[lecture notes in mathematics] functional analytic methods for evolution equations volume 1855 || an...

An Introduction to Parabolic MovingBoundary Problems

Alessandra Lunardi

Dipartimento di Matematica, Universita di Parma, Via D’Azeglio 85/A, 43100Parma, [email protected]

1 Introduction

In these lectures I will give an overview on a class of free boundary evolutionproblems, such as the free boundary heat equation,

ut = ∆u, t > 0, x ∈ Ωt,

u = 0,∂u

∂ν= −1, t > 0, x ∈ ∂Ωt,

(1.1)

and its generalizations. The unknowns are the open sets Ωt ⊂ Rn for t > 0,

and the real valued function u, defined for t > 0 and x ∈ Ωt. The initial setΩ0 and the initial function u(0, ·) = u0 : Ω0 → R are given data.

While this type of problems are well understood in space dimension N = 1(see e.g. [27] and the review paper [24]) much less is known in the multidi-mensional case. For instance, uniqueness of the classical solution to the initialvalue problem for (1.1) is still an open problem.

I will describe some different approaches to (1.1), which lead to severalproperties of the solutions. Moreover I shall describe some special solutions,whose behavior is of great help in understanding the nature of the problem.

Many results for (1.1) can be extended to evolution problems of the type

ut(t, x) = Lu(t, x) + f(t, x, u(t, x), Du(t, x)), t ≥ 0, x ∈ Ωt,

u(t, x) = g0(t, x), t ≥ 0, x ∈ ∂Ωt,

∂u

∂ν(t, x) = g1(t, x), t ≥ 0, x ∈ ∂Ωt,

u(0, x) = u0(x), x ∈ Ω0.

(1.2)

where the boundary data g0, g1 satisfy the transversality condition

G. Da Prato et al.: LNM 1855, M. Iannelli et al. (Eds.), pp. 371–399, 2004.c© Springer-Verlag Berlin Heidelberg 2004

372 Alessandra Lunardi

∂g0

∂ν(0, x) = g1(0, x), x ∈ ∂Ω0. (1.3)

Again, the unknowns are the open sets Ωt ⊂ RN for t > 0, and the function u.

The data are: the second order elliptic operator L =∑Ni,j=1 aij(t, x)Dij , the

functions f : [0, T ]×Rn×R×R

n → R, g0, g1 : [0, T ]×Rn → R, the (possibly

unbounded) initial set Ω0 ⊂ Rn, and the initial function u0 : Ω0 → R.

However, all the substantial difficulties of (1.2) are already present in prob-lem (1.1). Therefore, to simplify notation and to concentrate on the difficultiesdue to the structure of the problem, we shall consider mainly problem (1.1).It is motivated by models in combustion theory ([8, 26]) for equidiffusionalpremixed flames, where u = λ(T ∗ − T ), T is the temperature, T ∗ is the tem-perature of the flame, and λ > 0 is a normalization factor. It can be seen as thehigh activation energy limit of the regularizing problems ut = ∆u − βε(u) in[0, T ]×R

N , where T > 0 is fixed and βε(s) = β1(s/ε)/ε is supported in a smallinterval [0, ε]. In [9], Caffarelli and Vazquez used this regularization to proveexistence of global weak solutions to (1.1) for C2 initial data (Ω0, u0), un-der suitable geometric assumptions on u0. Such solutions may not be unique,and enjoy some regularity properties: the free boundary is locally Lipschitzcontinuous, and u is 1/2-Holder continuous with respect to time, Lipschitzcontinuous with respect to the space variables. I shall describe the approachof Caffarelli and Vazquez in §4.

In §5 I shall describe a completely different approach, due to Baconneauand myself [2], which leads to existence, uniqueness, and time C∞–regularityfor t > 0 of local smooth (very smooth!) solutions (with u bounded, in thecase of unbounded domains) to problem (1.1), and more generally to problem(1.2). It consists in transforming problem (1.1) into a fully nonlinear parabolicproblem in the fixed domain Ω0 for an auxiliary unknown w, for which theusual techniques of fully nonlinear parabolic problems in Holder spaces (seee.g. [22, Ch. 8]) give a local existence and uniqueness result. Coming back to(1.1), for Ω0 with C3+α+k boundary and u0 ∈ C3+α+k(Ω0) we obtain a localsolution with ∂Ωt ∈ C2+α+k and u(t, ·) in C2+α+k(Ωt), k = 0, 1.

