}} (r_{0}) sup_{(s,r)∈[0,t]×[r_{0},r_{1}]} ∫_{|x|=r} ϕ^{2}(x, s)dσ] (4.1)

where r_{1} = (r_{0} + R^{∗})/2.

To prove this claim we set D_{n} = {x/|x − x_{n}| ≥ r_{0}}, and define for x ∈ D_{n}

g(x) = w_{∗}^{m}(ar_{0}) − w_{∗}^{m}(a|x − x_{n}|), a ≡ \frac{2R^{∗}}{R^{∗} + r_{0}} > 1.

Let us multiply equation (3.12) by ψ_{n}(x, t)g(x) and integrate on D_{n}. Since g vanishes on the boundary of this region, recalling that (G_{n})_{t} = ψ_{n}(ϕ_{n})_{t}, we find

\frac{∂}{∂t} ∫_{D_{n}} G_{n}(x, t)g(x)dx

= B_{n}(t) + ∫_{D_{n}} \frac{ψ_{n}^{2}}{2} Δg(x)dx + ∫_{D_{n}}\{−|∇ψ_{n}|^{2} + ψ_{n}^{2} − \frac{1}{m − 1}ϕ_{n}ψ_{n}\}g(x) dx. (4.2)

where

B_{n}(t) = −\frac{a∂w_{∗}^{m}}{∂r} (ar_{0}) ∫_{|x−x_{n}|=r_{0}} \frac{ψ_{n}^{2}}{2} dσ.

Now, Δw_{∗}^{m} = \frac{w_{∗}}{m−1} − w_{∗}^{m} ≥ 0 for |x| ≥ r_{0} if r_{0} is sufficiently close to R^{∗}, hence we assume Δg(x) ≤ 0 on D_{n}. Using these observations and (4.2) we get

\frac{∂}{∂t} ∫_{D_{n}} G_{n}(x, t)g(x)dx ≤ B_{n}(t) + ∫_{D_{n}}\{ψ_{n}^{2} − \frac{1}{m − 1}ϕ_{n}ψ_{n}\}g(x) dx.

On the other hand, recalling the definition of ψ_{n}, estimate (3.10) then reads

0 ≤ − ∫ \frac{∂G_{n}}{∂t} + \frac{1}{m − 1}ϕ_{n}ψ_{n} − \frac{2}{m − 1}G_{n}.  (4.3)

We obtain then that

∫_{D_{n}} \frac{∂G_{n}}{∂t} (g(x) + w_{∗}^{m}(ar_{0}))

≤ B_{n}(t)+ \frac{1}{m − 1} ∫_{D_{n}} ϕ_{n}ψ_{n}(w_{∗}^{m}(ar_{0})−g(x))+∫_{D_{n}}\{ψ_{n}^{2}g(x)− \frac{2w_{∗}^{m}(ar_{0})}{m − 1}G_{n}.\}. (4.4)

Here we observe that w_{∗}^{m}(ar_{0})−g(x) = 0 on F_{n} = {a|x−x_{n}| ≥ R^{∗}}. Now, given t^{∗} > 0 we have that v_{n} → w uniformly on [0, t^{∗}] × ℝ^{N}. Thus, if r_{0} is sufficiently close to R^{∗} we obtain that ψ_{n}^{2} ≤ \frac{1}{(m−1)}G_{n} for |x| ≥ r_{0} and 0 < t < t^{∗} for large n. Hence

ψ_{n}^{2}g(x) − \frac{2w_{∗}^{m}(ar_{0})}{m − 1}G_{n} ≤ −\frac{(w_{∗}^{m}(ar_{0}) + g(x))}{(2(m − 1))}G_{n}.

on this region. Substituting this information into relation (4.4), we obtain the following differential inequality for all sufficiently large n.

