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MULTI-SCALE MULTI-PROFILE GLOBAL SOLUTIONS OFPARABOLIC EQUATIONS IN RN

THIERRY CAZENAVE, FLAVIO DICKSTEIN, AND FRED B. WEISSLER

Abstract. This paper explores certain concepts which extend the notions of

(forward) self-similar and asymptotically self-similar solutions. A self-similar

solution of an evolution equation has the property of being invariant withrespect to a certain group of space-time dilations. An asymptotically self-

similar solution approaches (in an appropriate sense) a self-similar solution to

first order approximation for large time. Such solutions have a definite long-time asymptotic behavior, with respect to a specific time dependent spatial

rescaling. After reviewing these fundamental concepts and the basic knownresults for heat equations on RN , we examine the possibility that a global

solution might not be asymptotically self-similar. More precisely, we show

that the asymptotic form of a solution can evolve differently along differenttime sequences going to infinity. Indeed, there exist solutions which are as-

ymptotic to infinitely many different self-similar solutions, along different time

sequences, all with respect to the same time dependent rescaling. We exhibitan explicit relationship between this phenomenon and the spatial asymptotic

behavior of the initial value under a related group of dilations. In addition, we

show that a given solution can exhibit nontrivial asymptotic behavior alongdifferent time sequences going to infinity, and with respect to different time

dependent rescalings.

1. Introduction

This article concerns the long-time asymptotic behavior of global solutions ofa certain class of parabolic equations in RN . One reason for studying the long-time asymptotic behavior of solutions of evolution equations is the hope and theexpectation that it will be simpler to describe than the transcient behavior. Also, inthe case where an equation models a real world phenomenon, the observed behavioroften corresponds to the long-time asymptotic behavior of a solution, especially ifone waits until the transcient effects have subsided. In general, solutions of apartial differential evolution equation live in an infinite dimensional Banach space.In many cases they are attracted to a finite dimensional manifold and can thereforebe described asymptotically by a finite dimensional system. It can even happenthat all solutions in a certain class approach the same asymptotic form for largetime. Many existing results in the literature are of this nature. On the other hand,the recent work of the authors described in this article shows that a large class ofsolutions do not conform to this paradigm.

The study of long-time asymptotic behavior of evolution equations is a vast sub-ject. The results which concern us here, both the previously known work and the

1991 Mathematics Subject Classification. 35K05, 35K58, 35B40.Key words and phrases. heat equation, asymptotic behavior, rescaling, decay rate, dilation

properties, asymptotically self-similar solutions, dynamical systems.

1

2 THIERRY CAZENAVE, FLAVIO DICKSTEIN, AND FRED B. WEISSLER

recent work we describe, are based on scaling properties of the evolution equation.It is well-known that for a large number of linear and nonlinear equations the setof solutions is invariant under some group of space-time dilations. This phenom-enon gives rise to the study of self-similar solutions, i.e. those solutions whichare themselves invariant under the same group of dilations. Often, the self-similarsolutions represent a more general behavior in that they attract a larger class ofsolutions. In other words, it can happen that a solution is asymptotic for large time,i.e. as t → ∞, to a self-similar solution. Such solutions are called asymptoticallyself-similar.

Self-similar and asymptotically self-similar solutions can be studied from thepoint of view of dynamical systems. In the class of problems we describe, we areconcerned with global solutions which decay to 0 as t → ∞. However, after asuitable change of variables (using so called “self-similar variables”), self-similarsolutions become stationary solutions of a new equation, and asymptotically self-similar solutions become solutions that converge to a stationary solution. In otherwords, a solution is asymptotically self-similar if, in terms of self-similar variables,its ω-limit set is a singleton.

In this article we describe some recent work motivated by the question as towhether or not asymptotically self-similar behavior is generic. In terms of dynamicalsystems, this amounts to studying, in self-similar variables, the ω-limit set of generalsolutions. Is the typical ω-limit set a singleton or do most solutions have a biggerω-limit set?

The authors have shown that asymptotically self-similar behavior is not generic.There exist solutions which are asymptotic along different time sequences tn →∞to different self-similar solutions. In terms of self-similar variables, the ω-limit setof such a solution contains more than one element. In fact, there exist solutionswhose ω-limit set, in terms of self-similar variables, is a closed ball in an infinitedimensional Banach space. This is the extreme opposite of a solution whose ω-limitis a singleton, i.e. an asymptotically self-similar solution.

There is a different way in which a solution can fail to be asymptotically self-similar. If a solution is asymptotically self-similar, then its norm in a suitableBanach space must decay at the same rate as the self-similar solution. However, wewill see that there exist solutions which decay at different rates along different timesequences tn →∞. Consequently, such a solution is not asymptotically self-similar,but it can be asymptotically self-similar along different time sequences tn →∞ todifferent self-similar solutions, with respect to different rescalings.

The construction of these non-asymptotically self-similar solutions is based ona careful study of the relationship between the spatial asymptotic behavior of theinitial value and the space-time asymptotic behavior of the resulting solution of theevolution equation. Under appropriate conditions, the limit function obtained by asequence of spatial dilations applied to the initial value of a solution enables one todetermine the limit obtained by applying a corresponding sequence of space-timedilations to the solution.

We consider here the following specific linear and nonlinear heat equations,

ut −∆u+ a|u|αu = 0, (1.1)

in RN where α > 0 and

a = 1,−1 or 0. (1.2)

MULTI-SCALE MULTI-PROFILE SOLUTIONS 3

The purpose of this article is to give a clear and relatively nontechnical expositionof the ideas and results which have appeared in previous work by the authors [8, 9,11, 12, 13, 14]. We also present and prove some new results.

The rest of the paper is organized as follows. In Section 2 we recall the defini-tion and properties of (forward) self-similar solutions of (1.1). Then in Section 3we recall the notion of asymptotically self-similar solutions, giving several equiv-alent formulations. We also include a review of the literature on asymptoticallyself-similar solutions. At the end of Section 3 we make some formal observationswhich provide the intuition needed to understand the generalizations of asymptot-ically self-similar solutions which we introduce in the subsequent sections. Next,in Sections 4 and 5 we discuss the first extension of the notion of asymptoticallyself-similar solutions: multi-profile solutions. In other words, we describe how andunder what circumstances a given solution can be asymptotically self-similar todifferent self-similar solutions along different time sequences. In this first case, onlyone rescaling is considered. Also in this case, the fixed time flow generated by thisequation, combined with an appropriate spatial rescaling, generates a chaotic dis-crete dynamical system. (See Section 6.) The generator of this dynamical system isin fact the renormalization group map used in [4]. We alert the reader that startingin the latter part of Section 4 and throughout Sections 5 and 6, we specialize to thesituation where α > 2/N . In this case, the results for the linear and nonlinear heatequations are analogous. At the end of Section 5, we make a brief remark about thelimiting case α = 2/N . In Sections 7 and 8, we discuss how a given solution of (1.1)can exhibit nontrivial asymptotic limits along different time sequences, and withrespect to different rescalings. In particular, in Section 7 we construct a positivesolution of the linear heat equation (i.e. (1.1) with a = 0) which has no definitedecay rate: it realizes all possible decay rates, along different time sequences goingto infinity. In Section 8, we show that a single solution of the linear heat equationcan exhibit nontrivial time-asymptotic behavior with respect to certain rescalingsthat do not even leave that equation invariant. The results described in Sections 7and 8 show that the linear heat equation has solutions with unexpectedly compli-cated long-time asymptotic behavior. In Section 9, we return to the question ofmulti-profile solutions with respect to one specific rescaling in the case a = 1 andα < 2/N . As motivation for our results, we begin this section by recalling thebasic properties of the well-known “very singular solution” of (1.1), which is in factself-similar. It is important to note that the results in this section are genuinelynonlinear, and have no analogue in the linear case a = 0. Finally, in Section 11, wegive the proofs of certain results stated in this paper which have not appeared inany of our previous articles.

We mention that related work has been done for other equations, including theporous medium equation [40, 6], the Navier-Stokes system [10] and Schrodinger ’sequation [16, 45, 23].

2. Self-similar solutions

It is well-known that the Cauchy problem for (1.1) is locally well-posed in thespace C0(RN ) of continuous functions vanishing at infinity. We denote by S(t) theresulting semiflow on C0(RN ), i.e.

S(t)u0 = u(t), (2.1)


where u(t) is the solution of (1.1) such that u(0) = u0. Given u0 ∈ C0(RN ),S(t)u0 ∈ C0(RN ) is defined for all t ≥ 0 in the maximal interval of existence ofu. If a = 0, then S(t) is of course the heat semigroup et∆. If a = 1, then all thesolutions of (1.1) are global in time, and so S(t)u0 is defined for all u0 ∈ C0(RN )and all t ≥ 0.