Both approaches lead to loss of regularity, in the sense that we are not ableto show that the regularity of the initial datum is preserved throughout theevolution. Although this is natural for global weak solutions, as the first exam-ple in section 2 shows, it not wholly satisfactory for local classical solutions.One could expect that in a small time interval the solution remains at leastas regular as the initial datum: this is what happens in other free boundaryproblems of parabolic type (see e.g. [11, 12] for boundary conditions of Stefantype), and in problem (1.2) for special initial data.

An interesting situation in which there is no loss of regularity, at least insmall time intervals, is the case of initial data near special smooth solutions,such as stationary solutions, self-similar solutions, travelling waves. In partic-ular, C2+α initial data near any smooth stationary solution (Ω,U) of (1.2),with bounded Ω, were considered in the paper [3], where we studied stabil-ity of smooth stationary solutions, establishing a linearized stability principle

An Introduction to Parabolic Moving Boundary Problems 373

for (1.2) in the time independent case. Since self-similar solutions to (1.1)become stationary solutions to a problem of the type (1.2) after a suitablechange of coordinates, we can also consider initial data for (1.1) near self-similar solutions. In the unbounded domain case, C2+α initial data near aplanar travelling wave solution of (1.1) have been considered in [4]. The waveturns out to be orbitally stable, but the discussion is not trivial, because indimension N ≥ 2 this is a very critical case of stability.

Another example without loss of regularity was considered by Andreucciand Gianni in [1], where a two-phase version of (1.2) was studied in a stripfor C2+α initial data far from special solutions but satisfying suitable mono-tonicity conditions: u0 is assumed to be strictly monotonic in the directionorthogonal to the strip. This allows to take u as new independent variable forsmall t, an old trick already used by Meirmanov in the Stefan problem andalso in a two-dimensional version of problem (1.2), see [23].

Together with loss of regularity, the other big question about problem(1.2) is uniqueness of the classical solution. Indeed, the uniqueness resultsavailable up to now concern only particular situations, such as radially sym-metric solutions of (1.1), studied in [13], and solutions in cylinders or strips,for initial data which are monotonic in the direction of the axis of the cylinder(see [23] in dimension N = 2 and [17] in any dimension), or in the directionorthogonal to the axis of the strip (see [1], for the two-phase case). The abovementioned papers [3, 4] give also uniqueness results in the parabolic Holderspace C1+α/2,2+α, but only for solutions close to the special solutions consid-ered. The paper [2] gives uniqueness of very regular solutions, in the spaceC3/2+α/2,3+α.

Therefore, the problem of wellposedness for general initial data in dimen-sion bigger than 1 still remains open, even in the simplest case (1.1).

2 The One Dimensional Case

Let us consider a one dimensional version of problem (1.1). To avoid techni-cal difficulties we consider the simplest situation as far as local existence isconcerned, i.e. the case where the initial set and the unknown sets Ωt are lefthalflines,

ut(t, x) = uxx(t, x), t ≥ 0, x < s(t),

u(t, s(t)) = 0, ux(t, s(t)) = −1, t ≥ 0.(2.1)

The initial conditions are

s(0) = s0, u(0, x) = u0(x), x ≤ s0, (2.2)

with u0 smooth enough (u0 ∈ C2+θ((−∞, s0]), 0 < θ < 1)), satisfying thecompatibility conditions


(a) u0(s0) = 0, (b) u′0(s0) = −1. (2.3)

The trivial change of coordinates

ξ = x− s(t),

reduces (2.1) to a fixed boundary problem in (−∞, 0]. Let u denote the func-tion u in the new variables, i.e. u(t, ξ) = u(t, x). Then u has to satisfy

ut(t, ξ)− s(t)uξ(t, ξ) = uξξ(t, ξ), t ≥ 0, ξ ≤ 0,u(t, 0) = 0, uξ(t, 0) = −1, t ≥ 0,u(0, ξ) = u0(ξ) = u0(ξ + s0), ξ ≤ 0.

(2.4)

This system may be immediately decoupled using the boundary conditions:the first boundary condition u(t, s(t)) ≡ 0 gives ut(t, s(t))+ s(t)uξ(t, s(t)) ≡ 0,and since uξ(t, s(t)) = −1 = 0 we get

s(t) = ut(t, s(t)) = uξξ(t, s(t)) = uξξ(t, 0). (2.5)

Replacing in (2.4) we get

ut(t, ξ) = uξξ(t, 0)uξ(t, ξ) + uξξ(t, ξ).

The final problem for u is therefore

ut(t, ξ) = uξξ(t, 0)uξ(t, ξ) + uξξ(t, ξ), t ≥ 0, ξ ≤ 0,uξ(t, 0) = −1, t ≥ 0,u(t,−∞) = 0, t ≥ 0,u(0, ξ) = u0(ξ) = u0(ξ + s0), ξ ≤ 0.