Y′_{n}(t) ≤ B_{n}(t) + W_{n}(t) − cY_{n}(s), 0 < t < t^{∗},

where c is a positive constant depending only on m and

Y_{n}(t) = ∫_{D_{n}} G_{n}(x, t)(w_{∗}^{m}(ar_{0}) + g(x)),

W_{n}(t) = \frac{1}{m − 1} ∫_{D_{n}\F_{n}} ϕ_{n}ψ_{n}(w_{∗}^{m}(ar_{0}) − g(x)).

It follows that

Y_{n}(s) ≤ Y_{n}(0)e^{−ct} + e^{−ct} ∫_{0}^{t} e^{cs}(B_{n}(s) + W_{n}(s))ds.  (4.5)

We will estimate the right hand side of (4.5). First, we see that B_{n}(s) → B(s) and W_{n}(s) → W (s) uniformly on compact sets where

B(s) = −\frac{a∂w_{∗}^{m}}{∂r} (ar_{0}) ∫_{|x|=r_{0}} m^{2}w_{∗}^{2(m−1)}(r_{0})\frac{ϕ^{2}}{2} dσ,

and

W (s) = \frac{m}{m − 1} ∫_{r_{0}≤|x|≤ \frac{r_{0}+R^{∗}}{2}} ϕ^{2} (w_{∗}^{m−1}(|x|) (w_{∗}^{m}(ar_{0}) − g(x)).

Now,

(w^{m})′(r_{0})w^{2m−2}(r_{0}) = \frac{2}{m − 1}(w^{\frac{m−1}{2}} )′(r_{0})w^{m}(r_{0})w^{\frac{3m−3}{2}} (r_{0}),

and w(ar_{0}) < w(r_{0}) , for r_{0} close enough to R^{∗}, so that

|B(s)| ≤ Cw^{m}(r_{0})w^{\frac{3m−3}{2}} (r_{0}) ∫_{|x|=r_{0}} ϕ^{2}dσ.

On the other hand,

|W (s)| ≤ C(w_{∗}^{m−1}(r_{0})w_{∗}^{m}(ar_{0})(\frac{R^{∗} − r_{0}}{2} ) sup_{r∈[r_{0}, \frac{r_{0}+R^{∗}}{2} ]} ∫_{|x−x_{i}|=r} ϕ^{2}dσ

Since \frac{∂}{∂r} w∗^{\frac{m−1}{2}} (R^{∗}) < −c < 0, we have

\frac{R^{∗} − r_{0}}{2} ≤ Cw∗^{\frac{m−1}{2}} (r_{0})

for some C > 0 depending only on m, provided that r_{0} is sufficiently close to R^{∗}. From these facts and (4.5) we see that for given ε > 0 and t > 0,

Y_{n}(t) ≤ Y_{n}(0)e^{−ct} + ϵ + Cw_{∗}^{m}(r_{0})w∗^{\frac{3m−3}{2}} (r_{0}) sup_{[0,t]} sup_{r∈[r_{0}, \frac{r_{0}+R^{∗}}{2} ]} ∫_{|x|=r} ϕ^{2}dσ,

for all sufficiently large n, where we have used again that w_{∗}(ar_{0}) < w_{∗}(r_{0}). Finally it is easily checked that 1 ≤ \frac{w_{∗}(r_{0})}{w_{∗}(ar_{0})} ≤ C for some constant C indepen-

dent of r_{0} close to R^{∗}. Since g(x) = w_{∗}^{m}(ar_{0}) for |x| ≥ \frac{r_{0}+R^{∗}}{2} , the resulting inequality (4.1) then follows. Step 2: As a second step in the proof we claim that in the expansion (3.17), we have D_{i} = 0 for i = 1, …, k.

We have that ∫ G_{n}(x, t)dx ≤ \frac{1}{m^{2}−1} ∫_{0}^{t} ∫ H_{n} + C, for certain number C independent of t. Let us fix a number r_{0} close to R^{∗} as in the first step and set r_{1} = \frac{r_{0}+R^{∗}}{2} . We also write

B_{r_{1}} = {x/|x| ≤ r_{1}}.