It is also well-known that equation (1.1) has the following invariance property:u ∈ C([0,∞), C0(RN )) is a solution of (1.1) if and only if

λσu(λ2t, λx) (2.2)

is also a solution for all λ > 0, where σ = 2/α in the nonlinear case a 6= 0.Furthermore, the corresponding self-similar change of variables is given by

v(s, y) = tσ2 u(t, x), (2.3)

wheres = log t, y =

x√t, (2.4)

i.e.v(s, y) = e

σs2 u(es, ye

s2 ). (2.5)

It follows that u ∈ C((0,∞), C0(RN )) is a solution of (1.1) if and only if v ∈C(R, C0(RN )) is a solution of

vs −∆v − 12y · ∇v − σ

2v + a|v|αv = 0. (2.6)

Note that v(0) = u(1) and that the Cauchy problem for (2.6), in the same way asfor (1.1), is locally well-posed in C0(RN ). Note also that

‖v(s)‖L∞ = tσ2 ‖u(t)‖L∞ .

In particular, ‖v(s)‖L∞ is bounded as s→∞ if and only if tσ2 ‖u(t)‖L∞ is bounded

as t → ∞. Typical examples of such solutions include self-similar and asymptoti-cally self-similar solutions.

A solution U ∈ C((0,∞), C0(RN )) of (1.1) is self-similar if

U(t, x) ≡ λσU(λ2t, λx),

for all λ > 0, where σ = 2/α in the case a 6= 0. Setting λ = t−12 , one immediately

sees that U is self-similar if and only if

U(t, x) ≡ t−σ2 f( x√

t

), (2.7)

where f = U(1) is called the profile of the self-similar solution U . (A solutiongiven by (2.7) is in fact a forward self-similar solution. In this paper, we will notbe concerned with backward self-similar solutions.) Direct substitution of (2.7)into (1.1) shows that the profile f must satisfy the equation

−∆f − 12y · ∇f − σ

2f + a|f |αf = 0. (2.8)

Note that if we apply the transformation (2.5) to the self-similar solution U of (1.1),the resulting solution V of (2.6) is given by V (s, y) ≡ f(y). In other words, asolution U of (1.1) is self-similar if and only if the corresponding solution V of (2.6)is stationary, i.e. a solution of (2.8).


It is clear that a self-similar solution with nontrivial profile f will never be inC([0,∞), C0(RN )). On the other hand, if U(t, x) is the self-similar solution givenby (2.7), and if U(t)→ U0 as t→ 0, in any reasonable sense, then formally

U0(x) = |x|−σ limr→∞

rσf( x|x|r). (2.9)

Therefore, if a self-similar solution U has an initial value U0, it must be homoge-neous, i.e.

U0(x) ≡ λσU0(λx),

for all λ > 0 and all x 6= 0. Conversely, if the limit on the right hand side of (2.9)exists in some reasonable sense, then the self-similar solution U has an initial valueU0, given by (2.9). This fundamental observation was first made (in the context ofthe Navier-Stokes system) by Giga and Miyakawa [26] and by Cannone, Meyer andPlanchon [5].

Of course, it is not immediately clear under what conditions a self-similar solutionof (1.1) exists, and if it exists, what the properties of its profile can be. At this pointwe simply mention some of the standard approaches used to prove the existence ofself-similar solutions. The first basic approach is to prove the existence of a solutionto the profile equation (2.8). This problem has often been studied using variationalmethods. If one looks only for radially symmetric solutions, then (2.8) reduces toan ordinary differential equation, which can be studied by shooting methods andphase plane methods. A second approach is to study the Cauchy problem for (1.1)in a space which includes homogeneous functions. Solutions with homogeneousinitial values will then be self-similar. It is this latter approach which led to theideas described in Sections 4, 5 and 9 below.

3. Asymptotically self-similar solutions

A solution u ∈ C([0,∞), C0(RN )) of (1.1) is asymptotically self-similar if thereexists a self-similar solution U of the form (2.7) such that

tσ2 ‖u(t)− U(t)‖L∞ −→

t→∞0. (3.1)

In other words, the difference between the given solution u and the self-similarsolution U decays faster as t → ∞ than either solution separately (provided U isnontrivial). Condition (3.1) is equivalent under the transformation to self-similarvariables (2.3)-(2.4) to

v(s) −→s→∞

f, (3.2)

in L∞(RN ), where f is the profile of U . Thus we see explicitly that asymptoticallyself-similar solutions u of (1.1), correspond to solutions v of (2.6) whose ω-limitset contains precisely one element, i.e. the profile of the self-similar solution. Theproperty (3.2) can also be written in the following form, which uses the originaltime variable, but the “self-similar” space variable:

tσ/2u(t, ·√t) −→t→∞

f, (3.3)

in L∞(RN ). Finally, condition (3.3) is equivalent to

λσu(λ2t, λx) −→λ→∞

U(t, x) (3.4)


uniformly for x ∈ RN and 0 < t1 ≤ t ≤ t2 < ∞. This can be seen by settingt = λ2τ in (3.3) and letting λ→∞. After a spatial dilation this gives (3.4) with treplaced by τ .

Remark 3.1. The above definition in fact includes two different possibilities.(i) In the first case, the solution u and the self-similar solution U are both solu-

tions of (1.1) with the same value of a. If a 6= 0, then σ = 2/α. In this casewe say simply that u is asymptotically self-similar.

(ii) The other possibility is that u is a solution of (1.1) with a 6= 0 and U is aself-similar solution of the linear heat equation, i.e. (1.1) with a = 0. Thiscan happen if the solution u decays fast enough so that the nonlinear termin (1.1) becomes insignificant in determining the asymptotic behavior. In thiscase we say that u is asymptotically self-similar to the solution U of the linearheat equation.

We now give a sampling of the known results about asymptotically self-similarsolutions of equation (1.1). This subject alone has a rather extensive literature.The reader can consult the papers cited below for further results, and for referencesto related papers. In addition, the book [25] is a good reference for the study ofasymptotically self-similar solutions for a variety of equations.

In the case a = 0, so that S(t) = et∆, it is easy to check that the fundamental

solution u(t, x) = (4πt)−N2 e−

|x|24t is self-similar with respect to the scaling (1.4)

with σ = N with profile

G(y) = (4π)−N2 e−

|y|24 . (3.5)

It is straighforward to see that if u0 ∈ L1(RN ), then the resulting solution of (1.1),i.e. u(t) = et∆u0, is asymptotically self-similar to the self-similar solution withprofile cG where c =

∫u0. (See [30, 8].) Also, if 0 < σ < N then et∆| · |−σ is a

self-similar solution of (1.1) with profile e∆| · |−σ. Moreover, if u0 ∈ C0(RN ) is suchthat

|x|σu0(x) −→|x|→∞

c, (3.6)

(for some c 6= 0), then the resulting solution of (1.1), u(t) = et∆u0, is asymptoticallyself-similar to the solution et∆(c| · |−σ). It is worth noting that the condition (3.6)implies that

λσu0(λ·) −→λ→∞

c| · |−σ, (3.7)

uniformly for |x| ≥ r0 > 0.We turn to the case a = 1. One observes that there is a spatially constant

self-similar solution, i.e. the solution of the resulting ordinary differential equa-tion obtained from (1.1) by deleting the diffusion term, U(t, x) = (σ/2t)

σ2 , where

σ = 2/α. Gmira and Veron [27] prove that if the initial value is nonnegative anddecreases slowly enough as |x| → ∞, i.e. (3.6) holds with c =∞ and σ = 2/α, thenthe resulting solution of (1.1) is asymptotic as t→∞ to this spatially constant self-similar solution. Since the profile, the constant function (σ/2)

σ2 , is not in C0(RN ),

the convergence (3.1) is replaced by uniform convergence on “parabolic” sets of theform |x| ≤ Ct for all C > 0. Kamin and Peletier [31] show that if α > 2/N and ifthe initial value u0 ≥ 0 in C0(RN ) satisfies (3.6), for some finite c > 0 and σ = 2/α,then the resulting solution of (1.1) is asymptotically self-similar to an appropriateself-similar solution of the same equation. Its profile has the same spatial decay as


u0, as described by (3.6) with the same c > 0. This result is also true in the caseα = 2/N (see [30, 7]).

