Nonlinear diffusion with a bounded stationary level surface

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Nonlinear diffusion with a bounded stationary level surface

Rolando Magnanini

, Shigeru Sakaguchi

aDipartimento di Matematica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy

bDepartment of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan Received 2 February 2009; received in revised form 28 December 2009; accepted 28 December 2009

Available online 7 January 2010

Abstract

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of classC². Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a properC²-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the wholeR^N of some substance whose density is initially a characteristic function of the complement of a domain with boundedC²boundary, and obtain similar results.

These results are also extended to the heat flow in the sphereS^Nand the hyperbolic spaceH^N.

©2010 Elsevier Masson SAS. All rights reserved.

Résumé

Nous considérons la diffusion non linéaire d’une substance dans un récipient (pas nécessairement borné) avec frontière bornée de classeC². Supposons qu’initialement, le récipient soit vide et, à sa frontière, la densité de la substance soit gardée à tout moment égale à 1. Nous montrons que, si le récipient contient un sous-domaineC²propre à la frontière duquel la substance est gardée à tout moment à densité constante, alors la frontière du récipient doit être une sphère. Nous considérons aussi la diffusion non linéaire dans toutR^Nd’une substance dont la densité est initialement une fonction caractéristique du complémentaire d’un domaine ayant la frontière bornée etC², et nous obtenons des résultats semblables. Ces résultats sont aussi généralisés au cas du flux de chaleur dans la sphèreS^Net l’espace hyperboliqueH^N.

©2010 Elsevier Masson SAS. All rights reserved.

MSC:primary 35K60; secondary 35B40, 35B25

Keywords:Nonlinear diffusion equation; Overdetermined problems; Stationary level surfaces

✩ This research was partially supported by Grants-in-Aid for Scientific Research (B) (15340047 and20340031) and a Grant-in-Aid for Exploratory Research (18654027) of Japan Society for the Promotion of Science, and by a Grant of the Italian MURST.

* Corresponding author.

E-mail addresses:magnanin@math.unifi.it (R. Magnanini), sakaguch@amath.hiroshima-u.ac.jp (S. Sakaguchi).

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.12.001

1. Introduction 1.1. Background

In the paper [12], we considered the solutionu=u(x, t )of the following initial–boundary value problem for the heat equation:

u_t=u inΩ×(0,+∞), (1.1)

u=1 on∂Ω×(0,+∞), (1.2)

u=0 onΩ× {0}, (1.3)

whereΩis a bounded domain inR^N withN2, and we obtained the following symmetry result.

Theorem A. (See [12].) LetΩ be a bounded domain inR^N, N2,satisfying the exterior sphere condition and suppose thatDis a domain, with boundary∂D,satisfying the interior cone condition, and such thatD⊂Ω.

Assume that the solutionuof problem(1.1)–(1.3)is such that

u(x, t )=a(t ), (x, t )∈∂D×(0,+∞), (1.4)

for some functiona:(0,+∞)→(0,+∞).ThenΩmust be a ball.

We recall some terminology from [12]. A surface satisfying (1.4) is said to be a stationary isothermic surface;

Ω satisfies theexterior sphere conditionif for everyy∈∂Ω there exists a ballBr(z)such thatBr(z)∩Ω= {y}, whereBr(z)denotes an open ball centered atz∈R^Nand with radiusr >0;Dsatisfies theinterior cone conditionif for everyx∈∂Dthere exists a finite right spherical coneKxwith vertexxsuch thatKx⊂DandKx∩∂D= {x}.

In order to better understand the background of the present paper, we outline the proof of Theorem A improved by a result in [13]. The proof is essentially based on three ingredients.

The first one is a result of Varadhan [21] which states that, as t→0⁺,the function −4tlogu(x, t ) converges uniformly onΩto the functiond(x)², where

d(x)=dist(x, ∂Ω), x∈Ω.

To apply this result one needs the boundary∂Ω to be also the boundary of the exteriorR^N\Ω. The assumption that Ω satisfies the exterior sphere condition is sufficient for that to happen. Hence, by (1.4) there existsR >0 satisfying

d(x)=R for everyx∈∂D. (1.5)

The second ingredient consists of abalance lawproved in [10] and [11] (see [12] for another proof). It states that, in any domainGinR^N,a solutionv=v(x, t )of the heat equation is zero at some pointx0∈Gfor everyt >0 if and only if

∂Br(x0)

v(x, t ) dS_x=0, for everyr∈

0,dist(x0, ∂G)

andt >0. (1.6)

We use (1.6) in two different ways. In the former one, we chooseG=Ω andv=u_xi, i=1, . . . , N,and obtain, by some manipulations, that the gradient∇uis zero at some pointx₀∈Ω for everyt >0 if and only if

∂Br(x0)

(x−x0)u(x, t ) dSx=0, for everyr∈

0, d(x0)

andt >0. (1.7)

This condition helps us show that both∂Dand∂Ωmust be analytic. Indeed, with the aid of the interior cone condition forD, by combining (1.7) and (1.5) with the short-time behavior ofudescribed in Varadhan [21], we can see that for every pointx0∈∂Dthere exists a timet0>0 satisfying∇u(x0, t0) =0; this implies that∂Dis analytic. Thus, by using the exterior sphere condition forΩagain, we can conclude that∂Ωis analytic and parallel to∂D.