(2.6)

I did not write the condition u(t, 0) = 0 in (2.6) because it is containedin the other conditions. Indeed, writing the differential equation ut(t, ξ) =uξξ(t, 0)uξ(t, ξ)+uξξ(t, ξ) at ξ = 0 and replacing uξ(t, 0) = −1 we get ut(t, 0) =0 and hence u(t, 0) = constant; the constant is 0 because the initial datum u0

satisfies the compatibility condition u0(s0) = 0.(2.6) is a nonlinear problem with a nonstandard nonlinearity, which de-

pends on the spatial second order derivative of the solution, evaluated at ξ = 0.However, it can be solved, at least locally in time, with the usual techniques ofnonlinear parabolic problems. Here we follow the abstract evolution equationstechnique, but other approaches work as well.

First, we transform easily problem (2.6) into a homogeneous at the bound-ary problem. This is made just to adapt our problem to the general theoryof evolution equations in Banach spaces. It is sufficient to introduce the newunknown

w(t, ξ) = u(t, ξ) + eξ, t ≥ 0, ξ ≤ 0.

Since ut = wt, uξ = wξ − eξ, uξξ = wξξ − eξ, then w has to solve


wt(t, ξ) = (wξξ(t, 0)− 1)(wξ(t, ξ)− eξ) + wξξ(t, ξ)− eξ, t ≥ 0, ξ ≤ 0,wξ(t, 0) = 0, t ≥ 0,w(0, ξ) = w0(ξ) = u0(ξ + s0) + eξ, ξ ≤ 0.

(2.7)It is convenient to set problem (2.7) in the space

X = BUC((−∞, 0])

of the uniformly continuous and bounded functions, endowed with the supnorm. The nonlinear function in the right hand side of the equation is welldefined in the subspace

D = ϕ ∈ BUC2((−∞, 0]) : ϕ′(0) = 0.

D is endowed with the C2 norm, and it is dense in X .The function

F : D → X, F (ϕ)(ξ) = (ϕ′′(0)− 1)(ϕ′(ξ)− eξ) + ϕ′′(ξ)− eξ, ξ ≤ 0

is smooth, and for each w0 ∈ D we have

F ′(w0)(ψ)(ξ) = (w′′0 (0)− 1)ψ′(ξ)+ψ′′(0)(w′

0(ξ)− eξ)+ψ′′(ξ), ψ ∈ D, ξ ≤ 0.

Therefore, F ′(w0) is an elliptic operator (Aψ = ψ′′ + (w′′0 (0) − 1)ψ′) plus

a perturbation (Bψ = ψ′′(0)(w′0(ξ) − eξ)) defined in D and not defined in

any intermediate space between D and X . The operator A : D(A) = D →generates an analytic semigroup in X , and its graph norm is equivalent to thenorm of D. Moreover B : D → X is a compact operator because its range isone dimensional. It follows (see e.g. [10]) that A + B : D → X generates ananalytic semigroup in X .

Now we may apply a local existence and uniqueness theorem for evolutionequations in Banach spaces.

Theorem 2.1. Let X, D be Banach spaces, with D continuously embedded inX. Let O = ∅ be an open set in D, let F : O → X be a C2 function. Assumethat for each w ∈ O the operator A = F ′(w) : D(A) = D → X generates ananalytic semigroup in X.

Fix α ∈ (0, 1). For every w0 ∈ D such that F (w0) ∈ DA(α,∞) there existδ > 0 and a unique solution w ∈ Cα([0, δ];D) ∩ C1+α([0, δ];X) of problem

w′(t) = F (w(t)), 0 ≤ t ≤ δ,

w(0) = w0,(2.8)

Moreover u′(t) ∈ DA(α,∞) for each t, and t → ‖u′(t)‖DA(α,∞) is bounded in[0, δ].


We recall that DA(α,∞) = (X,D)α,∞ is the real interpolation space be-tween X and D defined by

DA(α,∞) = w ∈ X : [w]α = sup0<t<1

t−α‖etAw − x‖ < ∞,

‖w‖DA(α,∞) = [w]α + ‖w‖.

Here ‖ · ‖ denotes the norm in X .The proof of theorem 2.1 relies on maximal regularity results in Holder

spaces for linear evolution equations in Banach spaces, due to [25].

Theorem 2.2. Let A : D(A) → X be the generator of an analytic semigroupetA in X. Fix α ∈ (0, 1) and T > 0. Let f ∈ Cα([0, δ], X), with 0 < δ ≤ T ,and x ∈ D(A) be such that Ax + f(0) ∈ DA(α,∞).