Still in the case a = 1, Gmira and Veron [27] show that if α > 2/N andu0 ∈ L1(RN ) (not necessarily nonnegative), then the resulting solution of (1.1)is asymptotically self-similar, with respect to the scaling (2.2) with σ = N , tothe self-similar solution of the linear heat equation whose profile is cG for an ap-propriate constant c. Kamin and Peletier [31] show that if (3.6) holds for some2/α < σ < N , then the resulting solution of (1.1) is asymptotically self-similar tothe self-similar solution et∆(c| · |−σ) of the linear heat equation. See [30, 7] for thecase σ = N and α > 2/N .

If a = 1 and α < 2/N , there is a unique positive self-similar solution of (1.1),called the “very singular solution” whose profile decays exponentially [43, 2, 20].Escobedo, Kavian and Matano [22] (see also [21]) show that if the initial valueu0 ≥ 0 in L1(RN ) is nontrivial and (3.6) holds with σ = 2/α and c = 0, thenthe resulting solution of (1.1) is asymptotically self-similar to the very singularsolution. It is also shown in [22] that if u0 ≥ 0 is in L1(RN ) and (3.6) holds withσ = 2/α and c > 0, then the resulting solution is asymptotically self-similar to aself-similar solution of the same equation whose profile has the same spatial decayas u0, as described by (3.6) with the same c > 0. See also [35, 30, 7]. For otherapproaches to the same problem, we refer the reader to Bricmont and Kupiainen [3]and Wayne [41].

The above results for a = 1, in [31, 22, 30], as well as the case a = 0, have allbeen extended in [7] to allow a more general version of (3.6), i.e. the limiting formas |x| → ∞ of |x|σu0(x) can depend on the direction. More precisely, the conditionon the initial value u0 ∈ C0(RN ) is given by

|x|σu0(x)− η(x) −→|x|→∞

0, (3.8)

where now η ∈ C0(RN \ 0) is a nontrivial homogeneous function of degree 0. Interms of the spatial dilations of u0, in analogy with formula (3.7), condition (3.8)implies that

λσu0(λ·) −→λ→∞

z, (3.9)

uniformly for |x| ≥ r0 > 0, where z ∈ C0(RN \ 0) is a homogeneous functionof degree −σ, z(x) = η(x)|x|−σ. It is also shown in [7] that if α ≥ 2/N , then nopositivity condition on u0 or on the limit as |x| → ∞ is required.

If a = −1 and α ≤ 2/N , then there are no global positive solutions of (1.1), sowe do not consider this case at all. (See [24, 29, 34, 1, 42, 32].)

In the case a = −1 and α > 2/N , a smallness condition on the initial value isneeded in order to guarantee that the resulting solution is global. Most of the resultson asymptotically self-similar solutions in this case are equally valid with a = 1instead of a = −1, with the same smallness hypothesis. Moreover, no positivityassumption is required. Combining the results of Cazenave and Weissler [15] andSnoussi, Tayachi and Weissler [38], we know that if u0 is small in some appropriatenorm, and if (3.8) holds with 2/α ≤ σ < N and some nontrivial η ∈ C0(RN \ 0),then the resulting solution of (1.1) is asymptotically self-similar to a self-similarsolution U(t) whose profile has the same decay properties described in (3.8). If2/α = σ, then U(t) is a self-similar solution of the same equation (1.1). If 2/α <σ < N , then U(t) is of the form et∆(η(·)| · |−σ). In fact, the formulations in [15]


and [38] use a condition slightly different from (3.8). Note that if (3.8) holds withσ < N and η 6≡ 0, then u0 is not in L1(RN ). Bricmont, Kupiainen and Lin [4] showin one space dimension and with integer α (with uα+1 replacing |u|αu in (1.1)),that if u0 is small in an appropriate sense (which implies small in L1(RN )), thenthe resulting solution is asympotically self-similar to a solution of (1.1) with a = 0with profile cG for an appropriate constant c. Wayne [41] uses invariant manifoldtheory to show (in a special case) that the higher order asymptotics are governed bya finite-dimensional dynamical system. See also [37, 38] which treat heat equationswith more general nonlinearities.

Finally, still in the case a = −1, if 2/N < α < 4/(N − 2) then there existsa unique positive self-similar solution U(t) of (1.1) whose profile has exponentialdecay (see [28, 44, 18, 46]). It thus follows from Kawanago [33] that given a positiveφ ∈ C0(RN ) with sufficiently rapid exponential decay as |x| → ∞, there exists aunique λ > 0 such that the solution of (1.1) with initial value u0 = λφ is asymp-totically self-similar to the solution U .

We conclude this section with some observations which lay the groundwork forthe rest of this paper. Many of the results cited above have the following form:if the initial value u0 of a solution u ∈ C([0,∞), C0(RN )) of (1.1) satisfies thecondition (3.8) for some η ∈ C0(RN \ 0) homogeneous of degree 0, η 6≡ 0, thenu is asymptotically self-similar to a self-similar solution U whose profile f has thesame asymptotic spatial behavior as the initial value, i.e.

|x|σf(x)− η(x) −→|x|→∞

0.

It follows that the profile f satisfies

λσf(λ·) −→λ→∞

z,

uniformly for |x| ≥ r0 > 0, where z(x) = η(x)|x|−σ, and so by (2.9) the self-similarU has initial value U0 = z. In other words, the initial value u0 of the asymptoticallyself-similar solution u, the profile and the initial value of the self-similar solutionU , all have the same spatial asymptotic behavior. We conclude formally that if theinitial value u0 of a solution u ∈ C([0,∞), C0(RN )) of (1.1) is spatially asymptoticto a function z, homogeneous of degree −σ, then the solution u is asymptoticallyself-similar to a self-similar solution U with initial value U0 = z. In other words,on a formal level, if (3.9) holds, then so do (3.3) and (3.4).

To understand this relationship further, recall that if u0 ∈ C0(RN ) is the initialvalue of a solution u ∈ C([0,∞), C0(RN )), then λσu0(λ·) is the initial value of thesolution λσu(λ2t, λx). Thus (3.4) reads

S(t)[λσu0(λ·)] −→λ→∞

U(t). (3.10)

Since U is a self-similar solution with initial value U0 = z, it is tempting to writeU(t) = S(t)z. In order to justify this, it is necessary to extend the semiflow S(t)generated by (1.1) to include homogeneous functions such as z. If this can be done,proving (3.10) amounts to proving continuous dependence of S(t) (for a given t)with respect to the convergence in (3.9). For example, (3.4) with t = 1 becomes

S(1)[λσu0(λ·)] −→λ→∞

S(1)z. (3.11)


We emphasize that this last observation is, for the moment, purely formal. Wehave not explained how S(t)z is defined, and we have not specified the nature ofthe convergence in (3.9) that is required.

Indeed, we have carefully avoided the question of the uniqueness of a self-similarsolution whose profile is spatially asymptotic to a given nontrivial homogeneousfunction. Without going into details, we mention that in some cases (for exampleif a = 1 and α > 2/N) it is unique . In other cases (for example if a = 1 andα < 2/N , or if a = −1 and 2/N < α < 4/(N − 2)) it will be unique only if certainother conditions are specified.

These formal observations form the basis for the notion of multi-profile solutions,which is the object of the upcoming sections.

4. Multi-profile solutions: definitions

We now turn to the question of the existence of solutions with a more complicatedasymptotic behavior than asymptotically self-similar. In terms of the equation (2.6)we wish to known whether there exist solutions v whose ω-limit set contains morethan one element. Recall that the ω-limit set of v is defined by

f ∈ C0(RN ); ∃sn →∞ such that v(sn) −→n→∞

f in C0(RN ). (4.1)

Just as (3.2) can be reformulated as (3.3), one can reformulate the set (4.1) interms of a solution u ∈ C([0,∞), C0(RN )) of (1.1). Such a solution is uniquelydetermined by its initial value u(0) = u0 ∈ C0(RN ). Thus for any u0 ∈ C0(RN )and any σ > 0 we define the ω-limit set

ωσ(u0) = f ∈ C0(RN ); ∃tn →∞ such that

tσ2n u(tn, ·

√tn) −→

n→∞f in C0(RN ), (4.2)

where u is the solution of (1.1) with the initial value u0. It is straightforward toverify that if u and v are related by the transformation (2.3)-(2.4), then ωσ(u0)given by (4.2) is identical to the set defined in (4.1). One interpretation of (4.2) isthat ωσ(u0) contains all possible asymptotic forms of u which are visible under theself-similar change of variables (2.3)-(2.4). On the other hand, to interpret ωσ(u0)as an ω-limit set in the sense of dynamical systems, the formulation (4.1) needs tobe used.

