# Symmetry properties of positive solutions of parabolic

## Transcript Of Symmetry properties of positive solutions of parabolic

Symmetry properties of positive solutions of parabolic equations: a survey

P. Pola´ˇcik

School of Mathematics, University of Minnesota Minneapolis, MN 55455

Abstract

This survey is concerned with positive solutions of nonlinear parabolic equations. Assuming that the underlying domain and the equation have certain reﬂectional symmetries, the presented results show how positive solutions reﬂect the symmetries. Depending on the class of solutions considered, the symmetries for all times or asymptotic symmetries are established. Several classes of problems, including fully nonlinear equations on bounded domains, quasilinear equations on RN , asymptotically symmetric equations, and cooperative parabolic systems, are examined from this point of view. Applications of the symmetry results in the study of asymptotic temporal behavior of solutions are also shown.

Contents

1 Introduction: basic problems, results and some history

2

1.1 Equations on bounded domains . . . . . . . . . . . . . . . . . . . 3

1.2 Equations on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Cooperative systems . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 General notation

10

3 Fully nonlinear equations on bounded domains

10

3.1 Solutions on (0, ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Solutions: (−∞, T ) and linearized problems . . . . . . . . . . . . 14

3.3 Asymptotically symmetric equations . . . . . . . . . . . . . . . . 17

4 Quasilinear equations on RN

19

4.1 Solutions on (0, ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Solutions on (−∞, T ) and linearized problems . . . . . . . . . . . 21

5 Cooperative systems

23

1

6 On the proofs: a comparison of bounded and unbounded do-

mains

25

7 Applications and some open problems

29

1 Introduction: basic problems, results and some

history

A conventional wisdom says that “parabolic ﬂows reduce complexity.” Although it should not be taken too seriously and universally, there are good examples where it manifests itself. Asymptotic symmetry of solutions, which is one of the main topics of this survey, is such an example. It shows a tendency of positive solutions of certain parabolic equations to “improve their symmetry” as time increases, becoming “symmetric in the limit” as t → ∞.

Historically, ﬁrst studies of symmetry properties of positive solutions of parabolic equations were carried out after similar properties for elliptic equations had been long understood. These studies brought about interesting qualitative results for parabolic equations, but at the same time they opened new perspectives for looking at the earlier results for elliptic equations. Viewing solutions of elliptic equations as equilibria (steady states) of the corresponding parabolic equations, one naturally tries to understand how their symmetry ﬁts in the broader picture of the parabolic semiﬂow. For example, examining heteroclinic orbits between symmetric equilibria, one naturally asks if they are symmetric as well. Generalizing, one is subsequently lead to the problem of symmetry of entire solutions, that is, solutions deﬁned for all times, positive and negative. Another symmetry problem is concerned with general solutions of parabolic equations, which are typically deﬁned for positive times only. If the parabolic semiﬂow admits a Lyapunov functional, which forces bounded solutions to converge to steady states, the symmetry of the latter translates to the asymptotic symmetry of the solutions of the parabolic equation. This immediately raises the question whether the asymptotic symmetry can be established regardless of the presence of any Lyapunov functional, even if the solution does not converge to a steady state. This question is even more interesting for time-dependent parabolic problems, whose solutions typically do not converge to equilibria and their temporal behavior can be very complicated. In this case, the asymptotic symmetrization in space is to be studied independently of the temporal structure. Once it is understood, however, it often proves very useful for studying the temporal behavior of the solutions.

The previous paragraph indicates the sort of problems this survey is devoted to. Considering parabolic equations with certain symmetry properties, we want to understand how their solutions reﬂect the symmetry. The key issues to be discussed are the asymptotic symmetry properties for the Cauchy problem, symmetry of the entire solutions (and related to this, symmetry of unstable spaces of entire solutions), and applications of these results. We also mention key ideas of the proofs and discuss diﬀerences of their use for diﬀerent type of

2

problems. To give a more speciﬁc overview of the results to be presented in this paper

and to put them in context with similar results on elliptic equations, we ﬁrst consider the following semilinear reaction-diﬀusion equation

ut = ∆u + f (t, u), x ∈ Ω, t ∈ J.

(1.1)

Here Ω is a domain in RN and f : R2 → R is a continuous function, which is Lipschitz continuous with respect to u. We take either J = (0, ∞) and consider a suitable initial-value problem, or J = (−∞, ∞) in case we want to consider entire solutions.

Although much simpler an equation than fully nonlinear ones examined is the forthcoming sections, for the purposes of the introduction (1.1) is suﬃciently representative and it allows us to discuss some key issues in a more rudimentary way. The simplicity of (1.1) consists mainly in the fact that, without any additional assumptions, the equation is invariant under any Euclidean symmetries the domain may have. This is true regardless of the temporal dependence of f on time, which is not restricted by any assumption like periodicity or almost periodicity.

We assume that either Ω = RN , or Ω is a bounded domain in RN which is symmetric with respect to the hyperplane

H0 := {x = (x1, . . . , xN ) ∈ RN : x1 = 0}

and convex in x1 (which means that with any two points (x1, x ), (x˜1, x ), diﬀering only at the ﬁrst component, Ω contains the line segment connecting them). The speciﬁc choice of the direction e1 for the reﬂectional symmetry is arbitrary, domains symmetric in other directions can be considered equally well. In case Ω is bounded, we complement the equation with Dirichlet boundary condition

u(x, t) = 0, x ∈ ∂Ω, t ∈ J.

(1.2)

If f is independent of t, steady states of (1.1), (1.2) are solutions of the

elliptic equation

∆u + f (u) = 0, x ∈ Ω,

(1.3)

complemented (in case Ω is bounded) with the Dirichlet condition

u(x) = 0, x ∈ ∂Ω.

(1.4)

Symmetry theorems for (1.1) have somewhat diﬀerent ﬂavors and hypotheses for bounded domains Ω and for Ω = RN , so we distinguish these case separately.