In the latter way of using (1.6), we choose two distinct pointsP , Q∈∂Ωand letp, q∈∂Dbe the points such that B_R(p)∩∂Ω= {P} and B_R(q)∩∂Ω= {Q}.

Thence, we consider the functionv=v(x, t )defined by

v(x, t )=u(x+p, t )−u(x+q, t ) for(x, t )∈B_R(0)×(0,+∞).

Sincevsatisfies the heat equation andv(0, t )=a(t )−a(t )=0 for everyt >0, it follows from (1.6) that t^−N4+1

BR(p)

u(x, t ) dx=t^−N4+1

BR(q)

u(x, t ) dx for everyt >0.

Therefore, by taking advantage of the boundary layer forufor short times, we lett→0⁺and by using a result in [13], we obtain that

C(N ) _N−1

j=1

1

R −κ_j(P )

−¹₂

=C(N ) _N−1

j=1

1

R−κ_j(Q)

−¹₂

, (1.8)

whereκ_j(x), j =1, . . . , N−1, denotes the j-th principal curvature of the surface∂Ω at the pointx∈∂Ω with respect to the inward unit normal vector to∂Ω, and whereC(N )is a positive constant depending only onN(see [13, Theorem 4.2]).

With (1.8) in hand, we are ready to use our third ingredient: Aleksandrov’s sphere theorem [1, p. 412]. (A special case of this theorem is the well-knownSoap-Bubble Theorem(see also [17]).) Since (1.8) implies thatN−1

j=1[_R¹ − κj(x)]is constant forx∈∂Ω, by applying Aleksandrov’s sphere theorem, we conclude that∂Ω must be a sphere (see [12] and [13] for details).

1.2. Main results

In the present paper, we extend and improve the results described in Section 1.1 to the case of certainnonlinear diffusion equations. It is evident that the introduction of a nonlinearity immediately rules out the use of our second ingredient, e.g. the balance law.

Since this was crucial to prove the necessary regularity of∂Ω,we will have to change our assumptions on the domain Ω.Thus, we shall assume Ω to be a domain (not necessarily bounded) in R^N, N 2, having bounded boundary of classC²,that is, ∂Ω consists ofm (m1)connected components S₁, . . . , S_m⊂∂Ω which are the boundaries of boundedC²-domainsG¹, . . . , G^minR^N, respectively. Thus

∂Ω= m

j=1

S_j and S_j=∂G^j for eachj∈ {1, . . . , m}. (1.9)

It should also be noticed that the lack of a balance law precludes the proof of property (1.8) (unless we find an alternative proof) and hence Aleksandrov’s sphere theorem cannot be put in action. We shall overcome this difficulty by a new and more direct proof of symmetry only based on our first ingredient (conveniently modified in Theorem 1.1) and Serrin’smethod of moving planes(see [18,16,19]). It is worth mentioning that our proof does not need Serrin’s corner lemmabut simply uses the strong maximum principle and Hopf boundary lemma (see Theorems 1.2 and 1.3).

We now set up our framework. We consider the unique bounded solutionu=u(x, t )of the nonlinear diffusion equation

ut=φ(u) inΩ×(0,+∞), (1.10)

subject to conditions (1.2) and (1.3). Hereφ:R→Ris such that

φ∈C²(R), φ(0)=0, and (1.11)

0< δ1φ(s)δ2 fors∈R, (1.12)

whereδ₁,δ₂are positive constants. By the maximum principle we know that 0< u <1 inΩ×(0,+∞).

Moreover, by applying the comparison principle tou(x, t+h)andu(x, t )forh >0, we get

u_t0 and hence φ(u)0 inΩ×(0,+∞). (1.13)

LetΦ=Φ(s)be the function defined by Φ(s)=

s

1

φ(ξ )

ξ dξ fors >0. (1.14)

Note that ifφ(s)≡s, thenΦ(s)=logs.

We extend Varadhan’s result to our setting by the following Theorem 1.1.Letube the solution of problem(1.10), (1.2)–(1.3).

Then, lim

t→0⁺−4t Φ u(x, t )

=d(x)² uniformly on every compact set inΩ.

The proof of this theorem is constructed by adapting well-known results of the theory of viscosity solutions [2,7,3, 4,9]. The techniques developed to prove Theorem 1.1 can be used to extend this result to the important case in which the homogeneous boundary condition (1.2) is replaced by the non-homogeneous one

u=f on∂Ω×(0,+∞), (1.15)

wheref=f (x)is a continuous function on∂Ω,bounded from above and away from zero by positive constants (see Theorem 3.7).

The following symmetry result corresponds to Theorem A and Theorem 3.1 in [14].

Theorem 1.2.LetDbe aC²domain inR^N satisfyingD⊂Ω. Assume that the solutionuof problem(1.10), (1.2)–

(1.3), satisfies(1.4).

Thenm=1and∂Ω must be a sphere.

WhenΩis limited to unbounded domains, we have

Theorem 1.3.LetDbe aC²unbounded domain inR^N satisfyingD⊂Ω.