Then the linear problem

u′(t) = Au(t) + f(t), 0 ≤ t ≤ δ,

u(0) = x,(2.9)

has a unique solution u ∈ C1+α([0, δ], X) ∩ Cα([0, δ], D(A)). Moreover, u′ isbounded with values in DA(α,∞), and there exists C such that

‖u‖C1+α([0,δ];X) + ‖Au‖Cα([0,δ];X) + sup0≤t≤δ ‖u′(t)‖DA(α,∞)

≤ C(‖f‖Cα([0,δ];X) + ‖x‖D(A) + ‖Ax + f(0)‖DA(α,∞)).(2.10)

The constant C is independent of f , x, and δ.

Once theorem 2.2 is established, it is not hard to prove theorem 2.1 by a fixedpoint argument. It is convenient to look for the local solution of problem (2.8)as a fixed point of the operator Γ in the set Y = w ∈ Cα([0, δ];D) : w(0) =w0, ‖w − w0‖Cα([0,δ];D) ≤ r defined by Γw = u = the solution to problem

u′(t) = Au(t) + F (w(t)), 0 ≤ t ≤ δ,

u(0) = w0.

The operator Γ maps Y into itself and it is a 1/2-contraction in Y providedδ and r are suitably chosen (roughly: r large and δ small). For the details see[21].

To apply theorem 2.1 we need to know Df ′(w0)(α,∞). It coincides withD∆(α,∞) where ∆ : D → X is the realization of the second order derivativein X with Neumann boundary condition. For α < 1/2 we have (see e.g. [22,thm. 3.1.30])

Df ′(w0)(α,∞) = C2α((−∞, 0]).


In our case w0 = u0(ξ+s0)+eξ ∈ C2+θ((−∞, 0]), and w′0(0) = 0. Therefore

w0 ∈ D and f(w0) ∈ Df ′(w0)(θ/2,∞). We may apply theorem 2.1 with α =θ/2, which gives local existence and uniqueness of a solution w such thatt → w(t, ·) ∈ C1+θ/2([0, δ];BUC(−∞, 0]) ∩ Cθ/2([0, δ];D), and t → wt(t, ·)is bounded with values in C2+θ((−∞, 0]). This implies that w belongs to theparabolic Holder space C1+θ/2,2+θ([0, δ]× (−∞, 0]).

Now we come back to the original problem. Recall that s = uξξ(t, 0) =wξξ(t, 0)− 1, so that

s(t) =∫ t

0

wξξ(σ, 0)dσ − t + s0,

u(t, ξ) = u(t, ξ − s(t)) = w(t, ξ − s(t))− eξ−s(t), ξ ≤ s(t).

Since w satisfies (2.7), then the couple (s, u) satisfies (2.1), with the onlyexception that the condition u(t, s(t)) = 0 is replaced by d/dt u(t, s(t)) = 0,i.e. u(t, s(t)) = constant. The constant to be 0 because of the compatibilitycondition u0(s0) = 0. So we get also u(t, s(t)) = u(0, s0) = 0 for each t ∈ [0, δ].

We have proved the following result.

Proposition 2.3. Let s0 ∈ R and let u0 ∈ C2+θ((−∞, s0]) satisfy the com-patibility conditions (2.3). Then there exists δ > 0 such that problem (2.1) hasa unique solution (s, u) such that s ∈ C1+θ/2([0, δ]) and u ∈ C1+θ/2,2+θ(Ω),where Ω = (t, x) : 0 < t < δ, x < s(t).

For another approach see [14].

Now I try to explain why the same approach does not work if the spacedimension is bigger than 1. Let us represent R

N as R × RN−1, and let us

write any element of RN as (x, y), with x ∈ R and y ∈ R

N−1. A naturalgeneralization of problem (2.1) is to look for Ωt of the type Ωt = (x, y) ∈RN : x < s(t, y) where s is an unknown function. Therefore, the free bound-

ary at time t is the graph of the function s(t, ·), the normal derivative is−(Dxu,Dyu) = ((1 + |Dys|2)−1/2,−Dys(1 + |Dys|2)−1/2) and problem (1.1)is rewritten as

ut = ∆u, t > 0, x ≤ s(t, y), y ∈ RN−1,

u(t, s(t, y), y) = 0,

ux(t, s(t, y), y) = − 1√1 + |Dys|2

t > 0, y ∈ RN−1.