We wish to emphasize the use of uniform convergence in the above definitions.One advantage of this choice is that uniform convergence in (3.1) is equivalentto uniform convergence in (3.2). The weaker notion of uniform convergence oncompact sets in (3.2) corresponds for (3.1) to uniform convergence on parabolicsets of the form |x| ≤ C

√t for all C > 0. This latter notion of convergence was

used by Gmira and Veron [27] in their study of asymptotically self-similar solutions.Since in [27] the asymptotic profile is a nonzero constant function, and thereforenot in C0(RN ), uniform convergence on the parabolic sets is natural. On the otherhand, if one considers limiting profiles in f ∈ C0(RN ), as we do in this paper,uniform convergence seems more natural and is certainly stronger.

Another point is the choice of the space C0(RN ) in definition (4.2). As is well-known, the Cauchy problem for (1.1) is locally well-posed in many spaces other thanC0(RN ), such as Lp spaces and Sobolev spaces. The main advantage of C0(RN ) is


that the initial value problem for (1.1) is locally well-posed independently of thepower α in the equation.

We refer the reader to Section 10 below for further remarks on the type ofconvergence used in the definition of ωσ(u0).

Next, we need to find conditions on u0 ∈ C0(RN ) which imply that ωσ(u0) isnon-empty and contains more than one element. To see how this might be done,we recall that the scaling invariance expressed by (2.2) can be translated in termsof the semiflow S(t). More precisely if σ > 0 in the case a = 0, and σ = 2/α in thecase a 6= 0, then

S(t)[λσu0(λ·)] = [λσS(λ2t)u0](λ·), (4.3)for all t, λ > 0 and all u0 ∈ C0(RN ). In particular, setting t = 1, we obtain

S(1)[λσu0(λ·)] = [λσS(λ2)u0](λ·) = λσu(λ2, λ·), (4.4)

where u is the solution of (1.1) with the initial value u0. It follows that

S(1)[λσu0(λ·)] = tσ2 u(t, ·

√t), (4.5)

where t =√λ and where u is the solution of (1.1) with the initial value u0. (The

variable t in (4.5) is not the same as the variable t in (4.3).) Therefore, the ω-limitset of u0 ∈ C0(RN ) is also given by

ωσ(u0) = f ∈ C0(RN ); ∃λn →∞ such that

S(1)[λσnu0(λn·)] −→n→∞

f in C0(RN ). (4.6)

This equivalent formulation of ωσ(u0) suggests that we should extend condition (3.9)by allowing different limits of λσu0(λ·) along different sequences λn → ∞. Thequestion arises as to the precise nature of the limit. The weaker the topology weuse for the limit, the more limit points we obtain and therefore the more infor-mation we obtain about the spatial asymptotic properties of the initial value. Onthe other hand, it is essential that the heat semigroup, or more generally S(t), hassome continuity properties with respect to this topology.

At this point we distinguish two cases. The simpler one is

0 < σ < N. (4.7)

In this case, | · |−σ ∈ S ′(RN ), and the natural choice for convergence is indeedS ′(RN ). Thus, for u0 ∈ C0(RN ) and 0 < σ < N we define

Ωσ(u0) = z ∈ S ′(RN ); ∃λn →∞ such that λσnu0(λn·) −→n→∞

z in S ′(RN ). (4.8)

Note that the convergence in (4.8) is weaker than the convergence in (3.8) or (3.9).In general Ωσ(u0) 6⊂ C0(RN ). For example, if u0 ∈ C0(RN ) satisfies (3.6) for somefinite c, then it is clear that λσu0(λx) → c|x|−σ in S ′(RN ) as λ → ∞; and soΩσ(u0) = c| · |−σ. On the other hand, as we will see later, Ωσ(u0) can containfunctions which are not homogeneous.

This definition is clearly inadequate if σ ≥ N , which corresponds to α ≤ 2/Nwhen α = 2/σ. The case α < 2/N is important if a = 1 in (1.1) and will be treatedin Section 9 below. We make a brief remark about the limiting case α = 2/N atthe end of the next section.

We now introduce a natural class of initial values u0 for the study of ωσ(u0)and Ωσ(u0), still under the assumption (4.7). As suggested above, we need a spacewhich includes homogeneous functions, and on which the semiflow S(t) generated


by (1.1) can be easily defined. Thus we define the Banach space Zσ, for 0 < σ < N ,by

Zσ = z ∈ S ′(RN ) ∩ L1loc(RN ); | · |σz(·) ∈ L∞(RN ), (4.9)

with the obvious norm ‖ · ‖Zσ . It is clear that ‖ · ‖Zσ is invariant under therescaling λσu0(λ·) and that if u0 ∈ C0(RN ) ∩ Zσ then Ωσ(u0) ⊂ Zσ. (Moreover,‖z‖Zσ ≤ ‖u0‖Zσ for all z ∈ Ωσ(u0).) It turns out that if a = 0 or a = 1, then for anyinitial value u0 ∈ Zσ there exists a unique solution u ∈ C((0,∞), C0(RN )) of (1.1)such that u(t)→ u0 in L1

loc(RN ) as t ↓ 0. (See for example [7].) Thus the semiflowS(t) defined on C0(RN ) by formula (2.1) extends naturally to Zσ ∪ C0(RN ), andin fact preserves Zσ ∩C0(RN ). If a = −1 and σ = 2/α, one can also define S(t) ina natural way on a sufficiently small ball in Zσ. More precisely, there exists δ > 0such that if u0 ∈ C0(RN )∩Zσ with ‖u0‖Zσ ≤ δ, then S(t)u0 is defined for all t ≥ 0(i.e. the corresponding solution of (1.1) is global). Moreover, the semiflow S(t)extends naturally to the ball of radius δ of Zσ for all t > 0. (See [9, Lemma 6.1]for the precise definition.)

Before proceeding to the next section, we mention a technical feature of thespace Zσ which is extremely useful. As a space of “L∞-type”, it is the dual of acorresponding space of “L1-type”, which happens to be separable. Hence, a closedball in Zσ is weak? compact, and metrizable. It follows that if u0 ∈ C0(RN ) ∩Zσ, then the convergence in the definition of Ωσ(u0) is equivalent to convergencein a compact metric space. This fact enables one to use standard techniques ofdynamical systems in compact metric spaces.

5. Multi-profile solutions: the case α > 2/N

The attentive reader should now be able to guess our first set of results. The keyobservation is formula (4.5). This shows how the dynamical system in Zσ of spatialdilations of the initial value u0 translates into the rescaled space-time behavior ofthe resulting solution u(t).

Let u0 ∈ C0(RN ) ∩ Zσ, with 0 < σ < N , and suppose that

λσnu0(λn·) −→n→∞

z in S ′(RN ),

i.e. z ∈ Ωσ(u0). Since e∆ is convolution with a Gaussian function, it followsimmediately that

e∆[λσnu0(λn·)] −→n→∞

e∆z

pointwise on RN . Since λσnu0(λn·)n≥1 is a bounded set in Zσ, the same is true fore∆[λσnu0(λn·)]n≥1, from which it follows without too much work that the aboveconvergence is uniform. Thus, by (4.6), e∆z ∈ ωσ(u0). On the other hand, supposef ∈ ωσ(u0). Again by (4.6), there exists a sequence λn with

e∆[λσnu0(λn·)] −→n→∞

f

uniformly. Since bounded sets in Zσ are weak? precompact, a subsequence ofλσnu0(λn·) must converge to something, call it z. Thus f = e∆z. These considera-tions lead to the following theorem.

Theorem 5.1 (Theorem 3.9 in [8]). Assume a = 0 in (1.1) and so S(t) = et∆.Let 0 < σ < N , suppose u0 ∈ C0(RN ) ∩ Zσ and let u(t) = et∆u0. It follows thatωσ(u0) is a nonempty, compact subset of C0(RN ), and

ωσ(u0) = e∆Ωσ(u0). (5.1)


Moreover,λσnu0(λn·) −→

n→∞z

in S ′(RN ) if and only iftσ2n u(tn, ·

√tn) −→

n→∞e∆z

uniformly on RN , where λn = t2n.

In the nonlinear case with a = 1, we have the following similar result.

Theorem 5.2 (Theorem 3.10 in [9]). Assume a = 1 and α > 2/N in (1.1) and let(S(t))t≥0 be the resulting semiflow defined by (2.1). Set σ = 2/α so that 0 < σ < N .Given any u0 ∈ C0(RN ) ∩ Zσ, let u(t) = S(t)u0. It follows that ωσ(u0) is anonempty, compact subset of C0(RN ), and

ωσ(u0) = S(1)Ωσ(u0), (5.2)

where S(1) is the natural extension of the semiflow to Zσ, as discussed at the endof the previous section. Moreover,


z


√tn) −→

n→∞S(1)z


The proof of Theorem 5.2 is analogous to the proof of Theorem 5.1, where S(1)replaces e∆. The only detail which needs to be checked is the continuity of S(1)from a ball in Zσ, with respect to convergence in S ′(RN ) (i.e. the complete metrictopology induced by the weak? topology), into C0(RN ).