Assume that, for any connected componentΓ of∂D, the solution uof problem(1.10), (1.2)–(1.3), satisfies the following condition:

u(x, t )=a_Γ(t ), (x, t )∈Γ ×(0,+∞), (1.16)

for some functionaΓ :(0,+∞)→(0,+∞).

Thenm=1and∂Ω must be a sphere.

Whenφ(s)=s andΩ is bounded, Theorem A is clearly stronger than Theorem 1.2, since in the former we can use the balance law to infer better regularity. Furthermore, the same techniques used for the proof of Theorem A also yield a more general version of it (see Theorem 2.1).

The paper is then organized as follows. In Section 2, we prove all our symmetry results: Theorems 1.2, 1.3 and 2.1.

In Section 3, with the help of the theory of viscosity solutions, we prove Theorem 1.1 and its extension, Theorem 3.7.

Section 4 is devoted to show similar results for the unique bounded solution of the Cauchy problem for nonlinear diffusion equations. In Section 5, we mention that this kind of results also hold for the heat flow in the sphereS^Nand the hyperbolic spaceH^NwithN2.

2. Symmetry results

In this section, with the aid of Theorem 1.1, by applying the method of moving planes to problem (1.10), (1.2)–(1.3) directly, we prove Theorems 1.2 and 1.3.

Proof of Theorem 1.2. First of all, we consider the case whereΩis unbounded. In this caseΩis an exterior domain, that is, we have

Gⁱ∩G^j= ∅ ifi =j, i, j=1, . . . , m, and Ω=R^N\ _m

j=1

G^j

(see (1.9) for the definitions ofS_jandG^j). Set G=

m

j=1

G^j. (2.1)

ThenGis a bounded open set inR^Nhavingmconnected componentsG¹, . . . , G^m. Theorem 1.1 and the assumption (1.4) yield (1.5). Furthermore, with the aid of ourC²-smoothness assumption on∂D and∂Ω, we see that both∂Ω and∂D consist ofmconnected closed hypersurfaces and each component of∂Ω is parallel, at distanceR,to only one component of∂D.

We apply the method of moving planes to the open setG. The proof runs similarly to those of Serrin’s [18] — or Reichel’s [16] and Sirakov’s [19] for exterior domains — but with the major difference that, here, since the relevant overdetermination takes placeinsideΩ,Serrin’s corner lemma — an extension of Hopf boundary lemma to domains with corners — is not needed.

Letbe a unit vector inR^N, λ∈R,and letπ_λbe the hyperplanex·=λ.For largeλ, π_λwill be disjoint fromG; asλdecreases,π_λwill intersectGand cut off fromGan open capG_λ(on the same side ofλ→ +∞).

Denote byG_λthe reflection ofG_λ in the planeπ_λ.At the beginning,G_λwill be and remain inGuntil one of the following occurs:

(i) G_λbecomes internally tangent to∂Gat some pointP not onπλ; (ii) π_λreaches a position in which it is orthogonal to∂Gat some pointQ.

Letλ_∗ denote the (minimal) value of λat which the planeπ_λ reaches one of these positions and suppose that Gis not symmetric with respect toπ_λ∗. LetΩ be the connected component ofΩ∩ {x∈R^N: x· < λ_∗}whose boundary contains the pointsP orQin the respective cases (i) or (ii). Since, as already observed,∂Ωand∂Dconsist of connected closed pairwise parallel hypersurfaces, we can find pointsP^∗ andQ^∗ in∂D such that |P −P^∗| or

|Q−Q^∗|equalR, respectively, and we have thatP^∗∈ΩandQ^∗∈∂Ω∩π_λ∗.

Letx^λ=x+2[λ−(x·)]denote the reflection of a pointx∈R^N in the planeπ_λ.For(x, t )∈Ω×(0,∞), consider the functionw=w(x, t )defined by

w(x, t )=u x^λ∗, t

. Then it follows from (1.4) that

w(P^∗, t )=u(P^∗, t ) or ∂u

∂(Q^∗, t )=0 for allt >0, (2.2)

where in the second equality we have used the fact that the vectoris tangential also to∂DatQ^∗∈∂D.

Observe thatwandusatisfy

w_t=φ(w) and u_t=φ(u) inΩ×(0,+∞),

w=u on(∂Ω∩π_λ∗)×(0,+∞),

w <1=u on(∂Ω\π_λ∗)×(0,+∞), w=u=0 onΩ× {0}.

Hence, by the strong comparison principle,

w < u inΩ×(0,+∞). (2.3)

Indeed, (2.3) can be obtained by applying the strong comparison principle to the bounded solutionsW =φ(w)and U=φ(u)ofW_t=_ψ(W )¹ W andU_t=_ψ¹(U )U, respectively; here,ψis the inverse function ofφ.

If case (i) applies, (2.3) contradicts the first equality in (2.2), sinceP^∗∈Ω.If case (ii) applies, by using Hopf’s boundary point lemma, we can infer that

∂u

∂(Q^∗, t ) <0 for allt >0,

which contradicts the second equality in (2.2).