Then there is a constant D > 0 depending on r_{0} such that

|H_{n}| ≤ \frac{D}{η_{n}^{2}} |v_{n} − w|^{3} on B_{r_{1}},

from where it follows that ∫_{0}^{t} ∫_{B_{r_{1}}} H_{n} → 0 as n → ∞, for each fixed t. Then recalling that |H_{n}| ≤ CG_{n}, we find that for some C > 0 independent of r_{0} and all sufficiently large n,

∫_{B_{r_{1}}} G_{n}(x, t)dx ≤ C[\frac{1}{m^{2} − 1} ∫_{0}^{t} ∫_{ℝ^{N} \B_{r_{1}}} G_{n} + 1].

Then, passing to the limit, recalling that G_{n} converges uniformly in B_{r_{1}} × [0, t] to mw^{m−1}ϕ^{2}, we get

∫_{B_{r_{1}}} mw^{m−1}ϕ^{2}(x, t)dx ≤ lim sup_{n→∞} C(∫_{0}^{t} ∫_{ℝ^{N} \B_{r_{1}}} G_{n} + 1).

Now, from the expression (3.17) for ϕ, we obtain

ϕ(x, t) = ∑_{j=1}^{k} D_{j}e^{−λ_{j}t}ϕ_{i}(x) + O(1).

with O(1) uniformly bounded in time and space inside B_{r_{1}}. It follows that

∫_{A_{∇∞}} w^{m−1}ϕ^{2}(x, t)dx = ∑_{j=1}^{k} D_{j}^{2}e^{−2λ_{j}t} + O(1).

On the other hand, we can find numbers A and c which depend only on m so that

lim sup_{n→∞} ∫_{ℝ^{N} \B_{r_{1}}} G_{n}(x, s)dx

≤ A\{e^{−cs}+w∗^{\frac{3m−3}{2}} (r_{0}) sup\{∫_{|x|=r} ϕ^{2}(x, s)dσ : s ∈ [0, t], r ∈ [r_{0}, r_{1}], 1 ≤ i ≤ k\}. (4.6)

It can also be proved that if ϕ_{i} is an eigenfunction corresponding to a negative eigenvalue of (3.16), then |ϕ_{i}(x)| ≤ Cw(|x|)^{\frac{3−m}{2}} , hence

w^{\frac{3m−3}{2}} (r_{0}) sup\{∫_{|x|=r} ϕ^{2}dσ : s ∈ [0, t], r ∈ [r_{0}, r_{1}], 1 ≤ i ≤ k\}

≤ Cw(r_{0})^{\frac{m+3}{2}} ∑_{j=1}^{k} D_{j}^{2}e^{−2λ_{j}t} + O(1), (4.7)

where C is independent of r_{0}. Combining the above relations, we get then that for certain constant C

∑_{j=1}^{k} D_{j}^{2}e^{−2λ_{j}t} ≤ Cw(r_{0})^{\frac{m+3}{2}} ∑_{j=1}^{k} D_{j}^{2}e^{−2λ_{j}t} + O(1),

where C is independent of r_{0}. Since w(r_{0}) may be chosen arbitrarily small, we obtain a contradiction from this last relation for all t sufficiently large if any of the D_{j}’s was not zero. This completes the second step.

The result of the lemma follows now easily from combining steps one and two. □

5 Proof of Proposition 2.1

We define x̄_{in} = η_{n}(C_{1}, …, C_{N}). Then |x̄_{n}| ≤ Cη_{n} with C = C(m). Let us write w̄_{n}(x) = w_{∗}^{m}(x + x̄_{n}). We want to estimate the quantity

I_{n}(s) = ∫ (v_{n}(x, s)^{m} − w̄_{n}(x)^{m})(v_{n}(x, s) − w̄_{n}(x))dx

Let us consider r ∈ [0, R^{∗}], to be determined later, and set B_{r} = {|x| < r}. Then

I_{n}(s) = ∫_{B_{δ}}(v_{n}^{m} − w̄_{n}^{m})(v_{n} − w̄_{n})dx + ∫_{ℝ^{N} \B_{δ}}(v_{n}^{m} − w̄_{n}^{m})(v_{n} − w̄_{n})dx
= I_{n}^{1}(s) + I_{n}^{2}(s).