1.1 Equations on bounded domains

Reﬂectional symmetry of positive solutions of (1.3), (1.4) was ﬁrst established by Gidas, Ni and Nirenberg [37]. They proved that if Ω is as above (convex

3

and symmetric in x1) and smooth (of class C2), then each positive solution u of (1.3), (1.4) has the following symmetry and monotonicity properties:

u(−x1, x2, . . . , xN ) = u(x1, x2, . . . , xN ) (x ∈ Ω), ux1 (x1, x2, . . . , xN ) < 0 (x ∈ Ω, x1 > 0).

(1.5)

The method of moving hyperplanes, which is the basic geometric technique in their paper, was introduced earlier by Alexandrov [2] and further developed by Serrin [70] ([70] also contains a related result on radial symmetry). Generalizations and extensions of the symmetry result have been made by many authors. In particular, Li [50] extended it to fully nonlinear equations on smooth domains. Later Berestycki and Nirenberg [11] found a way of dealing with fully nonlinear equations on nonsmooth domains employing the elliptic maximum principle for domains with small measure (see also Dancer’s contribution [29], where semilinear equations on nonsmooth domains are treated using a diﬀerent method). There are many other related results, including further developments regarding symmetry of elliptic overdetermined problems, as considered in the original paper of Serrin [70], see for example [1, 73]. Additional references and more detailed overviews can be found in the surveys [7, 46, 58]. Let us also mention a more recent work by Da Lio and Sirakov [27], where the symmetry results are extended to viscosity solutions of general elliptic equations.

The proofs of the above results are based on the method of moving hyperplanes and various forms of the maximum principle. Below we shall indicate how these techniques are typically used. A diﬀerent approach employing a continuous Steiner symmetrization was used in [12] (see also the survey [46]). It applies to positive solutions of (1.3), (1.4) and, as it relies on the variational structure of the problem and not so much the maximum principle, it allows for extensions of the results of [37] in diﬀerent directions.

We remark that it is not always possible to generalize the above symmetry result to solutions which are merely nonnegative, rather than strictly positive, (counterexamples are easily constructed if N = 1). However, if N > 1 and some regularity assumptions are made on the domain, it can be proved that nonnegative solutions are necessarily strictly positive, hence symmetric (see [19, 33] for results of this sort). The issue whether some strict positivity assumption is needed or not will arise again in our discussion of parabolic equations below. Also the convexity of Ω in x1 is an important assumption without which the result is not valid in general (however, see [47] for a symmetry result involving some nonconvex domains).

If Ω is a ball, say Ω = B(0, r0) (0 is the center, r0 the radius), then the reﬂectional symmetry theorem can be applied in any direction which leads to the following radial symmetry result. Any positive solution u(x) of (1.3), (1.4) is radially symmetric (it only depends on r = |x|) and radially decreasing (ur(x) < 0 for r ∈ (0, r0)).

There are numerous application of the above symmetry results in further studies of positive solutions of (1.3), (1.4). For example, the radial symmetry property implies that positive solutions can be viewed as solutions of the

4

ordinary diﬀerential equation (ODE)

N −1 urr + r ur + f (u) = 0, r ∈ (0, r0),

(1.6)

and that ur(0) = 0. Thus one immediately gains ODE tools, like the shooting method, for the study of positive solutions. Problems on multiplicities and/or bifurcations of positive solutions become then a lot more elementary. The reﬂectional symmetry results do not lead to such dramatic simpliﬁcations of the problem, but they are still very useful, especially if there are several directions in which the domain is symmetric.

For parabolic problems, such as (1.1), (1.2), ﬁrst symmetry results of similar nature started to emerge much later. After a prelude [30] devoted to timeperiodic solutions, symmetry of general positive solutions of parabolic equations on bounded domains was considered in [4, 5, 44] and later in [6, 63]. With Ω as above (convex and symmetric in x1) and with suitable symmetry assumptions on the nonlinearity, symmetries of two classes of solutions were examined in these papers. Closer in spirit to the results for elliptic equations are symmetry theorems concerning entire solutions. A typical theorem in this category states that if u is a bounded positive solution of (1.1), (1.2) with J = R satisfying

inf u(x, t) > 0 (x ∈ Ω, t ∈ J),

t∈R

(1.7)

then u has the symmetry and monotonicity properties (1.5) for each t ∈ R:

u(−x1, x , t) = u(x1, x , t) (x = (x1, x ) ∈ Ω, t ∈ R), ux1 (x, t) < 0 (x ∈ Ω, x1 > 0, t ∈ R).

(1.8)

This result follows from more general theorems of [4, 6], although to be precise we would need to include additional compactness assumptions on u (in the context of the semilinear problem (1.1), (1.2), the boundedness of u alone is suﬃcient if, for example, Ω has Lipschitz boundary and t → f (0, t) is a bounded function).

In a diﬀerent type of symmetry results, nonnegative solutions of the CauchyDirichlet problem for (1.1) are considered. These of course cannot be symmetric, unless they start from a symmetric initial function. However, it can be shown that they “achieve” the symmetry in the limit as t → ∞. To formulate this more precisely assume that u is a bounded positive solution of (1.1), (1.2) with J = (0, ∞) such that for some sequence tn → ∞

lim inf u(x, tn) > 0 (x ∈ Ω).

n→∞

(1.9)

Then u is asymptotically symmetric in the sense that

lim (u(−x1, x , t) − u(x1, x , t)) = 0

t→∞

lim sup ux1 (x, t) ≤ 0

t→∞

(x ∈ Ω), (x ∈ Ω, x1 > 0).

(1.10)

5

If {u(·, t) : t ≥ 1} is relatively compact in C(Ω¯ ), then the asymptotic symmetry of u can be expressed in terms of its limit proﬁles, that is, elements of its ω-limit set,

ω(u) := {φ : φ = lim u(·, tn) for some tn → ∞},

where the limit is in C(Ω¯ ) (with the supremum norm). It can be easily shown that condition (1.9) is equivalent to the requirement that there exist at least one element of ω(u) which is strictly positive in Ω and (1.10) translates to all elements of ω(u) being symmetric (even) in x1 and monotone nonincreasing in x1 > 0. This result is proved in a more general setting in [63]. Condition (1.9) is a relatively minor strict positivity condition (note that it is not assumed to be valid for all sequences tn → ∞), which cannot be omitted in general (a counterexample can be found in [63]). It is not needed, however, if the domain is suﬃciently regular [44].