In conclusion,Gis symmetric for any direction∈R^N, and in view of the definition (2.1) ofG,m=1 andG must be a ball. Namely,Ω is the exterior of a ball and∂Ωmust be a sphere.

WhenΩis bounded, it suffices to apply the method of moving planes directly toΩ. 2

Proof of Theorem 1.3. With the aid of theC²smoothness assumption of both∂D and∂Ω, Theorem 1.1 and the assumption (1.16), together with the fact thatDis unbounded, yield that∂Ω and∂Dconsist ofmpairs of connected closed hypersurfaces being parallel to each other respectively. (WhenDis bounded,∂Dmay consist of two connected components being parallel to one component of∂Ω.) Hence, the proof runs similarly to that of Theorem 1.2, with the only difference that the components in each pair constituting ∂Ω∪∂D may be at different distance from one another. 2

We conclude this section with a more general version of Theorem A.

Theorem 2.1.LetΩ be a domain(not necessarily bounded)inR^N,N 2,satisfying the exterior sphere condition and suppose that ∂Ω is bounded. Let D be a domain withD⊂Ω, and let Γ be a connected component of∂D satisfying

dist(Γ, ∂Ω)=dist(∂D, ∂Ω). (2.4)

Suppose thatDsatisfies the interior cone condition onΓ. Assume that the solutionuof problem(1.1)–(1.3)satisfies (1.16).

Then∂Ω must be either a sphere or the union of two concentric spheres.

Proof. Because of the assumption (2.4), the proofs of Lemma 2.2 of [14] and Lemma 3.1 in [12] also work in this situation. Then, there exists a connected componentSof∂Ω such that bothΓ andSare analytic and these are parallel with distanceR=dist(Γ, ∂Ω);also,_N−1

j=1[_R¹−κ_j(x)]is constant forx∈S. SinceSis bounded, by applying Aleksandrov’s sphere theorem [1] to this equation, we see thatSandΓ are concentric spheres.

LetE be the annulus with∂E=S∪Γ. With the help of the analyticity ofu, by proceeding as in the proof of Theorem 3.1 in [14], we see that for anyi =j

−(xj−aj)∂u(x, t )

∂x_i +(xi−ai)∂u(x, t )

∂x_j =0 inΩ×(0,+∞),

where the pointa=(a1, . . . , a_N)∈R^Nis the center of the sphereS. Henceumust be radially symmetric with respect toa. 2

3. Short-time behavior of solutions of nonlinear diffusion equations

In this section, with the help of the theory of viscosity solutions, we prove our keystone result, Theorem 1.1. We begin with some preliminaries.

Lemma 3.1.Letw=φ(u),whereuis the solution of (1.10), (1.2)–(1.3). Forj=1,2,letwj solve the problem:

(w_j)_t=δ_jw_j inΩ×(0,+∞), (3.1)

w_j=φ(1) on∂Ω×(0,+∞), (3.2)

wj=0 onΩ× {0}. (3.3)

Then

w₁ww₂ inΩ×(0,+∞).

Proof. Sincew_t=φ(u)w,from (1.12) and (1.13) we have:

δ₁ww_tδ₂w inΩ×(0,+∞). (3.4)

Hence, by the comparison principle we get our claim. 2 Now, letΨ =Ψ (s)be the inverse function ofΦ. Then

s=Φ Ψ (s)

=

Ψ (s)

1

φ(ξ ) ξ dξ and

Ψ (s)=φ Ψ (s)

Ψ(s), (3.5)

by differentiating ins.

As in Freidlin and Wentzell [5], for 0< ε <1, define the functionu^ε=u^ε(x, t )by u^ε(x, t )=u(x, εt ) for(x, t )∈Ω×(0,+∞).

Thenu^εsatisfies u^ε_t =εφ

u^ε

inΩ×(0,+∞), u^ε=1 on∂Ω×(0,+∞), u^ε=0 onΩ× {0}.

Moreover, the functionv^ε=v^ε(x, t )defined by v^ε(x, t )= −εΦ

u^ε(x, t )

for(x, t )∈Ω×(0,+∞) is such thatu^ε=Ψ (−ε⁻¹v^ε)and, by (3.5), we have that

v_t^ε=εφv^ε−∇v^ε2 inΩ×(0,+∞), (3.6)

v^ε=0 on∂Ω×(0,+∞), (3.7)

v^ε= +∞ onΩ× {0}, (3.8)

whereφ=φ(Ψ (−ε⁻¹v^ε)).

Lemma 3.2.It holds that for(x, t )∈Ω×(0,+∞) δ₁

δ₂· 1

4td(x)²lim inf

ε→0⁺ v^ε(x, t )lim sup

ε→0⁺

v^ε(x, t )δ₂ δ₁· 1

4td(x)²,

where these limits asε→0⁺are uniform in every compact set contained inΩ×(0,+∞).