We have

I_{n}^{1}(s) ≤ C[∫_{B_{r}}(v_{n}^{m} − w_{n}^{m})(v_{n} − w_{n})dx + ∫_{B_{r}}(w̄_{n}^{m} − w_{n}^{m})(w̄_{n} − w_{n})dx].

Now, from 3.9, we get

∫_{B_{r}}(v_{n}^{m} − w_{n}^{m})(v_{n} − w_{n})dx ≤ Cη_{n}^{2} ∫_{B_{r}} G_{n}(x, s)dx,

again with C = C(m). Corollary 4.1 then implies that if r is chosen close enough to R∗, depending on m, and s ≥ s^{∗}, with s^{∗} also depending only on m then for all n sufficiently large

C ∫_{B_{r}} G_{n}(x, s)dx < \frac{1}{8}.

Also,

C(w̄_{n}^{m} − w_{n}^{m})(w̄_{n} − w_{n}) ≤ Kη_{n}^{2}

with K depending on m and k only. Therefore, taking r closer to R^{∗}, if necessary, we get

C ∫_{B_{r}}(w̄_{n}^{m} − w_{n}^{m})(w̄_{n} − w_{n})dx ≤ \frac{η_{n}^{2}}{8}

if n is large enough. Putting these two estimates together we see that if we choose T ≥ s^{∗}, then

I_{n}^{1}(s) ≤ \frac{η_{n}^{2}}{4}

for all s ∈ [T, 2T] provided that n is sufficiently large. On the other hand, we recall that

v_{n}(x, s) = w_{n}(x) + η_{n}ϕ_{n}(x, s),

so that we can write

v_{n}(x, s) = w_{n}(x) + η_{n} ∑_{i=1}^{N} C_{i} \frac{∂w_{∗}}{∂x_{i}} (x) + η_{n}θ(x, s) + (ϕ_{n}(x, s) − ϕ(x, s)),

where θ decays exponentially in compact sets of A.

Now, since r has been already fixed it follows that w̄_{n}(x) = w_{∗}(x − x_{n} + x̄_{n}) and

lim_{n→∞} η_{n}^{−2} ∫_{ℝ^{N} \B_{r}}(v_{n}^{m} − w̄_{n}^{m})(v_{n} − w̄_{n})dx = m ∫_{ℝ^{N} \B_{r}} w^{m−1}(x)θ^{2}(x, s) dx

uniformly on s on compact subsets of (0, ∞). Since θ(x, s) decays exponentially we have that there are positive numbers A and a, depending only on m, such that

I_{n}^{2}(s) ≤ η_{n}^{2}Ae^{−at}.

Consequently we have

I_{n}(s) ≤ η_{n}^{2}(\frac{1}{4} + Ae^{−aT})

for all s ∈ [T, 2T]. Making T larger if necessary (depending only on m) we obtain that the quantity between brackets is less than 1/2. This concludes the proof of the Proposition. □

Acknowledgements C. Cortázar and M. Elgueta were supported by grant 1000625 from Fondecyt. M. del Pino was supported by grant 8000010 from Lineas Complementarias Fondecyt and by grant from Fondap Matematicas Aplicadas.

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Carmen Cortázar (e-mail: ccortaza@mat.puc.cl) Manuel Elgueta (e-mail: melgueta@mat.puc.cl) Departamento de Matemáticas Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile Manuel del Pino Departamento de Ingeniería Matemática and CMM Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile e-mail: delpino@dim.uchile.cl