There are connections between the two types of symmetry results, the asymptotic symmetry for the Cauchy-Dirichlet problem and the symmetry for entire solutions. Oftentimes, if the nonlinearity is suﬃciently regular, the limit proﬁles of a solution of the Cauchy-Dirichlet problem can be shown to be given by entire solutions of suitable limit parabolic problems. Thus if the limit proﬁles are all positive (i.e., (1.9) holds for any sequence tn → ∞), the symmetry of entire solutions for the limit problems can be used to establish the asymptotic symmetry of positive solutions of the original Cauchy-Dirichlet problem. This is how the asymptotic symmetry is proved in [4, 6]. A diﬀerent approach to asymptotic symmetry is used in [63]. It is based on direct estimates, not relying on any limit equation, thus the regularity and positivity requirements on the nonlinearity and the solutions are signiﬁcantly relaxed compared to the earlier results. A yet diﬀerent approach was used in the original paper [44]. While it requires more regularity of the nonlinearity and the domain, it does not assume any strict positivity condition. Also it has an interesting feature in that it shows that the symmetry of a positive solution u improves with time in the sense that a quantity which can be thought of as a measure of symmetry increases strictly along any positive solution which is not symmetric from the start.

See Section 3 for precise formulations of the above results in the context of fully nonlinear parabolic equations. Results on entire solutions given there also include a statement on the symmetry of unstable spaces of positive solutions. In the special case when the positive solution is a steady state of a time-autonomous equation, the statement says that the eigenfunctions of the linearization around the steady state corresponding to negative eigenvalues are all symmetric.

When considering the asymptotic symmetry of solutions, several natural questions come to mind. For example, can the asymptotic symmetry be proved if the equations itself is not symmetric, but rather is merely asymptotically symmetric as t → ∞? One could for example think of equations (1.1) with an extra term added, say

ut = ∆u + f (t, u) + g(t, u), x ∈ Ω, t > 0,

where g(t, u) → 0 as t → ∞ for each u. Then one can also consider relaxing

6

other conditions, like the assumption of positivity of the solutions, and only require them to be satisﬁed asymptotically. Sometimes such problems can be addressed in a relatively simple manner. Indeed, as in the discussion above, if the limit proﬁles of a solution considered can be shown to be given by positive entire solutions of a limit equation, then, the limit equations being symmetric by assumption, one can apply to these entire solution the symmetry results discussed above. This gives the asymptotic symmetry of the original solution. However, in a general setting such a simple argument may not be applicable, a simple possible reason being that the original solution does not have a strictly positive inferior limit as t → ∞ at each point x ∈ Ω. In that case, not only is the treatment of asymptotically symmetric problems more complicated, the result may not be true in the form one could expect. Asymptotically symmetric problems are considered in the recent work [35]. We include statements of the main theorems and some discussion in Section 3.3.

1.2 Equations on RN

Let us now take Ω = RN . In their second symmetry paper [38], a sequel to

[37], Gidas, Ni and Nirenberg considered elliptic equations on RN including the

following one

∆u + f (u) = 0, x ∈ RN .

(1.11)

They assumed that f (0) = 0 and made other hypotheses on the behavior of f (u)

near u = 0. They proved that each positive solution u(x) of (1.11) which decays

to 0 as |x| → ∞ at a suitable rate has to be radially symmetric around some

ξ ∈ RN and radially decreasing away from ξ. Later it was proved by Li and Ni

[52] that a mere decay (with no speciﬁc rate) is suﬃcient for the symmetry of u if

f (0) = 0 and f is nonpositive near zero (under the stronger condition f (0) < 0,

this result was also proved in [51]). In both [51] and [52], general fully nonlinear

equations satisfying suitable symmetry assumptions are treated. Many other

extensions of the symmetry results are available. For example, one can consider

some degenerate equations [71] on RN or diﬀerent types of unbounded domains

[8, 9, 10, 69]. Again we refer the reader to the surveys [7, 58] for more details

and references.

Contrary to elliptic equations, symmetry results for parabolic equations on

RN did not appear so soon after the ﬁrst results on bounded domains. The fact that the possible center or hyperplane of symmetry in RN is not ﬁxed a

priori (unlike on bounded domains) adds an interesting ﬂavor to the symmetry

problem and is the cause of major diﬃculties. Already in the simple autonomous

case, the problem is by no means trivial. Consider for example the Cauchy

problem

ut = ∆u + f (u), u = u0,

x ∈ RN , t > 0, x ∈ RN , t = 0,

(1.12)

where f is of class C1, f (0) = 0, and u0 is a positive continuous function on RN decaying to 0 at |x| = ∞. Assume the solution u of (1.12) is global, bounded,

7

and localized in the sense that

sup u(x, t) → 0 as |x| → ∞.

t≥0

(1.13)

It is not clear whether u is asymptotically radially symmetric around some center, even if it is known that its ω-limit set ω(u) consists of steady states, each of them being radially symmetric about some center. It is not obvious whether all the functions in ω(u) share the same center of symmetry and, in fact, that is not true in general. A counterexample can be found in [68] where equations (1.12) with N ≥ 11, f (u) = up, and p suﬃciently large are considered. The proof of the existence of a solution with no asymptotic center of symmetry, as given there, depends on the fact that the steady states, in particular the trivial steady state, are stable in some weighted norms but are unstable in L∞(RN ) (see [40, 41, 67]). If, on the other hand, one makes the assumption f (0) < 0, which in particular implies that u ≡ 0 is asymptotically stable in L∞(RN ), then bounded solutions satisfying (1.13) do symmetrize as t → ∞: they actually converge to a symmetric steady state. This convergence result is proved in [15], under slightly stronger hypotheses (exponential decay of the solution at spatial inﬁnity); for more speciﬁc nonlinearities proofs can also be found in [26, 34]. The proofs of these convergence theorems depend heavily on energy estimates and are thus closely tied to the autonomous equations.