(2.11)

The initial conditions are now

s(0, y) = s0(y), y ∈ RN−1, u(0, x, y) = u0(x, y), x ≤ s0(y), y ∈ R

N−1,(2.12)

where s0, u0 are smooth enough and satisfy the compatibility conditions


(a) u0(s0(y), y) = 0, (b) Dxu0(s0(y), y) = −1/√

1 + |Dys0(y)|2. (2.13)

The change of coordinates which fixes the boundary is again

ξ = x− s(t, y), (2.14)

and it reduces (2.11) to a fixed boundary problem in (−∞, 0]× RN−1. Let u

denote the function u in the new variables, i.e. u(t, ξ, y) = u(t, x, y). Then uhas to satisfy

ut −Dtsuξ = ∆u− 2〈Dyuξ, Dys〉+ uξξ|Dys|2 − uξ∆ys, ξ ≤ 0, y ∈ RN−1,

u(t, 0, y) = 0, uξ(t, 0, y) = − 1√1 + |Dys|2

, t ≥ 0, y ∈ RN−1,

u(0, ξ, y) = u0(ξ, y) = u0(ξ + s0(y), y), ξ ≤ 0, y ∈ RN−1.

(2.15)The identity u(t, s(t, y), y) ≡ 0 may be still used to get an expres-

sion for st: differentiating with respect to time we get ut(t, s(t, y), y) +ux(t, s(t, y), y)st(t, y) ≡ 0, so that st(t, y) =−ut(t, s(t, y), y)/ux(t, s(t, y), y) =−∆u(t, s(t, y), y)/ux(t, s(t, y), y). Rewriting in terms of u we get

st = −∆u− 2〈Dyuξ, Dys〉+ uξξ|Dys|2 − uξ∆ys

uξ,

where all the derivatives of u are evaluated at (t, 0, y). Recalling thatuξ(t, 0, y) = −1/

√1 + |Dys(t, y)|2, we get

st = ∆ys− 2〈D2ysDs,Ds〉

1 + |Dys|2+ (∆u(t, 0, y) + uξξ(t, 0, y)|Dys|2)

√1 + |Dys|2.

This expression for st may be replaced in (2.15), to get a complicated systemfor (u, s),

ut = uξ

(∆ys− 2

〈D2ysDs,Ds〉

1 + |Dys|2+

+(∆u(t, 0, y) + uξξ(t, 0, y)|Dys|2)√

1 + |Dys|2)

+

+∆u− 2〈DyuξDys〉+ uξξ|Dys|2 − uξt > 0, ξ ≤ 0,y ∈ R

N−1,

st = ∆ys− 2〈D2ysDs,Ds〉

1 + |Dys|2+

+(∆u(t, 0, y) + uξξ(t, 0, y)|Dys|2)√

1 + |Dys|2 t > 0,

u(t, 0, y) = 0, uξ(t, 0, y) = − 1√1 + |Dys|2

t > 0, y ∈ RN−1,

u(0, ξ, y) = u0(ξ, y) ξ ≤ 0, y ∈ RN−1.


Note that in dimension N = 1 all the derivatives with respect to y disappear,and the system reduces to (2.5)–(2.6). For N > 1 local solvability of thesystem, even for special initial data, is not clear.

3 Basic Examples

3.1 Self-similar Solutions in Bounded Domains

We give an example in which the domain Ωt = B(0, r(t)) shrinks to a singlepoint in finite time T . We look for radial self-similar solutions of the form

u(t, x) = (T − t)αf(η) , η =|x|

(T − t)β, Ωt = |x| < r(T − t)β, 0 ≤ t < T

(3.1)with f : [0,+∞) → R, r > 0. Replacing in (1.1), we see that α = β = 1/2,and that the function g(x) = f(|x|) has to be an eigenfunction of an Ornstein-Uhlenbeck type operator in the ball B(0, r),

∆g − 12〈x,Dg(x)〉 +

12g = 0,

with two boundary conditions, g = 0, ∂g/∂ν = −1. This amounts to say that

f ′′(η) +N − 1

ηf ′(η) +

12f(η) =

12ηf ′(η) for 0 < η ≤ r ;

f ′(0) = f(r) = 0 , f ′(r) = −1 .

(3.2)

Problem (3.2) is over determined. However, the following proposition holds[9].

Proposition 3.1. There exist a unique r > 0 and a unique C2 function f :[0, r] → R satisfying (3.2) and such that f(η) > 0 for 0 ≤ η < r. Moreover fis analytic.

To build the solution, first we find a solution g of the differential equa-tion with the initial condition g′(0) = 0. All the solutions of the differentialequation are linear combinations of two linearly independent solutions, one ofwhich is singular at η = 0 and the other one may be written as a power series,g(η) = c

∑∞n=0 anη

n. Replacing in (3.2) we get

a0 = 1,

a2n = − (2n− 1)!!4n+1(n + 1)!N(2 + N)(4 + N) · . . . · (2n + N)

, a2n+1 = 0, n ∈ N.