Theorems 5.1 and 5.2 generalize the notion of asymptotically self-similar solu-tions as described in Remark 3.1 (i). The following result does the same in the caseof Remark 3.1 (ii).

Theorem 5.3 (Section 5 in [9]). Assume a = 1 and α > 2/N in (1.1) and let(S(t))t≥0 be the resulting semiflow defined by (2.1). Fix 2/α < σ < N . Givenany u0 ∈ C0(RN ) ∩ Zσ, let u(t) = S(t)u0. It follows that ωσ(u0) is a nonempty,compact subset of C0(RN ), and

ωσ(u0) = e∆Ωσ(u0). (5.3)


n→∞z


√tn) −→

n→∞e∆z


In the case a = −1, we have the following result analogous to Theorem 5.2.

Theorem 5.4 (Section 6 in [9]). Assume a = −1 and α > 2/N in (1.1), and let(S(t))t≥0 be the resulting semiflow defined by (2.1). Set σ = 2/α so that 0 < σ < N .Let δ > 0 be sufficiently small so that S(t) extends naturally to a global semiflowon the ball of radius δ in Zσ, as described at the end of the previous section. Given


any u0 ∈ C0(RN ) ∩ Zσ with ‖u0‖Zσ ≤ δ, let u(t) = S(t)u0. It follows that ωσ(u0)is a nonempty, compact subset of C0(RN ), and

ωσ(u0) = S(1)Ωσ(u0). (5.4)


n→∞z


√tn) −→

n→∞S(1)z


Theorems 5.1–5.4 include as a special case the result that if the initial value isasymptotically homogeneous in space, then the resulting solution is asymptoticallyself-similar. Indeed, the initial value being asymptotically homogeneous in spacemeans that Ωσ(u0) = z for some homogeneous function z ∈ Zσ. Further, theresulting solution u being asymptotically self-similar means that ωσ(u0) = fwhere f is the profile of the self-similar solution. The above theorems imply thatthe second condition is a consequence of the first, where f is the profile of theself-similar solution with initial value z.

More generally, if we only suppose f ∈ ωσ(u0) so that f = e∆z (in the case ofTheorems 5.1 and 5.3) or f = S(1)z (in the case of Theorems 5.2 and 5.4) for somez ∈ Ωσ(u0), it follows that t

σ2n u(tn, ·

√tn) → f in C0(RN ) for some sequence tn →

∞. This is equivalent to tσ2n ‖u(tn) − U(tn)‖L∞ → 0, where U(t) = t−

σ2 f(x/

√t).

In general U(t) is not a solution of (1.1). For this to be true, it is necessaryand sufficient that z ∈ Ωσ(u0) be homogeneous (see Proposition 3.5 in [8] andProposition 3.5 in [9]). In this case the solution u of (1.1) is asymptotically self-similar along the sequence tn.

In addition, the above theorems show how one can answer the question raisedearlier, i.e. do there exist solutions such that ωσ(u0) contains more than one point.Indeed, formulas (5.1) through (5.4) show that, for u0 ∈ C0(RN ) ∩ Zσ, the largerthe set Ωσ(u0) the larger the set ωσ(u0). The next result shows that Ωσ(u0) canbe quite large.

Theorem 5.5 (Theorem 1.2 in [8]). Given 0 < σ < N and M > 0 there existsu0 ∈ C0(RN ) ∩ Zσ ∩ C∞(RN ) such that ‖u0‖Zσ = M and

Ωσ(u0) = z ∈ Zσ; ‖z‖Zσ ≤M. (5.5)

We sometimes refer to u0 in Theorem 5.5 as a “universal” initial value. Thefunction u0 encodes all the information needed to construct any other function inthe ball of Zσ whose radius is ‖u0‖Zσ . Since necessarily ‖z‖Zσ ≤ ‖u0‖Zσ for allz ∈ Ωσ(u0), it follows that Ωσ(u0) is as large as possible.

To give some intuitive idea about how to construct such a u0, we explain how toconstruct an explicit u0 such that Ωσ(u0) contains two different nonzero elements,i.e. | · |−σ and 1

2 | · |−σ. The idea is to let u0 be equal, alternately, to | · |−σ and

to 12 | · |

−σ on interspersed sequences of disjoint expanding shells around the origin.As a result, for a fixed compact set E in RN \ 0, different dilations of u0 will beequal, alternately, to | · |−σ and to 1

2 | · |−σ on E.

More precisely, let (an)n≥1 ⊂ (0,∞) be an increasing sequence such thatan+1

an−→n→∞

∞,


and set

u0(x) =∞∑k=1

1[a24k,a

24k+1]|x|−σ +

12

∞∑k=1

1[a24k+2,a

24k+3]|x|−σ. (5.6)

Note that, given any x ∈ RN , the above sum contains at most one nonzero term,and that u0 ∈ Zσ with ‖u0‖Zσ = 1. If we let λn = a4na4n+1, then given anycompact subset E of RN \ 0, λσnu0(λnx) = |x|−σ on E for all sufficiently largen. This implies that Ωσ(u0) contains | · |−σ. Similarly, if µn = a4n+2a4n+3, thenµσnu0(µnx) = 1

2 |x|−σ on E for all sufficiently large n. In particular, Ωσ(u0) also

contains 12 | · |

−σ.This construction can be modified to produce u0 with ‖u0‖Zσ = 1 such that

Ωσ(u0) contains any given countable set in the unit ball of Zσ. The specific u0 weconstruct is not continuous, but this can easily be rectified using cut-off functions.This suffices to prove Theorem 5.5, since the unit ball of Zσ contains a weak? densesequence.

One immediate consequence of Theorem 5.5 is that Ωσ(u0) can contain bothhomogeneous and nonhomogeneous functions. The following theorem shows thatthere exists u0 such that Ωσ(u0) contains no homogeneous function. (Since thistheorem and the resulting corollaries have not appeared in our earlier papers, wegive the proofs in Section 11.)

Theorem 5.6. Given 0 < σ < N , there exists a radially symmetric and decreasinginitial value u0 ∈ C0(RN ) ∩Zσ ∩C∞(RN ) such that Ωσ(u0) is nonempty and doesnot contain any homogeneous function of any degree.

Combining Theorems 5.1 and 5.2 with Theorem 5.5 and 5.6, we obtain thefollowing two corollaries.

Corollary 5.7. Let 0 < σ < N .

(i) Given M > 0 and u0 as in Theorem 5.5, let u(t) = et∆u0. It follows that

ωσ(u0) = e∆z ∈ Zσ; ‖z‖Zσ ≤M. (5.7)

In particular, given any homogeneous z ∈ Zσ with ‖z‖Zσ ≤M , there exists asequence tn →∞ such that u(t) = et∆u0 is asymptotic along the sequence tnto the self-similar solution et∆z, i.e.

tσ2n ‖u(tn)− etn∆z‖L∞ → 0,

as n→∞.(ii) If u0 is as in Theorem 5.6 and u(t) = et∆u0, then

lim inft→∞

tσ2 ‖u(t)− U(t)‖L∞ > 0,

for every self-similar solution U(t) of the linear heat equation. Moreoverωσ′(u0) = 0 for all 0 < σ′ < σ and ωσ

′(u0) = ∅ for all σ < σ′ < N .

Corollary 5.8. Assume a = 1 and α > 2/N in (1.1) and let (S(t))t≥0 be thecorresponding semiflow. Set σ = 2/α so that 0 < σ < N .

(i) Given M > 0 and u0 as in Theorem 5.5, let u(t) = S(t)u0. It follows that

ωσ(u0) = S(1)z ∈ Zσ; ‖z‖Zσ ≤M. (5.8)


In particular, given any homogeneous z ∈ Zσ with ‖z‖Zσ ≤M , there exists asequence tn →∞ such that u(t) = S(t)u0 is asymptotic along the sequence tnto the self-similar solution S(t)z, i.e.

tσ2n ‖u(tn)− S(tn)z‖L∞ → 0,

as n→∞.(ii) If u0 is as in Theorem 5.6 and u(t) = S(t)u0, then

lim inft→∞

tσ2 ‖u(t)− U(t)‖L∞ > 0, (5.9)

for every self-similar solution U(t) of (1.1) with a = 1. Furthermore ωσ′(u0) =

0 for all 0 < σ′ < σ and ωσ′(u0) = ∅ for all σ < σ′ < N . Moreover, prop-

erty (5.9) also holds if U(t) is any self-similar solution of the linear heatequation.