The symmetry problem for nonautonomous parabolic equations on Ω = RN , such as (1.1), was addressed in [61, 62]. The asymptotic symmetry for solutions of the Cauchy problems as well as symmetry for all times for entire solutions is established in these papers, see Section 4 for the statements. It is worthwhile to mention that these result are available for quasilinear equations only, not for fully nonlinear as in the case of bounded domains. The technical reasons for this will be brieﬂy explained in Section 4.

1.3 Cooperative systems

We shall now discuss extensions of the symmetry results to a class of parabolic systems. A model problem is the following cooperative system of reactiondiﬀusion equations

ut = D(t)∆u + f (t, u), (x, t) ∈ Ω × (0, ∞).

(1.14)

Here D(t) = diag(d1(t), . . . , dn(t)) is a diagonal matrix whose diagonal entries are continuous functions bounded above and below by positive constants, and f = (f1, . . . , fn) : [0, ∞) × Rn → Rn is a continuous function which is Lipschitz continuous in u ∈ Rn and which satisﬁes the cooperativity condition ∂fi(t, u)/∂uj ≥ 0 whenever i = j and the derivative exists (which is almost everywhere by the Lipschitz continuity). We couple (1.14) with the Dirichlet

boundary conditions

ui(x, t) = 0, (x, t) ∈ ∂Ω × (0, ∞), i = 1, . . . , n.

(1.15)

8

In case the diﬀusion coeﬃcients D and the nonlinearity are time-independent and only steady state solutions are considered, we are lead to the elliptic system

D∆u + f (u) = 0, x ∈ Ω,

(1.16)

with Dirichlet boundary conditions. Symmetry properties of positive solutions for such elliptic cooperative sys-

tems were established by Troy [75], then by Shaker [72] (see also [24]) who considered equations on smooth bounded domains. In [31], de Figueiredo removed the smoothness assumption on the domain in a similar way as Berestycki and Nirenberg [11] did for the scalar equation. For cooperative systems on the whole space, a general symmetry result was proved by Busca and Sirakov [16] (an earlier more restrictive result can be found in [32]). The cooperativity hypotheses which is assumed in all these references cannot be removed. Without it, neither is the maximum principle applicable nor do the symmetry result hold in general (see [17] and [72] for counterexamples).

Parabolic cooperative systems, such as (1.14), (1.15), were considered in [36], where the asymptotic symmetry of positive solutions is proved (see Section 5 for the results). A new diﬃculty that arises when dealing with parabolic systems, as opposed to scalar parabolic equations or elliptic systems, is that diﬀerent components of the positive solution may be very small at diﬀerent times. This situation has to be handled carefully using Harnack type estimates which were developed in [36] for this purpose.

Similar symmetry results for parabolic systems on RN can be proved using ideas form [36] and [61], but they are not documented in literature.

1.4 Applications

When it comes to applications of the symmetry results in further qualitative studies of parabolic equations, the matters are more complicated than in the case of elliptic equations. Even when dealing with radially symmetric solutions of (1.1), the analogue of (1.6) is

N −1 ut = urr + r ur + f (t, u) = 0,

(1.17)

which is still a PDE. Even worse, studying positive solutions of the Cauchy problem, we only have the asymptotic symmetry results, hence (1.17) can only be valid asymptotically, if one can make a sense of that. Nonetheless, the symmetry theorems have proved very useful for further studies of positive solutions of parabolic problems. For example, they have been used in the proofs of convergence results for some autonomous and time-periodic equations We will sketch the proof of such a convergence theorem in Section 7. In that section we also discuss some open problems related to symmetry properties of positive solutions and indicate possible directions of further research.

We would like to emphasize that we have devoted this survey exclusively to symmetry properties related to the positivity of solutions. There are many

9

other results in literature where symmetry is shown to be a consequence of other properties of solutions, like stability (see, for example, [57, 59] and references therein) or being a minimizer for some variational problems (see [55] and reference therein). Diﬀerent types of parabolic symmetry results can also be found in [25, 45, 56, 69].

2 General notation

The following general notation is used throughout the paper. For x0 ∈ RN

and r > 0, B(x0, r) stands for the ball centered at x0 with radius r. For a set Ω ⊂ RN and functions v and w on Ω, the inequalities v ≥ 0 and w > 0 are

always understood in the pointwise sense: v(x) ≥ 0, w(x) > 0 (x ∈ Ω). For a function z, z+, z− stand for the positive and negative parts of z, respectively:

z+(x) = (|z(x)| + z(x))/2 ≥ 0, z−(x) = (|z(x)| − z(x))/2 ≥ 0.

If D0, D are subsets of Rm with D0 bounded, the notation D0 ⊂⊂ D means D¯0 ⊂ D; diam(D) stands for the diameter of D; and |D| for the (Lebesgue) measure of D (if D is measurable). In each section of the paper, Ω is a ﬁxed domain in RN and we denote

:= sup{x1 : (x1, x ) ∈ Ω for some x ∈ RN−1} ≤ ∞,

Ωλ := {x ∈ Ω : x1 > λ},

Hλ := {x ∈ RN : x1 = λ}, Γλ := Hλ ∩ Ω¯ .

(2.1)

By Pλ we denote the reﬂection in the hyperplane Hλ. Note that if Ω is convex

in x1 and symmetric in the hyperplane H0, then Pλ(Ωλ) ⊂ Ω for each λ ∈ [0, ). For a function z(x) = z(x1, x ) deﬁned on Ω, let zλ and Vλz be deﬁned by

zλ(x) = z(Pλx) = z(2λ − x1, x ), Vλz(x) = zλ(x) − z(x) (x ∈ Ωλ).

(2.2)

3 Fully nonlinear equations on bounded domains

In this section we consider fully nonlinear parabolic problems of the form

ut = F (t, x, u, Du, D2u), x ∈ Ω, t ∈ J,

u = 0,

x ∈ ∂Ω, t ∈ J.

(3.1) (3.2)

Here Ω ⊂ RN is a bounded domain and J is either (0, ∞) or (−∞, T ) for some T ≤ ∞. Taking J = (0, ∞) we have the Cauchy-Dirichlet problem in mind, although usually we do not write down the initial condition explicitly (it does not play a role in our analysis). Included in the case J = (−∞, T ) are entire solutions (T = ∞), but the results we state apply also to T < ∞.