Proof. We observe that the following hold:

δ1sφ(s)δ2s fors0, (3.9)

−δ1logs−Φ(s)−δ2logs for 0< s1, (3.10)

e^s/δ1Ψ (s)e^s/δ2 for − ∞< s0. (3.11)

Letw^ε_j=w^ε_j(x, t ) (j=1,2)be the functions defined by w^ε_j(x, t )=wj(x, εt ),

where thew_j’s are defined in Lemma 3.1. With the aid of (3.9) and (3.10), it follows from Lemma 3.1 that

−εδ1log w^ε₂

δ₁

v^ε−εδ2log w^ε₁

δ₂

inΩ×(0,+∞).

By the result of Varadhan [21], we see that, asε→0⁺,the functions−εδ_jlogw^ε_jconverge to the function _4t¹d(x)² uniformly on every compact set contained inΩ×(0,+∞), since each scaled function_{φ (1)}¹ w_j(x, δ_j⁻¹t )solves problem (1.1)–(1.3). Our claim then follows at once. 2

The next lemma easily follows from Lemma 3.2.

Lemma 3.3.For any compact setKinΩ×(0,+∞), there exist three positive constantsε₀, c₁,andc₂(0< c₁c₂) depending onKsuch that

0< c₁v^εc₂ inK, for0< εε0.

The key point in the proof of Theorem 1.1 is to obtain the following gradient estimate which we shall prove at the end of this section.

Lemma 3.4.For any compact setKinΩ×(0,+∞), there exist two positive constantsε1(ε1ε0)andc3depending onK,such that

∇v^εc3 inK, for0< εε1.

Then, by combining Lemmas 3.3 and 3.4 with Gilding’s result [6], we obtain the following uniform Hölder esti- mate.

Lemma 3.5.For any compact setKinΩ×(0,+∞), there exist two positive constantsε₂(ε₂ε₁)andc₄depending onK,such that

v^ε(x, t )−v^ε(x, s)c4|t−s|¹² for any pair(x, t ), (x, s)∈K and for0< εε2.

Theorem 3.6.The following limit

εlim→0+v^ε(x, t )= 1 4td(x)²

holds uniformly on every compact set inΩ×(0,+∞).

Proof. Lemmas 3.3, 3.4, and 3.5 together with Ascoli–Arzelà’s theorem and the Cantor diagonal process yield a positive vanishing sequence of numbers ε_n and a continuous function v=v(x, t ) in Ω×(0,+∞)such that, as n→ ∞, thev^εn’s converge tovuniformly on every compact set contained inΩ×(0,+∞).Hence, by Lemma 3.2,

δ₁ δ₂· 1

4td(x)²v(x, t )δ₂ δ₁· 1

4td(x)² for(x, t )∈Ω×(0,+∞). (3.12)

Define a functionV =V (x, t )onR^N×(0,+∞)by V (x, t )=v(x, t ) ifx∈Ω,

0 ifx /∈Ω.

Since bothd²and its gradient vanish on∂Ω, (3.12) yields thatV is continuous onR^N×(0,+∞), differentiable at any point on∂Ω×(0,+∞), and that bothV and∇V vanish on∂Ω×(0,+∞). Also, (3.12) yields that limt→0⁺v(x, t )= +∞ifx∈Ω.

Therefore, by using the fact thatv^ε solves problem (3.6)–(3.8), with the help of Crandall, Ishii, and Lions [2], we see thatV is a viscosity solution of the following Cauchy problem:

V_t= −|∇V|² inR^N×(0,+∞),

V =0 on

R^N\Ω

× {0},

V = +∞ onΩ× {0}. (3.13)

Moreover, since a uniqueness result of Strömberg [20] tells us that the Hopf–Lax formula provides the unique viscosity solution of the Cauchy problem (3.13), we must have that, for any(x, t )∈R^N×(0,+∞),

V (x, t )=inf

ϕ(ξ )+|x−ξ|²

4t : ξ∈R^N

=(dist(x,R^N\Ω))²

4t ,

whereϕ=ϕ(ξ )is the lower semicontinuous initial data defined by ϕ(ξ )=

+∞ ifξ∈Ω, 0 ifξ /∈Ω.

By the uniqueness ofV, the whole sequence{v^ε}converges asε→0⁺, and we get our claim. 2

Proof of Theorem 1.1. The desired result follows by simply settingt=1 and thenε=t in Theorem 3.6. 2 By a simple argument, we can extend Theorem 1.1 to the important case of non-homogeneous boundary values.

Theorem 3.7.Letf=f (x)be a continuous function on∂Ω such that

0< b₁f (x)b₂ for allx∈∂Ω, (3.14)

for some positive constantsb₁andb₂.Letube the solution of problem(1.10),(1.15), (1.3).

Then, lim

t→0⁺−4t Φ u(x, t )

=d(x)² uniformly on every compact set inΩ.

Proof. Consider the unique bounded solutionsu^j=u^j(x, t ) (j=1,2)of the following initial–boundary value prob- lems:

u^j_t =φ u^j

inΩ×(0,+∞), u^j=b_j on∂Ω×(0,+∞), u^j=0 onΩ× {0}.

Then it follows from (3.14) and the comparison principle that

u¹uu² inΩ×(0,+∞). (3.15)

With the help of Theorem 1.1, we see that, as t→0⁺,the function −4t Φ(u^j(x, t ))converges to the function d(x)²uniformly on every compact set inΩ for eachj=1,2. Indeed, for eachj=1,2,we set

U=u^j

b_j, φ(s)˜ = 1

b_jφ(bjs) fors∈R, and Φ(s)˜ = s

1

φ˜(ξ )

ξ dξ fors >0.