Since g is strictly decreasing for η > 0 and goes to −∞ as η → +∞, thereexists a unique r > 0 such that g(r) = 0. Now we choose c = −1/g′(r). Thefunction f(η) = cg(η) satisfies all requirements of proposition 3.1.


The couple (Ωt, u) defined in (3.1) with α = β = 1/2, and f given byproposition 3.1, is a classical solution to (1.1) in the time interval [0, T ). Notethat u is smooth for t < T ; in particular Ω0 is the ball centered at 0 withradius r

√T and u(0, ·) ∈ C∞(Ω0), but at t = T the free boundary defined by

|x| = r√

T − t is only 1/2-Holder continuous with respect to time.In [15] and in [13] it is shown that all radial solutions to (1.1) shrink to

the origin in finite time and that they are asymptotically selfsimilar.

3.2 Travelling Waves in Unbounded Domains

Another important class of solutions are the travelling waves, with u(t, x) =f(x + cte), e being some unit vector, c = 0.

After a rotation we may assume that e = (1, 0, . . . , 0). A planar travellingwave is a solution such that Ωt is a halfspace delimited by the hyperplanex1 = −ct + c0, and u(t, x) = f(x1 + ct − c0) where f(ξ) is defined in thehalfline (−∞, c0] or [c0,+∞). Since the problem is invariant with respectto the change of variables x1 → −x1, we may choose to consider only lefthalflines (−∞, c0]. Moreover, having fixed c and f , such solutions differ onlyby a translation in the x1 variable. So we can look for solutions with c0 = 0.Then we find f(ξ) = −(ecξ − 1)/c.

So we have an example of smooth solutions with existence in the large.Stability of planar travelling waves is of physical interest. However, while

stability under one-dimensional perturbations is relatively easy (see e.g. [14,6]), stability for genuinely multidimensional perturbations comes out to be arather complicated problem. See [4] for the heat equation, and [7, 5, 19, 20]for other free boundary problems of this type.

4 Weak Solutions

We describe here the approach of Caffarelli and Vazquez which provides weaksolutions to (1.1), under suitable assumptions on the initial data. They lookfor an open set Ω ⊂ (0, T )× R

N with Lipschitz continuous lateral boundaryΓ and for a continuous nonnegative function u, vanishing on Γ , such that foreach test function φ ∈ C∞

0 ([0, T )× RN ) it holds

∫ ∫

Ω

u(φt + ∆φ)dt dx +∫

Ω0

uoφdx =∫

Γ

φ cosαdΣ, (4.1)

where α is the angle between the exterior normal vector to Γ and the hori-zontal hyperplane t = constant. Moreover, the free boundary should start atΓ0 = ∂Ω0, i.e. the section Γt = x : (t, x) ∈ Γ should converge to Γ0 in asuitable sense as t → 0.

If (Ωt, u) is a regular solution to (1.1) in the interval [0, T ], setting Ω =(t, x) : 0 < t < T, x ∈ Ωt, multiplying the equation by φ and integratingover Ω we obtain (4.1).


The assumptions on the initial data (Ω0, u0) are the following: Ω0 is anopen set with uniformly C2 boundary Γ0, u0 ∈ C2(Ω0) has positive values inΩ0 and vanishes in Γ0; moreover |Du0| ≤ 1 and there exists a number λ > 0such that

∆u0 + λ|Du0| ≤ 0.

The idea to construct such weak solutions is to follow backwards the pro-cedure used by the physicists to obtain the free boundary problem (1.1) asa model in combustion theory. Problem (1.1) is approximated by a family ofproblems in [0, T ]× R

N ,

Dtuε = ∆uε − βε(uε), 0 < t ≤ T, x ∈ RN ,

uε(0, x) = uε0(x), x ∈ RN ,

(4.2)

where βε(u) = 1εβ(uε

), and β : R → R is a nonnegative regular function sup-

ported in [0, 1], increasing in (0, 1/2) and decreasing in (1/2, 1), with integralequal to 1/2. uε0 is suitable smooth approximation of the initial datum u0

(more precisely, it is an approximation by convolution of the extension of u0

which vanishes outside Ω0).The general theory of parabolic problems gives existence and uniqueness

of a global bounded solution to (4.2).Then the first step is to prove that the family uε : ε > 0 is bounded

in the space C1/2,1([0, T ] × RN ), and this is done using classical techniques

of parabolic equations, under the only assumption that u0 is a nonnegativefunction with bounded gradient. It follows that there is a sequence εn → 0such that uεn converges uniformly on each compact subset of [0, T ]× R

N toa continuous function u, which in addition belongs to C1/2,1([0, T ]× R

N ).The candidate to be a weak solution is the couple (Ω, u) where Ω =

(t, x) ∈ (0, T )×RN : u(t, x) > 0. Note that if K is a compact set contained

in Ω, then there is C > 0 such that for ε small enough we have uε ≥ C; henceβε(uε) = 0 for ε small and u is a weak solution to the heat equation in K, sothat it is smooth.