We make the following assumptions

10

P. Pola´ˇcik

School of Mathematics, University of Minnesota Minneapolis, MN 55455

Abstract

This survey is concerned with positive solutions of nonlinear parabolic equations. Assuming that the underlying domain and the equation have certain reﬂectional symmetries, the presented results show how positive solutions reﬂect the symmetries. Depending on the class of solutions considered, the symmetries for all times or asymptotic symmetries are established. Several classes of problems, including fully nonlinear equations on bounded domains, quasilinear equations on RN , asymptotically symmetric equations, and cooperative parabolic systems, are examined from this point of view. Applications of the symmetry results in the study of asymptotic temporal behavior of solutions are also shown.

Contents

1 Introduction: basic problems, results and some history

2

1.1 Equations on bounded domains . . . . . . . . . . . . . . . . . . . 3

1.2 Equations on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Cooperative systems . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 General notation

10

3 Fully nonlinear equations on bounded domains

10

3.1 Solutions on (0, ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Solutions: (−∞, T ) and linearized problems . . . . . . . . . . . . 14

3.3 Asymptotically symmetric equations . . . . . . . . . . . . . . . . 17

4 Quasilinear equations on RN

19

4.1 Solutions on (0, ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Solutions on (−∞, T ) and linearized problems . . . . . . . . . . . 21

5 Cooperative systems

23

1

6 On the proofs: a comparison of bounded and unbounded do-

mains

25

7 Applications and some open problems

29

1 Introduction: basic problems, results and some

history

A conventional wisdom says that “parabolic ﬂows reduce complexity.” Although it should not be taken too seriously and universally, there are good examples where it manifests itself. Asymptotic symmetry of solutions, which is one of the main topics of this survey, is such an example. It shows a tendency of positive solutions of certain parabolic equations to “improve their symmetry” as time increases, becoming “symmetric in the limit” as t → ∞.

Historically, ﬁrst studies of symmetry properties of positive solutions of parabolic equations were carried out after similar properties for elliptic equations had been long understood. These studies brought about interesting qualitative results for parabolic equations, but at the same time they opened new perspectives for looking at the earlier results for elliptic equations. Viewing solutions of elliptic equations as equilibria (steady states) of the corresponding parabolic equations, one naturally tries to understand how their symmetry ﬁts in the broader picture of the parabolic semiﬂow. For example, examining heteroclinic orbits between symmetric equilibria, one naturally asks if they are symmetric as well. Generalizing, one is subsequently lead to the problem of symmetry of entire solutions, that is, solutions deﬁned for all times, positive and negative. Another symmetry problem is concerned with general solutions of parabolic equations, which are typically deﬁned for positive times only. If the parabolic semiﬂow admits a Lyapunov functional, which forces bounded solutions to converge to steady states, the symmetry of the latter translates to the asymptotic symmetry of the solutions of the parabolic equation. This immediately raises the question whether the asymptotic symmetry can be established regardless of the presence of any Lyapunov functional, even if the solution does not converge to a steady state. This question is even more interesting for time-dependent parabolic problems, whose solutions typically do not converge to equilibria and their temporal behavior can be very complicated. In this case, the asymptotic symmetrization in space is to be studied independently of the temporal structure. Once it is understood, however, it often proves very useful for studying the temporal behavior of the solutions.

The previous paragraph indicates the sort of problems this survey is devoted to. Considering parabolic equations with certain symmetry properties, we want to understand how their solutions reﬂect the symmetry. The key issues to be discussed are the asymptotic symmetry properties for the Cauchy problem, symmetry of the entire solutions (and related to this, symmetry of unstable spaces of entire solutions), and applications of these results. We also mention key ideas of the proofs and discuss diﬀerences of their use for diﬀerent type of

2

problems. To give a more speciﬁc overview of the results to be presented in this paper

and to put them in context with similar results on elliptic equations, we ﬁrst consider the following semilinear reaction-diﬀusion equation

ut = ∆u + f (t, u), x ∈ Ω, t ∈ J.

(1.1)

Here Ω is a domain in RN and f : R2 → R is a continuous function, which is Lipschitz continuous with respect to u. We take either J = (0, ∞) and consider a suitable initial-value problem, or J = (−∞, ∞) in case we want to consider entire solutions.

Although much simpler an equation than fully nonlinear ones examined is the forthcoming sections, for the purposes of the introduction (1.1) is suﬃciently representative and it allows us to discuss some key issues in a more rudimentary way. The simplicity of (1.1) consists mainly in the fact that, without any additional assumptions, the equation is invariant under any Euclidean symmetries the domain may have. This is true regardless of the temporal dependence of f on time, which is not restricted by any assumption like periodicity or almost periodicity.

We assume that either Ω = RN , or Ω is a bounded domain in RN which is symmetric with respect to the hyperplane

H0 := {x = (x1, . . . , xN ) ∈ RN : x1 = 0}

and convex in x1 (which means that with any two points (x1, x ), (x˜1, x ), diﬀering only at the ﬁrst component, Ω contains the line segment connecting them). The speciﬁc choice of the direction e1 for the reﬂectional symmetry is arbitrary, domains symmetric in other directions can be considered equally well. In case Ω is bounded, we complement the equation with Dirichlet boundary condition

u(x, t) = 0, x ∈ ∂Ω, t ∈ J.

(1.2)

If f is independent of t, steady states of (1.1), (1.2) are solutions of the

elliptic equation

∆u + f (u) = 0, x ∈ Ω,

(1.3)

complemented (in case Ω is bounded) with the Dirichlet condition

u(x) = 0, x ∈ ∂Ω.

(1.4)

Symmetry theorems for (1.1) have somewhat diﬀerent ﬂavors and hypotheses for bounded domains Ω and for Ω = RN , so we distinguish these case separately.

1.1 Equations on bounded domains

Reﬂectional symmetry of positive solutions of (1.3), (1.4) was ﬁrst established by Gidas, Ni and Nirenberg [37]. They proved that if Ω is as above (convex

3

and symmetric in x1) and smooth (of class C2), then each positive solution u of (1.3), (1.4) has the following symmetry and monotonicity properties:

u(−x1, x2, . . . , xN ) = u(x1, x2, . . . , xN ) (x ∈ Ω), ux1 (x1, x2, . . . , xN ) < 0 (x ∈ Ω, x1 > 0).