Then it follows that

φ˜(s)=φ(bjs) fors∈R, Φ(s)˜ =Φ(bjs)−Φ(bj) fors >0, (3.16) and

U_t=φ(U )˜ inΩ×(0,+∞), (3.17)

U=1 on∂Ω×(0,+∞), (3.18)

U=0 onΩ× {0}. (3.19)

Thus, applying Theorem 1.1 toUyields that, ast→0⁺,the function−4tΦ(U (x, t ))˜ converges to the functiond(x)² uniformly on every compact set inΩ. Hence, with the aid of the second equality of (3.16), this means that, ast→0⁺, the function−4t Φ(u^j(x, t ))converges to the functiond(x)²uniformly on every compact set inΩ.

On the other hand, sinceΦis increasing ins >0, we have from (3.15) that

−4t Φ u¹

−4t Φ(u)−4t Φ u²

inΩ×(0,+∞),

which implies that, as t→0⁺, the function −4t Φ(u(x, t )) converges to the function d(x)² uniformly on every compact set inΩ. 2

Proof of Lemma 3.4. We use Bernstein’s technique (see [3,7,4,9]). Let r,τ andT be positive numbers such that τ <2τ < T andK⊂B_r(0)× [2τ, T].Takeζ∈C^∞(B_2r(0)×(τ, T])satisfying

0ζ1 and ζt0 inB2r(0)×(τ, T],

ζ =1 onB_r(0)× [2τ, T], and suppζ⊂B_2r(0)×(τ, T].

In the sequel of this proof, we will use the constantsε0, c1andc2of Lemma 3.3 relative to the compact setB2r(0)× [τ, T].

Consider the functionz=z(x, t )defined by

z=ζ²∇v^ε2−λv^ε, (3.20)

whereλ >0 is a constant to be determined later, and 0< εε₀.Suppose that(x₀, t₀)is a point inB_2r(0)×(τ, T] satisfying

ζ (x₀, t₀) >0 and max

B_2r(0)×[τ,T]z=z(x₀, t₀).

At(x₀, t₀)we then have

zt0, zx_i=0, and z0, (3.21)

and hence

0z_t−εφ Ψ

−ε⁻¹v^ε z.

The following inequality holds at(x₀, t₀)for some positive constantsA₁andA₂independent of(x₀, t₀)andε:

λ∇v^ε2A₁∇v^ε2+A₂ζ∇v^ε3−2ζ²∇v^ε2φΨv^ε−εφζ²∇²v^ε2. (3.22) It is a consequence of (3.21) and some lengthy calculations that, for the reader’s convenience, will be carried out in Appendix A.

Now, we want to bound the third and fourth summand on the right-hand side of (3.22). The bound for the latter summand,

−εφζ²∇²v^ε2−εδ1ζ²∇²v^ε2,

easily follows from (1.12). In order to bound the former one, we use the fact that φ∈C²(R)and Lemma 3.3, the algebraic inequality 2aba²+b²,and the key inequality

0< Ψ

−ε⁻¹v^ε

=Ψ (−ε⁻¹v^ε)

φ 1

δ1

e⁻

vε

εδ2 1

δ1

e⁻

c1

εδ2, (3.23)

which follows from (3.5), (1.12) and (3.11). With these three ingredients, we show that

−2ζ²∇v^ε2φΨv^ε 1 δ₁e⁻

c1 εδ2ζ²

A3∇v^ε4+∇²v^ε2 , at(x₀, t₀),for some positive constantA₃independent of(x₀, t₀)andε.

Set

M= max

B2r(0)×[τ,T]ζ∇v^ε, λ= M²+1 2(c2+1),

and chooseε_∗in(0, ε0]so small to obtain that A₃

δ₁e⁻

c1

εδ2 1

4(c2+1) and 1 δ₁e⁻

c1 εδ2 εδ1

for allε∈(0, ε_∗].Then, with these choices of constants, from (3.22) and the aforementioned bounds on the second- order derivatives ofv^ε,we have that

M²+1

4(c2+1)∇v^ε2A1∇v^ε2+A2M∇v^ε2 (3.24)

at(x0, t0),for anyε∈(0, ε_∗].

Thus, if∇v^ε(x₀, t₀) =0,from (3.24) we get M²+1

4(c2+1)A1+A2M,

which yields the desired gradient estimate at once. If∇v^ε(x0, t0)=0, instead, we use the definition (3.20) ofzto infer that

M²maxz+λmaxv^ελmaxv^ε M²+1

2(c2+1)c2M² 2 +1

2,

sincez(x0, t0)= −λv^ε(x0, t0) <0.Therefore,M1 and this completes the proof. 2

Remark.Lions, Souganidis, and Vázquez [9] consider the pressure equation for the porous medium equation:

(v_m)_t=(m−1)vmv_m+ |∇v_m|² form >1,

and consider the asymptotic behavior asm→1⁺. They get the interior gradient estimate forv_mindependent ofmby a technique similar to ours. We follow the outline of their proof but we use inequality (3.23) in order to overcome the difficulty caused byφ=φ(Ψ (−ε⁻¹v^ε))in Eq. (3.6).