Corollaries 5.7 and 5.8 give explicit examples of solutions of heat equations whoseω-limit set, in self-similar variables, is nonempty but is not made up of a single sta-tionary solution. As these results show, this can happen in two essentially differentways. If u0 is as in part (i) of Corollary 5.7 or Corollary 5.8, then the solutionu(t) of (1.1) is not asymptotically self-similar as t → ∞ because it is asymptoticalong different time sequences to many different self-similar solutions. In fact, theωσ(u0) is as large as it can possibly be and contains both infinitely many stationarysolutions of (2.6) and infinitely many elements which are not stationary solutions.On the other hand, if u0 is as in part (ii) of Corollary 5.7 or Corollary 5.8, thenthe solution u(t) of (1.1) is not asymptotically self-similar along any time sequence(much less as t → ∞) because ωσ(u0) contains no stationary solutions of (2.6).In fact, the initial value we construct is such that Ωσ(u0) is a periodic orbit withrespect to the group of spatial dilations, so that ωσ(u0) consists precisely of onenontrivial periodic orbit of (2.6).

We close this section with the remark that when a = 1, the situation if α = 2/Nis very similar to the case α > 2/N , but the proofs are more involved. In particular,we can no longer work in S ′(RN ). For details, we refer the reader to Section 3 in [9].

6. Chaotic behavior of solutions

Another way of understanding the long-time asymptotic behavior of the solutionsof (1.1) is to study the discrete dynamical system generated by the flow at a fixedtime t0. We consider simultaneously the cases a = 0 and a = 1. If a = 0, thenS(t) = et∆ and 0 < σ < N is arbitrary. If a = 1 then α > 2/N , σ = 2/α (so that0 < σ < N) and S(t) is the corresponding nonlinear semiflow. To compensate forthe fact that S(t)u0 → 0 as t → ∞, we consider instead an appropriate dilationof S(t0). By “appropriate” we mean determined by the scaling properties of theequation, i.e.

Fz = (1 + t0)σ2 [S(t0)z](·

√1 + t0), (6.1)

for z ∈ Zσ∩C0(RN ). Note that if f is the profile of a self-similar solution as definedby (2.7), it is immediate to verify that Ff = f . (In fact, f ∈ C0(RN ) ∩ Zσ is theprofile of a self-similar solution if and only if Ff = f for all t0 > 0.) We have thefollowing result.


Theorem 6.1 (Corollary 3.13 in [8] and Corollary 3.14 in [9]). Under the aboveassumptions, fix t0 > 0, M > 0, let F de defined by (6.1) and set KM = S(1)z ∈Zσ; ‖z‖Zσ ≤M.

(i) KM is a compact subset of C0(RN ) and F (KM ) = KM(ii) The discrete dynamical system generated by F on KM is chaotic. In other

words (Devaney [17]), periodic points of F are dense in KM , F is topologicallytransitive and F has sensitive dependence on initial conditions.

Theorem 6.1 might seem unexpected. Indeed, the map F is precisely the renor-malization group map defined in Bricmont, Kupiainen and Lin [4]. It is used in [4]to prove that a certain class of solutions of (1.1) are asymptotically self-similar.On the other hand, Theorem 6.1 shows that in a broader context this mappingis chaotic. For a more recent example of chaotic behavior of solutions to a linearparabolic semigroup, see Emamirad, Goldstein and Goldstein [19].

7. Multi-scale solutions

At first glance, Theorems 5.1 through 5.4 seem to give a complete descriptionof the time asymptotic behavior of a solution of (1.1) in terms of the spatial as-ymptotic behavior of its initial value. Indeed, as we observed, ωσ(u0) contains allpossible asymptotic forms of u which are visible under the self-similar change ofvariables (2.3)-(2.4). It turns out that this description is not complete. For ex-ample, if 0 ∈ ωσ(u0) for some 0 < σ < N , then t

σ2n u(tn, ·

√tn) → 0 in C0(RN )

along some sequence tn → ∞. This leaves open the possibility that there exists

σ′ > σ such that tσ′2n u(tn, ·

√tn) has a nonzero limit as n→∞. In other words, to

completely describe the long-time asymptotic behavior of a solution u, we mightneed to consider the sets ωσ(u0) for various values of σ.

This idea is developed in [12] in the context of the linear heat equation, i.e.equation (1.1) with a = 0. In that article, the ideas used in the constructionof the universal initial value (Theorem 5.5 above) are pushed further in order toconstruct an initial value u0 ∈ C0(RN ) ∩ Zσ ∩ C∞(RN ), for a given 0 < σ < N ,which is “universal” in the sense that Ωσ(u0) and ωσ(u0) are as large as possible,as given by formulas (5.5) and (5.7), and for which in addition, ωσ

′(u0) = C0(RN )

for countably many values of σ′ with σ < σ′ < N . Moreover, the results in [12]show that there exists an initial value u0 ∈ C0(RN ) for which ωσ(u0) = C0(RN )for all σ ∈ (0, N) ∩Q.

As mentioned just after the statement of Theorem 5.5, the basic idea in theconstruction of these initial values is that they should be carefully specified on anever expanding sequence of disjoint shells. The behavior of the initial value on ashell which is “far away” from the origin can be brought in “close” to the originby a dilation. Thus, a given sequence of dilations applied to such an initial valuecan reproduce the behavior of that initial value on a sequence of distant shells. Bychoosing different behaviors of the initial value on different sequences of expandingshells, one obtains different limits in Zσ of the initial value under different sequencesof dilations. Nothing prevents one from specifying the initial value on various shellsso that dilations with different values of σ produce the desired function “close” tothe origin. This is how one can construct a single initial value u0 for which ωσ(u0)is nontrivial for countably many different values of σ.


We next state and prove a result based on a modification of this construction.Perhaps more fundamental than describing the long-time asymptotic behavior ofa solution under various rescalings is the simpler question of the long-time decayrate of a solution. The next theorem shows that there exists a positive solutionof the linear heat equation which decays at all possible rates, along different timesequences going to infinity.

Theorem 7.1. There exists a positive, radially symmetric and decreasing initialvalue u0 ∈ C0(RN ) ∩ C∞(RN ) such that for all 0 < σ < N and all 0 ≤ c < ∞,there exists a sequence tn →∞ such that t

σ2n ‖etn∆u0‖L∞ → c as n→∞.

An earlier version of this result appeared as Theorem 1 in [11]. The constructionof the initial value u0 in [11] was based on the same idea of specifying u0 on everexpanding disjoint shells. However, the placement of the shells was such that thevarious sequences of dilations of u0 converged to cδ, rather than to an element ofZσ. Based on the more recent article [45], where the analogous result for the linearSchrodinger equation is proved, it is possible to simplify the proof of Theorem 7.1.The initial value can be chosen as an infinite linear combination of Gaussian func-tions, which is therefore radially symmetric and decreasing. The resulting proof isso elementary that we give it below in complete detail.

Proof of Theorem 7.1. Let Gt be the Gauss kernel,

Gt(x) = (4πt)−N2 e−

|x|24t . (7.1)

Note that G1 = G defined by (3.5). As is well known,

et∆Gs = Gt+s, (7.2)

for all t, s > 0. We let

u0(x) =∞∑j=1

ajGsj , (7.3)

where the sequences (sj)j≥1 and (aj)j≥1 are defined by

s1 = e, sj+1 = exp exp sj for j ≥ 1, (7.4)

andaj = s

N2j (log sj)−1, j ≥ 1. (7.5)

Since∞∑j=1

ajs−N2 −kj <∞, (7.6)

for all k ≥ 0, we see that the series (7.3) is norm convergent in C0(RN ), and so isthe series of any arbitrary space derivative. Thus u0 ∈ C0(RN ) ∩ C∞(RN ). It isalso clear that u0 is radially symmetric and decreasing.

Next, given 0 < σ < N and c > 0, we set

tn = [(4π)N2 c]−

2N−σ s

NN−σn (log sn)−

2N−σ , (7.7)

and we claim thattσ2n ‖etn∆u0‖L∞ −→

n→∞c. (7.8)


Indeed, it follows from (7.2) that

tσ2n e

tn∆u0 =∞∑j=1

ajtσ2nGtn+sj = vn + wn + zn, (7.9)

where

vn =n−1∑j=1

ajtσ2nGtn+sj ,

wn = antσ2nGtn+sn ,

zn =∞∑n+1

ajtσ2nGtn+sj .