(1.5)

The method of moving hyperplanes, which is the basic geometric technique in their paper, was introduced earlier by Alexandrov [2] and further developed by Serrin [70] ([70] also contains a related result on radial symmetry). Generalizations and extensions of the symmetry result have been made by many authors. In particular, Li [50] extended it to fully nonlinear equations on smooth domains. Later Berestycki and Nirenberg [11] found a way of dealing with fully nonlinear equations on nonsmooth domains employing the elliptic maximum principle for domains with small measure (see also Dancer’s contribution [29], where semilinear equations on nonsmooth domains are treated using a diﬀerent method). There are many other related results, including further developments regarding symmetry of elliptic overdetermined problems, as considered in the original paper of Serrin [70], see for example [1, 73]. Additional references and more detailed overviews can be found in the surveys [7, 46, 58]. Let us also mention a more recent work by Da Lio and Sirakov [27], where the symmetry results are extended to viscosity solutions of general elliptic equations.

The proofs of the above results are based on the method of moving hyperplanes and various forms of the maximum principle. Below we shall indicate how these techniques are typically used. A diﬀerent approach employing a continuous Steiner symmetrization was used in [12] (see also the survey [46]). It applies to positive solutions of (1.3), (1.4) and, as it relies on the variational structure of the problem and not so much the maximum principle, it allows for extensions of the results of [37] in diﬀerent directions.

We remark that it is not always possible to generalize the above symmetry result to solutions which are merely nonnegative, rather than strictly positive, (counterexamples are easily constructed if N = 1). However, if N > 1 and some regularity assumptions are made on the domain, it can be proved that nonnegative solutions are necessarily strictly positive, hence symmetric (see [19, 33] for results of this sort). The issue whether some strict positivity assumption is needed or not will arise again in our discussion of parabolic equations below. Also the convexity of Ω in x1 is an important assumption without which the result is not valid in general (however, see [47] for a symmetry result involving some nonconvex domains).

If Ω is a ball, say Ω = B(0, r0) (0 is the center, r0 the radius), then the reﬂectional symmetry theorem can be applied in any direction which leads to the following radial symmetry result. Any positive solution u(x) of (1.3), (1.4) is radially symmetric (it only depends on r = |x|) and radially decreasing (ur(x) < 0 for r ∈ (0, r0)).

There are numerous application of the above symmetry results in further studies of positive solutions of (1.3), (1.4). For example, the radial symmetry property implies that positive solutions can be viewed as solutions of the

4

ordinary diﬀerential equation (ODE)

N −1 urr + r ur + f (u) = 0, r ∈ (0, r0),

(1.6)

and that ur(0) = 0. Thus one immediately gains ODE tools, like the shooting method, for the study of positive solutions. Problems on multiplicities and/or bifurcations of positive solutions become then a lot more elementary. The reﬂectional symmetry results do not lead to such dramatic simpliﬁcations of the problem, but they are still very useful, especially if there are several directions in which the domain is symmetric.

For parabolic problems, such as (1.1), (1.2), ﬁrst symmetry results of similar nature started to emerge much later. After a prelude [30] devoted to timeperiodic solutions, symmetry of general positive solutions of parabolic equations on bounded domains was considered in [4, 5, 44] and later in [6, 63]. With Ω as above (convex and symmetric in x1) and with suitable symmetry assumptions on the nonlinearity, symmetries of two classes of solutions were examined in these papers. Closer in spirit to the results for elliptic equations are symmetry theorems concerning entire solutions. A typical theorem in this category states that if u is a bounded positive solution of (1.1), (1.2) with J = R satisfying

inf u(x, t) > 0 (x ∈ Ω, t ∈ J),

t∈R

(1.7)

then u has the symmetry and monotonicity properties (1.5) for each t ∈ R:

u(−x1, x , t) = u(x1, x , t) (x = (x1, x ) ∈ Ω, t ∈ R), ux1 (x, t) < 0 (x ∈ Ω, x1 > 0, t ∈ R).

(1.8)

This result follows from more general theorems of [4, 6], although to be precise we would need to include additional compactness assumptions on u (in the context of the semilinear problem (1.1), (1.2), the boundedness of u alone is suﬃcient if, for example, Ω has Lipschitz boundary and t → f (0, t) is a bounded function).

In a diﬀerent type of symmetry results, nonnegative solutions of the CauchyDirichlet problem for (1.1) are considered. These of course cannot be symmetric, unless they start from a symmetric initial function. However, it can be shown that they “achieve” the symmetry in the limit as t → ∞. To formulate this more precisely assume that u is a bounded positive solution of (1.1), (1.2) with J = (0, ∞) such that for some sequence tn → ∞

lim inf u(x, tn) > 0 (x ∈ Ω).

n→∞

(1.9)

Then u is asymptotically symmetric in the sense that

lim (u(−x1, x , t) − u(x1, x , t)) = 0

t→∞

lim sup ux1 (x, t) ≤ 0

t→∞

(x ∈ Ω), (x ∈ Ω, x1 > 0).

(1.10)

5

If {u(·, t) : t ≥ 1} is relatively compact in C(Ω¯ ), then the asymptotic symmetry of u can be expressed in terms of its limit proﬁles, that is, elements of its ω-limit set,

ω(u) := {φ : φ = lim u(·, tn) for some tn → ∞},

where the limit is in C(Ω¯ ) (with the supremum norm). It can be easily shown that condition (1.9) is equivalent to the requirement that there exist at least one element of ω(u) which is strictly positive in Ω and (1.10) translates to all elements of ω(u) being symmetric (even) in x1 and monotone nonincreasing in x1 > 0. This result is proved in a more general setting in [63]. Condition (1.9) is a relatively minor strict positivity condition (note that it is not assumed to be valid for all sequences tn → ∞), which cannot be omitted in general (a counterexample can be found in [63]). It is not needed, however, if the domain is suﬃciently regular [44].