4. On the Cauchy problem

LetΩ be a domain given in (1.9) and consider the unique bounded solutionu=u(x, t )of the following Cauchy problem:

u_t=φ(u) inR^N×(0,+∞), and u=χ_RN\Ω onR^N× {0}, (4.1)

whereχ_RN\Ω denotes the characteristic function of the setR^N\Ωandφsatisfies the assumptions (1.11)–(1.12). The purpose of this section is to prove the following result.

Theorem 4.1.Theorems1.1,1.2, and1.3also hold for the unique bounded solutionuof the Cauchy problem(4.1).

Let us start with two lemmas.

Lemma 4.2.There exist a smallδ >0and aC²-functionf=f (ξ )onRsatisfying φ(f )f

+1

2(ξ+2δ)f=0 and f<0 inR, and

1> f (−∞) > f (0) >0> f (+∞) >−∞. (4.2)

Proof. It suffices to show that there exists aC²-functionh=h(ξ )onRsatisfying φ(h)h

+1

2ξ h=0 and h<0 inR, and (4.3)

1> h(−∞) > h(0) >0> h(+∞) >−∞. (4.4)

Indeed, settingf (ξ )=h(ξ+2δ)for sufficiently smallδ >0 gives the desired solutionf.

The assumptions (1.11)–(1.12) guarantee existence and uniqueness,on the wholeR,of the solution(h, H )of the Cauchy problem for the system of ordinary differential equations

h= H

φ(h), H= −1 2ξ H

φ(h), and

h(0), H (0)

=(h0, H0) (4.5)

(obtained by lettingH=φ(h)hin (4.3)); hereh₀>0 andH₀<0 are given numbers. Also, by uniqueness we infer thatH <0 onRand henceh<0 onR.

Thus, with the help of (1.12), by integrating the second equation in (4.5), we have that H₀exp

−ξ² 4δ2

H (ξ )H₀exp

−ξ² 4δ1

<0 and

H₀ δ₁ exp

−ξ² 4δ2

h(ξ )H₀ δ₂ exp

−ξ² 4δ1

<0 forξ∈R; hence

h0+H₀ δ₁

ξ

0

exp

−η² 4δ2

dηh(ξ )h0+H₀ δ₂

ξ

0

exp

−η² 4δ1

dη, forξ >0,and

h₀+H₀ δ₂

ξ

0

exp

−η² 4δ1

dηh(ξ )h₀+H₀ δ₁

ξ

0

exp

−η² 4δ2

dη,

forξ <0.By lettingξ → +∞andξ→ −∞, respectively, we get h0+H₀

δ₁

π δ2h(+∞)h0+H₀ δ₂

π δ1,

h₀−H0

δ2

π δ₁h(−∞)h₀−H0

δ1

π δ₂.

Therefore, (4.4) is obtained by setting h₀= δ₁^3/2

2(δ₁^3/2+δ₂^3/2) and H₀= − δ1δ2

√π (δ^3/2₁ +δ₂^3/2) in the last two formulas. 2

Lemma 4.3.There exists a constantc₀>0satisfying c0u <1 on∂Ω×(0,1].

Proof. First of all, by the strong maximum principle

0< u <1 inR^N×(0,+∞). (4.6)

Consider the signed distance functiond^∗=d^∗(x)ofx∈R^Nto the boundary∂Ω defined by d^∗(x)=

dist(x, ∂Ω) ifx∈Ω,

−dist(x, ∂Ω) ifx /∈Ω. (4.7)

Since∂ΩisC²and compact, there exists a numberρ >0 such thatd^∗(x)isC²-smooth on a compact neighborhood N of the boundary∂Ω given by

N =

x∈R^N: −ρd^∗(x)ρ .

We set now w(x, t )=f

t⁻¹²d^∗(x)

for(x, t )∈R^N×(0,+∞). (4.8)

Then, it follows from a straightforward computation and the properties off that w_t−φ(w)= −1

tf

t⁻¹²d^∗

−δ+√

t φ(f )d^∗

inN ×(0,+∞).

Notice that if√

t <_δ ^δ

2max_N|d^∗|, thenw_t−φ(w) <0. Hence, since∂N is compact, in view of Lemma 4.2, (4.8), and (4.1), we observe that there exists a smallτ >0 satisfying

w_t−φ(w) <0=u_t−φ(u) inN×(0, τ],

wu on∂N×(0, τ],

wu onN× {0}.

Here, we note that u and w are regarded as continuous mappings from [0, τ] toL¹(N) and, by taking into ac- count (4.2), the initial condition is satisfied by their limits ast→0⁺inL¹(N).

Therefore it follows from the comparison principle thatwuinN ×(0, τ]. In particular, we have uf (0) (>0) on∂Ω×(0, τ].

Combining this with (4.6) completes the proof. 2

Proof of Theorem 4.1. Consider the unique bounded solutionsu^±=u^±(x, t )of the following initial–boundary value problems:

u^±_t =φ u^±

inΩ×(0,+∞), u⁺=1 and u⁻=c₀ on∂Ω×(0,+∞),

u^±=0 onΩ× {0}.