It is straightforward to check that

‖wn‖L∞ = tσ2n an[4π(tn + sn)]

N2 −→n→∞

c. (7.10)

Thus to prove the claim it suffices to show that ‖vn‖L∞ + ‖zn‖L∞ → 0. This isclear, since

‖un‖L∞ ≤ tσ2n

n−1∑j=1

aj(tn + sj)−N2 ≤ nan−1t

−N−σ2n −→

n→∞0, (7.11)

and

‖zn‖L∞ ≤ tσ2n

∞∑j=n+1

aj(tn + sj)−N2 ≤ t

σ2n

∞∑j=n+1

ajs−N2j

= tσ2n

∞∑j=n+1

(log sj)−1 = tσ2n

∞∑j=n

e−sj −→n→∞

0.

(7.12)

This establishes the claim and therefore proves the theorem.

It is important to observe that the argument which starts with formula (7.7) infact shows that cG ∈ ωσ(u0). Thus, Theorem 7.1 not only shows that there existsa solution of the heat equation that realizes all possible decay rates, but also forwhich ωσ(u0) 6= ∅ for all σ ∈ (0, N). It is natural to ask how large ωσ(u0) can be forall the values of σ between 0 and N . It turns out that the ideas used in [11] to proveTheorem 7.1 can be pushed much further. In [13] we construct an initial value inu0 ∈ C0(R) (i.e. with N = 1) such that ωσ(u0) = C0(R) for all σ ∈ (0, 1) ∩Q, andfor almost all σ ∈ (0, 1), ωσ(u0) is equal to the set of all multiples of all derivativesof the Gaussian kernel G.

Finally, we remark that up until now, all results in this section have concernedonly solutions of the linear heat equation. We therefore mention a similar result forthe nonlinear heat equation, i.e. (1.1) with a = 1. For this purpose, we consider thecorresponding flow S(t) and the corresponding ω-limit sets ωσ(u0). Although it isnot stated in [12], it follows from the same considerations that given 0 < σ < Nand a sequence (σn)n≥1 ⊂ (σ,N), there exists an initial value u0 ∈ C0(RN ) ∩ Zσwith ωσ(u0) = S(1)z ∈ Zσ; ‖z‖Zσ ≤ 1 and ωσn(u0) = C0(RN ) for all n ≥ 1.


8. Nonparabolic multi-profile, multi-scale solutions

Up until now we have looked at space-time dilations which leave equation (1.1)invariant and studied the asymptotic behavior of all solutions with respect to theserescalings. We have shown in fact that for a given solution, we need to considerall these rescalings to describe the complete asymptotic behavior of the solution.One might think that this tells the whole story. Indeed, space-time dilations thatdo not leave the equation (1.1) invariant would seem to be irrelevant. Surprisingly,this turns out to be false. Let us see why.

Let u be a solution of the linear heat equation, i.e. ∂tu = ∆u and let

vλ(t, x) = λσu(λ2t, λ2βx), (8.1)

for λ > 0, where σ, β > 0 and β is not necessarily equal to 1/2. Then ∂tvλ =λ2−4β∆vλ. If we now suppose that as λ → ∞ the functions vλ converge in somereasonable sense to w, we obtain formally that

∆w = 0 if β < 1/2,∂tw = 0 if β > 1/2.

(8.2)

In the first case, since the only harmonic function in C0(RN ) is 0, the only asymp-totic limit we can expect from this rescaling would be 0. On the other hand, thereis no obvious reason to ignore the second case. Our results bear this out.

As we will see below, if β > 1/2 and 0 < σ < N , there exists a solution of (1.1)with a = 0 for which t

σ2 u(t, xtβ) converges in C0(RN ) along some sequence tn →∞

to (for example) a Gaussian function. It follows that tσ2 u(t, x

√t) cannot have

any limit in C0(RN ) along the sequence tn. Thus the asymptotic behavior of thissolution cannot be completely described without considering nonparabolic scalingsof the type (8.1).

The construction of initial values u0 ∈ C0(RN ) producing this type of solutionsuses the same techniques already described and is quite general. Indeed, solutionsexist for which the corresponding ω-limit set ωσ,β(u0) (see Theorem 8.2 below) isall of C0(RN ) for countably many different σ and β. Moreover, if v0 differs from u0

on a compact set, then the resulting solution v has the same properties. Therefore,the set of initial values giving rise to such solutions is highly nontrivial and thisphenomenon cannot be ignored.

We now state precisely our basic results of this type. For σ, β > 0, we define

ωσ,β(u0) = f ∈ C0(RN ); ∃tn →∞ such that

tσ2n u(tn, · tβn) −→

n→∞f in C0(RN ), (8.3)

where u is the solution of (1.1) with a = 0.

Theorem 8.1 (Theorem 5.1 in [14]). Fix 0 < σ < N and β > 1/2 and let f ∈C0(RN ) be positive, radially symmetric and decreasing. There exists u0 ∈ C0(RN ),positive, radially symmetric and decreasing such that f ∈ ωσ,β(u0).

Theorem 8.2 (Corollary 6.3 in [14]). Given any countable subset S of (0, N), thereexists u0 ∈ C0(RN ) such that ωσ,β(u0) = C0(RN ) for all 0 < σ < N and β ≥ 1/2with σ/2β ∈ S.


9. Multi-profile solutions: the case α < 2/N

In this section we return to the situation of multi-profile limits with respect toa single rescaling, as treated in Section 5, but in the case of (1.1) with a = 1 andα < 2/N . In particular, σ = 2/α > N . Our results concern only nonnegativesolutions. The distinguishing feature of this case is that it is genuinely nonlinear.The results have no counterpart for the heat semigroup. For example, the homoge-neous function | · |−σ is not locally integrable. Furthermore, all solutions in C0(RN )of (1.1), and in particular self-similar solutions, decay faster than any positive so-lution of the linear heat equation. Indeed, ±(αt)−

1α are both solutions of (1.1),

and any initial value u0 ∈ C0(RN ) is bounded above and below by some ±(αt0)−1α .

The resulting solution is therefore bounded above and below by ±(α(t0 + t))−1α for

all t > 0.In this case, a result similar to Theorem 5.2 holds. However the operator S(1),

where (S(t))t≥0 is the semiflow generated by (1.1) on C0(RN ), must be replaced byanother operator, and the set Ωσ(u0), as well as the space Zσ, need to be redefinedto take into account the fact that we can no longer work in S ′(RN ).

In order to motivate the various definitions, we need to discuss what is known asthe “very singular solution” of (1.1), which is in fact self-similar. Following [20], weconsider the profile equation (2.8) as a nonlinear eigenvalue problem in the weighted

L2-space E = L2(RN , e|x|24 dx). The operator L = −∆− 1

2y · ∇ is self-adjoint in Ewith first eigenvalue N/2 and the equation (2.8) becomes

Lf − σ

2f + a|f |αf = 0. (9.1)

Taking the inner product with f yieldsN − σ

2‖f‖2E + a

∫RN|f |α+2e

|x|24 dx ≤ 0.

In the cases that we previously discussed in Section 5, where σ < N , it easily followsthat if a = 0 or a = 1, then the only solution of (2.8) in E is 0. Similarly, if a = −1,then the only “small” solution in E is also 0.

Under the present circumstances (i.e. σ > N and a = 1), it follows from varia-tional arguments [43, 20] that there exists a positive solution ϕ ∈ E of (2.8). Thissolution ϕ is unique [2] and plays the same role in our results in the case α < 2/Nas the solution 0 plays in the case α > 2/N . Furthermore, it is the profile of thevery singular solution

V (t, x) = t−1αϕ( x√

t

), (9.2)

where V satisfies V (t)−→t↓0

0 in L1loc(RN \ 0),

‖V (t)‖L1 −→t↓0∞.

(9.3)

In particular, the initial value of V (t) is equal to 0 in the formal sense definedby (2.9).

We now turn to the precise statements of our results. We set

Zσ = z ∈ L1loc(RN \ 0); | · |σz(·) ∈ L∞(RN ), (9.4)

with the obvious norm ‖ · ‖Zσ . This definition is similar to (4.9) except we donot assume z ∈ S ′(RN ). (In fact, the definition (9.4) with σ < N gives the same


space as (4.9).) Just as in the case (4.9), a closed ball of Zσ defined by (9.4) canbe endowed with the structure of a compact metric space equivalent to the weak?

topology on the ball. Furthermore, convergence of sequences for the weak? topologyon the ball is equivalent to convergence in D′(RN \0). In addition, it follows fromthe results in [36] that, given any u0 ∈ Zσ, u0 ≥ 0, there exists a unique solutionu ∈ C((0,∞), C0(RN )) of (1.1), u ≥ 0 such that u(t) → u0 in L1

loc(RN \ 0) and‖u(t)‖L1 →∞ as t ↓ 0. See Proposition 4.3 in [9]. Given u0 ∈ Zσ, u0 ≥ 0, we set

U(t)u0 = u(t), (9.5)

where u is the solution of (1.1) with the properties described just above. It isimportant to observe that if u0 ∈ C0(RN ) ∩ Zσ, u0 ≥ 0, then U(t)u0 6= S(t)u0,where S(t) is the semiflow on C0(RN ) induced by the equation (1.1). In particular,it follows from (9.3) that

U(t)0 = V (t), (9.6)where V is the very singular solution. Finally, if u0 ∈ C0(RN ) ∩ Zσ, u0 ≥ 0, then

Ωσ(u0) = z ∈ Zσ; ∃λn →∞ such that λσnu0(λn·) −→n→∞

z in D′(RN \ 0). (9.7)

This definition is analogous to (4.8) with S ′(RN ) replaced by D′(RN \ 0). Werecall that ωσ(u0) is defined by (4.2) and we emphasize the fact that u(t) in (4.2)is given by S(t)u0, not by U(t)u0.