There are connections between the two types of symmetry results, the asymptotic symmetry for the Cauchy-Dirichlet problem and the symmetry for entire solutions. Oftentimes, if the nonlinearity is suﬃciently regular, the limit proﬁles of a solution of the Cauchy-Dirichlet problem can be shown to be given by entire solutions of suitable limit parabolic problems. Thus if the limit proﬁles are all positive (i.e., (1.9) holds for any sequence tn → ∞), the symmetry of entire solutions for the limit problems can be used to establish the asymptotic symmetry of positive solutions of the original Cauchy-Dirichlet problem. This is how the asymptotic symmetry is proved in [4, 6]. A diﬀerent approach to asymptotic symmetry is used in [63]. It is based on direct estimates, not relying on any limit equation, thus the regularity and positivity requirements on the nonlinearity and the solutions are signiﬁcantly relaxed compared to the earlier results. A yet diﬀerent approach was used in the original paper [44]. While it requires more regularity of the nonlinearity and the domain, it does not assume any strict positivity condition. Also it has an interesting feature in that it shows that the symmetry of a positive solution u improves with time in the sense that a quantity which can be thought of as a measure of symmetry increases strictly along any positive solution which is not symmetric from the start.

See Section 3 for precise formulations of the above results in the context of fully nonlinear parabolic equations. Results on entire solutions given there also include a statement on the symmetry of unstable spaces of positive solutions. In the special case when the positive solution is a steady state of a time-autonomous equation, the statement says that the eigenfunctions of the linearization around the steady state corresponding to negative eigenvalues are all symmetric.

When considering the asymptotic symmetry of solutions, several natural questions come to mind. For example, can the asymptotic symmetry be proved if the equations itself is not symmetric, but rather is merely asymptotically symmetric as t → ∞? One could for example think of equations (1.1) with an extra term added, say

ut = ∆u + f (t, u) + g(t, u), x ∈ Ω, t > 0,

where g(t, u) → 0 as t → ∞ for each u. Then one can also consider relaxing

6

other conditions, like the assumption of positivity of the solutions, and only require them to be satisﬁed asymptotically. Sometimes such problems can be addressed in a relatively simple manner. Indeed, as in the discussion above, if the limit proﬁles of a solution considered can be shown to be given by positive entire solutions of a limit equation, then, the limit equations being symmetric by assumption, one can apply to these entire solution the symmetry results discussed above. This gives the asymptotic symmetry of the original solution. However, in a general setting such a simple argument may not be applicable, a simple possible reason being that the original solution does not have a strictly positive inferior limit as t → ∞ at each point x ∈ Ω. In that case, not only is the treatment of asymptotically symmetric problems more complicated, the result may not be true in the form one could expect. Asymptotically symmetric problems are considered in the recent work [35]. We include statements of the main theorems and some discussion in Section 3.3.

1.2 Equations on RN

Let us now take Ω = RN . In their second symmetry paper [38], a sequel to

[37], Gidas, Ni and Nirenberg considered elliptic equations on RN including the

following one

∆u + f (u) = 0, x ∈ RN .

(1.11)

They assumed that f (0) = 0 and made other hypotheses on the behavior of f (u)

near u = 0. They proved that each positive solution u(x) of (1.11) which decays

to 0 as |x| → ∞ at a suitable rate has to be radially symmetric around some

ξ ∈ RN and radially decreasing away from ξ. Later it was proved by Li and Ni

[52] that a mere decay (with no speciﬁc rate) is suﬃcient for the symmetry of u if

f (0) = 0 and f is nonpositive near zero (under the stronger condition f (0) < 0,

this result was also proved in [51]). In both [51] and [52], general fully nonlinear

equations satisfying suitable symmetry assumptions are treated. Many other

extensions of the symmetry results are available. For example, one can consider

some degenerate equations [71] on RN or diﬀerent types of unbounded domains

[8, 9, 10, 69]. Again we refer the reader to the surveys [7, 58] for more details

and references.

Contrary to elliptic equations, symmetry results for parabolic equations on

RN did not appear so soon after the ﬁrst results on bounded domains. The fact that the possible center or hyperplane of symmetry in RN is not ﬁxed a

priori (unlike on bounded domains) adds an interesting ﬂavor to the symmetry

problem and is the cause of major diﬃculties. Already in the simple autonomous

case, the problem is by no means trivial. Consider for example the Cauchy

problem

ut = ∆u + f (u), u = u0,

x ∈ RN , t > 0, x ∈ RN , t = 0,

(1.12)

where f is of class C1, f (0) = 0, and u0 is a positive continuous function on RN decaying to 0 at |x| = ∞. Assume the solution u of (1.12) is global, bounded,

7

and localized in the sense that

sup u(x, t) → 0 as |x| → ∞.

t≥0

(1.13)

It is not clear whether u is asymptotically radially symmetric around some center, even if it is known that its ω-limit set ω(u) consists of steady states, each of them being radially symmetric about some center. It is not obvious whether all the functions in ω(u) share the same center of symmetry and, in fact, that is not true in general. A counterexample can be found in [68] where equations (1.12) with N ≥ 11, f (u) = up, and p suﬃciently large are considered. The proof of the existence of a solution with no asymptotic center of symmetry, as given there, depends on the fact that the steady states, in particular the trivial steady state, are stable in some weighted norms but are unstable in L∞(RN ) (see [40, 41, 67]). If, on the other hand, one makes the assumption f (0) < 0, which in particular implies that u ≡ 0 is asymptotically stable in L∞(RN ), then bounded solutions satisfying (1.13) do symmetrize as t → ∞: they actually converge to a symmetric steady state. This convergence result is proved in [15], under slightly stronger hypotheses (exponential decay of the solution at spatial inﬁnity); for more speciﬁc nonlinearities proofs can also be found in [26, 34]. The proofs of these convergence theorems depend heavily on energy estimates and are thus closely tied to the autonomous equations.

The symmetry problem for nonautonomous parabolic equations on Ω = RN , such as (1.1), was addressed in [61, 62]. The asymptotic symmetry for solutions of the Cauchy problems as well as symmetry for all times for entire solutions is established in these papers, see Section 4 for the statements. It is worthwhile to mention that these result are available for quasilinear equations only, not for fully nonlinear as in the case of bounded domains. The technical reasons for this will be brieﬂy explained in Section 4.