The second step is to pass to the limit in the obvious equality∫ T

0

∫

RN

uε(φt + ∆φ)dt dx +∫

RN

uε0φdx =∫ T

0

∫

RN

βε(uε)φdt dx (4.3)

for every test function φ. The convergence of the left hand side is immediate.Concerning the right hand side, since βε(uε) is bounded in L1

loc, along a sub-sequence it converges to

∫ T0

∫RN φdµ where µ is some nonnegative measure.

So we have∫ ∫

Ω

u(φt + ∆φ)dt dx +∫

Ω0

u0φdx =∫ T

0

∫

RN

φdµ.

The measure µ has to be concentrated on the free boundary Γ . Indeed, itcannot be supported in Ω because there we have ut = ∆u. It cannot be


supported in the interior of N = (t, x) : u(t, x) = 0 because if φ is a testfunction with support in N then (4.3) becomes

∫ T

0

∫

RN

uε(φt + ∆φ)dt dx =∫ T

0

∫

RN

βε(uε)φdt dx

and the left hand side goes to 0 as ε → 0. It cannot be supported in (0, x) :x ∈ R

N because of (4.3) again.The third step is to prove that Γ is the graph of a Lipschitz continuous

function t = θ(x), defined for x ∈ Ω0, and that dµ = cosαdΣ where dΣ is thesurface measure on Γ . This is not trivial at all, and requires all the furtherassumptions on (Ω0, u0) listed above.

The last step consists to prove that at each point (x0, t0) ∈ Γ where thefree boundary is a differentiable surface and the exterior normal is not directedalong the time axis, the boundary condition ∂u

∂ν = −1 holds.An interesting feature of this approach is a comparison principle between

the self-similar solutions described in §3.1 and any weak solution obtainedby this procedure. Let (B(0, R), U) be the initial datum of any self-similarsolution obtained in §3.1, and let (Ω0, u0) be the initial datum of a weaksolution obtained by the procedure of this section, such that Ω0 ⊂ B(0, R)and u0(x) ≤ U(x) for each x ∈ Ω0. Then for each t the weak solution u staysbelow the self-similar solution (to be more precise, below the extension of theselfsimilar solution which vanishes outside the ball B(0, r(t))). This impliesthat Ωt extinguishes in finite time.

5 The Fully Nonlinear Equations Approach

We assume that Ω0 is an open set with uniformly C3+α boundary, and thatu0 ∈ C3+α(Ω0). Since we look for a regular solution up to t = 0, we assumealso that u0 satisfies the compatibility conditions

u0 = 0,∂u0

∂ν= −1 at ∂Ω0.

At the end of section 2 problem (1.1) was transformed to a fixed boundaryproblem in the special case where Ω0 = (x, y) ∈ R × R

N−1 : x < s0(y) isthe left hand side of the graph of a regular function s0, and we look forΩt of the same type, Ωt = (x, y) ∈ R × R

N−1 : x < s(t, y). In thatcase, the change of coordinates which fixes the boundary is trivial, beingjust (x, y) → (x− s(t, y), y). It transforms Ωt into the halfspace Ω = (ξ, y) :ξ < 0, y ∈ R

N−1. Then ∂Ωt may be seen as the range of a function definedin the boundary of Ω, (0, y) → (s(t, y), y).

Of course this is not possible for general Ω0, in particular when Ω0 isbounded. But we can do something similar.

We look for ∂Ωt as the range of a function ξ → ξ + s(t, ξ)ν(ξ), defined forξ ∈ ∂Ω0, where ν(ξ) is the exterior unit normal vector to ∂Ω0 at ξ, and the


unknown s(t, ξ) is the signed distance of the point x = ξ + s(t, ξ)ν(ξ) from∂Ωt.

By a natural change of coordinates we transform Ωt into Ω0, at least forsmall time. The change of coordinates is defined by ξ → ξ + s(t, ξ)ν(ξ) forξ ∈ ∂Ω0, and it is extended smoothly to a C2+α diffeomorphism ξ → ξ+Φ(t, ξ)in the whole of Ω0.