Theorem 9.1 (Theorem 4.15 in [9]). Let u0 ∈ C0(RN ) ∩ Zσ, u0 ≥ 0, u0 6≡ 0.With the above notation, it follows that ωσ(u0) is a nonempty, compact subset ofC0(RN ), and

ωσ(u0) = U(1)Ωσ(u0). (9.8)Moreover,


z

in D′(RN \ 0) if and only if

tσ2n u(tn, ·

√tn) −→

n→∞U(1)z


As the simplest application of Theorem 9.1, we recover the result of [21, 22]referred to in Section 3. Let u0 ∈ C0(RN ), u0 ≥ 0, u0 6≡ 0 satisfy |x|σu0(x) → 0as |x| → ∞, so that Ωσ(u0) = 0 (for example u0 ∈ S(RN )). It follows thatωσ(u0) = U(1)0 = ϕ and that u(t) = S(t)u0 is asymptotic to the very singularsolution, i.e.

tσ2 ‖u(t)− V (t)‖L∞ −→

t→∞0. (9.9)

This contrasts with the case α > 2/N treated in Section 5, where under the anal-ogous conditions we would have t

σ2 ‖u(t)‖L∞ → 0. This shows explicitly how the

very singular solution V (t) in the case α < 2/N replaces the trivial solution in thecase α > 2/N . Furthermore this shows another fundamental difference between thetwo cases α < 2/N and α > 2/N . In the case α > 2/N a solution with initial valueu0 ∈ S(RN ) decays faster than t−

σ2 , while if α < 2/N a solution with initial value

u0 ∈ S(RN ), u0 ≥ 0, u0 6≡ 0 decays precisely at the rate t−σ2 .

In addition, there exists a universal initial value in this case, i.e. there existsu0 ∈ C0(RN ) ∩ Zσ ∩ C∞(RN ), u0 ≥ 0 such that ‖u0‖Zσ = 1 and Ωσ(u0) =z ∈ Zσ; z ≥ 0, ‖z‖Zσ ≤ 1. The resulting solution has the property that it is


asymptotically self-similar to infinitely many different self-similar solutions alongdifferent time sequences. See [9, Corollary 4.17].

10. Remarks on the convergence in ωσ(u0)

In the definition (4.2) of ωσ(u0), we use uniform convergence. Assuming u0 ∈C0(RN )∩Zσ (as in Theorem 5.1, 5.2, 5.3, 5.4 and 9.1), it follows that the resultingsolution of (1.1) is bounded in Zσ. (See Section 8 in [7] for the cases a = 0and a = 1 and Lemma 6.1 in [9] for the case a = −1.) Consequently if u0 ∈C0(RN ) ∩ Zσ the L∞ convergence in (4.2) implies convergence for the weightednorm ‖(1 + | · |2)γu(·)‖L∞ for all 0 ≤ γ < σ/2. Therefore it is natural to ask ifconvergence also holds with γ = σ/2. Unfortunately, the situation is not as simpleas for the results cited in this paper.

In fact, for u0 ∈ C0(RN )∩Zσ, it is natural to consider ω-limit sets analogous tothe one defined by (4.2) but with convergence in both C0(RN ) and Zσ. This otherω-limit set would in principle be smaller than ωσ(u0) and have different properties.These questions are studied in [7, 8, 9], but will not be dealt with here. Sufficeit to say that there are reasonable sufficient conditions on u0 which guarantee thestronger convergence.

11. Proofs

In this section, we give the proofs of the results stated above which have notappeared in any of our previous papers.

Proof of Theorem 5.6. Let g ∈ C∞(R) be a positive, nonconstant 1-periodic func-tion. Replacing g by g + C with C sufficiently large, we may assume that

g′(t) ≤ σg(t),

for all t ∈ R. Letting u0(x) = |x|−σ g(log |x|), it follows that u0 ∈ Zσ is a radiallysymmetric, positive, decreasing function. We next consider a C∞, radially sym-metric, positive, decreasing function u0 which agrees with u0 for |x| ≥ 1. Sinceλσu0(x)(λx) ≡ |x|−σ g(log λ+ log |x|) for all λ > 0, we see that

Ωσ(u0) = z ∈ Zσ; ∃s ∈ [0, 1], z(x) = |x|−σ g(s+ log |x|).

Since g is nonconstant and bounded, we see that Ωσ(u0) does not contain anyhomogeneous function of any degree.

Proof of Corollary 5.7. Property (i) is Theorem 1.1 in [8].Proof of (ii): The first part follows from Proposition 3.5 in [8]. Since u0 ∈ Zσ,

it is immediate that ωσ′(u0) = 0 for all 0 < σ′ < σ. Finally, if σ′ > σ is such

that ωσ′(u0) 6= ∅, then it is immediate that 0 ∈ ωσ(u0). This is ruled out by

Theorem 5.6. (0 6∈ Ωσ(u0) because 0 is homogeneous.)

Proof of Corollary 5.8. Property (i) is Theorem 1.3 in [9].Proof of (ii): The first part follows from Proposition 3.5 in [9]. Indeed, suppose

to the contrary there exist a self-similar solution U(t) and a sequence tn →∞ suchthat t

σ2n ‖u(tn)−U(tn)‖L∞ → 0. It follows that f , the profile of U belongs to ωσ(u0).

Therefore, f = S(1)z for some z ∈ Ωσ(u0). Since f is the profile of the self-similarsolution U , it follows from Proposition 3.5 in [9] that z is homogeneous. This isruled out by Theorem 5.6.


Since u0 ∈ Zσ, it is immediate that ωσ′(u0) = 0 for all 0 < σ′ < σ. Moreover,

if σ′ > σ is such that ωσ′(u0) 6= ∅, then it is immediate that 0 ∈ ωσ(u0). This is

ruled out by Theorem 5.6. (0 6∈ Ωσ(u0) because 0 is homogeneous.)Finally, we suppose that V (t) = et∆z where z ∈ Zσ is homogeneous and that

there exists a sequence tn → ∞ such that tσ2n ‖u(tn) − V (tn)‖L∞ → 0. This means

that ψ ∈ ωσ(u0) where ψ = e∆z. Note that Ωσ(z) = z because z is homogeneous.Note also that Ωσ(ψ) = Ωσ(z) by Proposition 3.4 (v) in [8], so that Ωσ(ψ) = z.Therefore, it follows from Theorem 5.2 that ωσ(ψ) = S(1)z. On the other hand,since ψ ∈ ωσ(u0), we claim that

ωσ(ψ) ⊂ ωσ(u0). (11.1)

This contradicts formula (5.9) with the self-similar solution U(t) = S(t)z of (1.1)with a = 1. To prove the claim (11.1) we argue as follows. Let ψ ∈ ωσ(u0) and setv(t) = S(t)ψ. It follows from Proposition 3.8(i) in [9] that (1 + t)

σ2 v(t, ·

√1 + t) ∈

ωσ(u0) for all t ≥ 0. Now let f ∈ ωσ(ψ), so that there exist tn → ∞ such thattσ2n v(tn, ·

√tn) → f in C0(RN ) as n → ∞. It follows in a straightforward fashion

that (1 + tn)σ2 v(tn, ·

√1 + tn) → f in C0(RN ). Since ωσ(u0) is closed in C0(RN ),

this proves the claim.

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Universite Pierre et Marie Curie & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187,4 place Jussieu, 75252 Paris Cedex 05, France

E-mail address: [email protected]

Instituto de Matematica, Universidade Federal do Rio de Janeiro, Caixa Postal68530, 21944–970 Rio de Janeiro, R.J. Brazil


Universite Paris 13, CNRS UMR 7539 LAGA, 99 Avenue J.-B. Clement F-93430 Vil-

letaneuse France


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