1.3 Cooperative systems

We shall now discuss extensions of the symmetry results to a class of parabolic systems. A model problem is the following cooperative system of reactiondiﬀusion equations

ut = D(t)∆u + f (t, u), (x, t) ∈ Ω × (0, ∞).

(1.14)

Here D(t) = diag(d1(t), . . . , dn(t)) is a diagonal matrix whose diagonal entries are continuous functions bounded above and below by positive constants, and f = (f1, . . . , fn) : [0, ∞) × Rn → Rn is a continuous function which is Lipschitz continuous in u ∈ Rn and which satisﬁes the cooperativity condition ∂fi(t, u)/∂uj ≥ 0 whenever i = j and the derivative exists (which is almost everywhere by the Lipschitz continuity). We couple (1.14) with the Dirichlet

boundary conditions

ui(x, t) = 0, (x, t) ∈ ∂Ω × (0, ∞), i = 1, . . . , n.

(1.15)

8

In case the diﬀusion coeﬃcients D and the nonlinearity are time-independent and only steady state solutions are considered, we are lead to the elliptic system

D∆u + f (u) = 0, x ∈ Ω,

(1.16)

with Dirichlet boundary conditions. Symmetry properties of positive solutions for such elliptic cooperative sys-

tems were established by Troy [75], then by Shaker [72] (see also [24]) who considered equations on smooth bounded domains. In [31], de Figueiredo removed the smoothness assumption on the domain in a similar way as Berestycki and Nirenberg [11] did for the scalar equation. For cooperative systems on the whole space, a general symmetry result was proved by Busca and Sirakov [16] (an earlier more restrictive result can be found in [32]). The cooperativity hypotheses which is assumed in all these references cannot be removed. Without it, neither is the maximum principle applicable nor do the symmetry result hold in general (see [17] and [72] for counterexamples).

Parabolic cooperative systems, such as (1.14), (1.15), were considered in [36], where the asymptotic symmetry of positive solutions is proved (see Section 5 for the results). A new diﬃculty that arises when dealing with parabolic systems, as opposed to scalar parabolic equations or elliptic systems, is that diﬀerent components of the positive solution may be very small at diﬀerent times. This situation has to be handled carefully using Harnack type estimates which were developed in [36] for this purpose.

Similar symmetry results for parabolic systems on RN can be proved using ideas form [36] and [61], but they are not documented in literature.

1.4 Applications

When it comes to applications of the symmetry results in further qualitative studies of parabolic equations, the matters are more complicated than in the case of elliptic equations. Even when dealing with radially symmetric solutions of (1.1), the analogue of (1.6) is

N −1 ut = urr + r ur + f (t, u) = 0,

(1.17)

which is still a PDE. Even worse, studying positive solutions of the Cauchy problem, we only have the asymptotic symmetry results, hence (1.17) can only be valid asymptotically, if one can make a sense of that. Nonetheless, the symmetry theorems have proved very useful for further studies of positive solutions of parabolic problems. For example, they have been used in the proofs of convergence results for some autonomous and time-periodic equations We will sketch the proof of such a convergence theorem in Section 7. In that section we also discuss some open problems related to symmetry properties of positive solutions and indicate possible directions of further research.

We would like to emphasize that we have devoted this survey exclusively to symmetry properties related to the positivity of solutions. There are many

9

other results in literature where symmetry is shown to be a consequence of other properties of solutions, like stability (see, for example, [57, 59] and references therein) or being a minimizer for some variational problems (see [55] and reference therein). Diﬀerent types of parabolic symmetry results can also be found in [25, 45, 56, 69].

2 General notation

The following general notation is used throughout the paper. For x0 ∈ RN

and r > 0, B(x0, r) stands for the ball centered at x0 with radius r. For a set Ω ⊂ RN and functions v and w on Ω, the inequalities v ≥ 0 and w > 0 are

always understood in the pointwise sense: v(x) ≥ 0, w(x) > 0 (x ∈ Ω). For a function z, z+, z− stand for the positive and negative parts of z, respectively:

z+(x) = (|z(x)| + z(x))/2 ≥ 0, z−(x) = (|z(x)| − z(x))/2 ≥ 0.

If D0, D are subsets of Rm with D0 bounded, the notation D0 ⊂⊂ D means D¯0 ⊂ D; diam(D) stands for the diameter of D; and |D| for the (Lebesgue) measure of D (if D is measurable). In each section of the paper, Ω is a ﬁxed domain in RN and we denote

:= sup{x1 : (x1, x ) ∈ Ω for some x ∈ RN−1} ≤ ∞,

Ωλ := {x ∈ Ω : x1 > λ},

Hλ := {x ∈ RN : x1 = λ}, Γλ := Hλ ∩ Ω¯ .

(2.1)

By Pλ we denote the reﬂection in the hyperplane Hλ. Note that if Ω is convex

in x1 and symmetric in the hyperplane H0, then Pλ(Ωλ) ⊂ Ω for each λ ∈ [0, ). For a function z(x) = z(x1, x ) deﬁned on Ω, let zλ and Vλz be deﬁned by

zλ(x) = z(Pλx) = z(2λ − x1, x ), Vλz(x) = zλ(x) − z(x) (x ∈ Ωλ).

(2.2)

3 Fully nonlinear equations on bounded domains

In this section we consider fully nonlinear parabolic problems of the form

ut = F (t, x, u, Du, D2u), x ∈ Ω, t ∈ J,

u = 0,

x ∈ ∂Ω, t ∈ J.

(3.1) (3.2)

Here Ω ⊂ RN is a bounded domain and J is either (0, ∞) or (−∞, T ) for some T ≤ ∞. Taking J = (0, ∞) we have the Cauchy-Dirichlet problem in mind, although usually we do not write down the initial condition explicitly (it does not play a role in our analysis). Included in the case J = (−∞, T ) are entire solutions (T = ∞), but the results we state apply also to T < ∞.

We make the following assumptions

10