Then it follows from Lemma 4.3 and the comparison principle that

u⁻uu⁺ inΩ×(0,1]. (4.9)

By applying Theorem 3.7 tou^±, we have that, ast→0⁺,both functions−4t Φ(u^±(x, t ))converge to the function d(x)²uniformly on every compact set inΩ.

On the other hand, sinceΦis increasing ins >0, we have from (4.9) that

−4t Φ u⁻

−4t Φ(u)−4t Φ u⁺

inΩ×(0,1],

which implies that, as t →0⁺, the function −4t Φ(u(x, t )) converges to the function d(x)² uniformly on every compact set inΩ. This means that Theorem 1.1 also holds for the Cauchy problem (4.1).

Finally, proceeding as in Section 2, with the aid of the strong comparison principle for the Cauchy problem, we can easily show that Theorems 1.2 and 1.3 also hold for the Cauchy problem (4.1). 2

5. SphereSSS^N and hyperbolic spaceHHH^N

The purpose of this section is to show that similar results hold also for the heat flow in the sphereS^N and the hyperbolic spaceH^N withN2. In order to handleS^NandH^Ntogether, let us putM=S^NorM=H^N.

LetΩbe a domain inMwith boundedC²-smooth boundary∂Ω, and denote byLthe Laplace–Beltrami operator onM. Letu=u(x, t )be the unique bounded solution either of the following initial–boundary value problem for the heat flow:

u_t=Lu inΩ×(0,+∞), (5.1)

u=1 on∂Ω×(0,+∞), (5.2)

u=0 onΩ× {0}, (5.3)

or of the following Cauchy problem for the heat flow:

u_t=Lu inM×(0,+∞), and u=χ_M\Ω onM× {0}, (5.4) whereχ_M\Ω denotes the characteristic function of the setM\Ω.

Denote byd(x)=inf{d(x, y): y∈∂Ω}the geodesic distance betweenx and∂Ω, whered(x, y)is the geodesic distance between two pointsxandyinM. Then, with the aid of a result of Norris [15, Theorem 1.1, p. 82] concerning the short-time asymptotics of the heat kernel of Riemannian manifolds, we later prove

Theorem 5.1.Letube the solution either of problem(5.1)–(5.3)or of problem(5.4)inS^NorH^N. Then the function

−4tlogu(x, t )converges to the functiond(x)²ast→0⁺uniformly on every compact set inΩ. Theorem 5.1 yields the following symmetry results.

Theorem 5.2.Letube the solution either of problem(5.1)–(5.3)or of problem(5.4)inS^NorH^N.In the case ofS^N, assume thatΩ is contained in a hemisphere inS^N.

Then Theorems1.2and1.3hold forH^N and Theorem1.2holds forS^N in the sense that∂Ω must be a geodesic sphere inH^NorS^N,respectively.

Proof. By Theorem 5.1, each stationary isothermic surface of classC²inΩis then parallel to a connected component of∂Ωin the sense of the geodesic distance. The claims of our theorem can thus be proved by replacing the method of moving planes, used in Section 2 forR^N, by a straightforward adaptation of the method of moving closed and totally geodesic hypersurfaces forS^NorH^Ndeveloped by Kumaresan and Prajapat in [8]. 2

Proof of Theorem 5.1. Consider the signed distance functiond^∗=d^∗(x)ofx ∈Mgiven by the same definition as (4.7). Since ∂Ω isC²and compact, there exists a number ρ₀>0 such thatd^∗(x)isC²-smooth on a compact neighborhoodN of∂Ω given by

N =

x∈M: −ρ0d^∗(x)ρ0

. Set

N⁻=

x∈M:−ρ₀d^∗(x)0 (⊂N).

Letu⁻=u⁻(x, t )be the unique bounded solution of the following Cauchy problem:

u⁻_t =Lu⁻ inM×(0,+∞), and u⁻=χ_N− onM× {0}, (5.5) whereχ_N−denotes the characteristic function of the setN⁻.Moreover, for each 0< ρ < ρ0, we set

Nρ=

x∈M: −ρd^∗(x)ρ (⊂N).

For each 0< ρ < ρ₀, letu^ρ+=u^ρ+(x, t )be the unique bounded solution of the following Cauchy problem:

u^ρ_t⁺=Lu^ρ+ inM×(0,+∞), and u^ρ+=2χ_Nρ onM× {0}, (5.6) whereχ_Nρ denotes the characteristic function of the setNρ.

Letρ∈(0,min{1, ρ0})be arbitrarily small. Then, there exists a numbert_ρ>0 satisfying

u^ρ+>1 in∂Ω×(0, tρ]. (5.7)

Thus it follows from (5.7) and the comparison principle that for any(x, t )∈Ω×(0, tρ]

N⁻

p(t, x, y) dy=u⁻(x, t )u(x, t )u^ρ+(x, t )=2

Nρ

p(t, x, y) dy, (5.8)

wherep=p(t, x, y)denotes the heat kernel or the fundamental solution of the heat equation on the wholeM.