More precisely, let N be a neighborhood of ∂Ω0 such that each x ∈ Nmay be written in a unique way as x = ξ+sν(ξ) with ξ ∈ ∂Ω0 and s ∈ R. Forevery ξ ∈ N we denote by ξ′ the point of ∂Ω0 closest to ξ. For every ξ ∈ R

N

we set

ν(ξ) =

θ(ξ)ν(ξ′) if ξ ∈ N ,

0 otherwise,(5.1)

where θ is a C∞ function with support contained in N , such that θ ≡ 1 in aneighborhood of ∂Ω0, θ ≡ 0 outside a bigger neighborhood of ∂Ω0.

Then we set

Φ(t, ξ) =

θ(ξ)s(t, ξ′)ν(ξ′) if ξ ∈ N ,

0 otherwise.(5.2)

For t small, the mapping

ξ → x(t, ξ) = ξ + Φ(t, ξ), ξ ∈ RN , (5.3)

is a C2+α diffeomorphism from RN to itself, which maps Ω0 onto Ωt, and will

be used to change coordinates. At t = 0, Φ(0, ·) ≡ 0, so that the diffeomor-phism is the identity. Moreover for any t, Φ(t, ·) vanishes in R

N \ N , so thatthe diffeomorphism differs from the identity only in a small neighborhood of∂Ω0.

Note that if ∂Ω0 has a straight part Γ , up to translations and rotationsthe above change of coordinates coincides with (2.14) near Γ . Indeed, after atranslation and a rotation, we may assume that Γ = 0 × U , where U is anopen set in R

N−1, and that Ω0 lies locally in the left hand side of Γ . The signeddistance of ξ = (ξ1, . . . , ξN ) from Γ is just the first coordinate ξ1, Φ(t, ξ) =s(t, ξ2, . . . , ξN )e1, so that ξ + Φ(t, ξ) = (ξ1 + s(t, ξ2, . . . , ξN ), ξ2, . . . , ξN ) and(5.3) coincides with (2.14) near Γ .

Denoting by u the unknown u expressed in the new coordinates (t, ξ), i.e.

u(t, ξ) = u(t, ξ + Φ(t, ξ)), t ≥ 0, ξ ∈ Ω0, (5.4)

we get a fixed boundary system for (s, u), in the domain [0, δ]×Ω0 with smallδ,

ut − 〈Du, (I +tDΦ)−1Φt〉 = Lu, t ≥ 0, ξ ∈ Ω0, (5.5)

where [DΦ]ij = ∂Φi

∂ξj(t, ξ), the superscript t denotes the transposed matrix,

and L is the new expression of the laplacian, i.e.


L =N∑

i,j,h=1

∂ξj∂xi

∂ξh∂xi

Djh +N∑

i,j=1

∂2ξj∂x2i

Dj .

Here we denote by ξ = ξ(t, x) the inverse of (5.3).The main step is now the introduction of a new unknown w defined by

u(t, ξ) = u0(ξ) + 〈Du0(ξ), Φ(t, ξ)〉 + w(t, ξ). (5.6)

At t = 0 we have u(0, ξ) = u0(ξ), Φ(0, ·) ≡ 0, so that

w(0, ξ) = 0, ξ ∈ Ω0. (5.7)

(5.6) allows to decouple the system using the boundary condition u = 0 at∂Ωt, which gives

s(t, ξ) = w(t, ξ), t ≥ 0, ξ ∈ ∂Ω0, (5.8)

so thatΦ(t, ξ) = w(t, ξ′)ν(ξ), t ≥ 0, ξ ∈ Ω0. (5.9)

Strictly speaking, the above expression is incorrect because ξ′ is defined onlyfor ξ ∈ N . However, since ν(ξ) vanishes outside N , for any f defined in ∂Ω0

we keep the expression f(ξ′)ν(ξ) to denote the function which coincides withf(ξ′)ν(ξ) in N and with 0 outside N .

Replacing (5.9) in (5.5) we get

wt = F1(ξ, w,Dw,D2w) + F2(ξ, w,Dw)st, t ≥ 0, ξ ∈ Ω0, (5.10)

where F1, F2 are obtained respectively from

L(u0 + 〈Du0, Φ〉+ w) (5.11)

and from−〈Du0 − (I + DΦ)−1D(u0 + 〈Du0, Φ〉+ w), ν〉 (5.12)

replacing Φ = w(t, ξ′)ν(ξ).Equation (5.10) still contains st; to eliminate it we use again the identity

s = w at the boundary which gives st = wt. Replacing in (5.10) for ξ ∈ ∂Ω0

we get

st(1−F2(ξ, w,Dw)) = F1(ξ, w,Dw,D2w), t ≥ 0, ξ ∈ ∂Ω0.

At t = 0 we have w ≡ 0, and F2 vanishes at (ξ, 0, 0), so that, at least for tsmall, F2(·, w(t, ·), Dw(t, ·)) is different from 1 and it is possible to get st asa function of w,

st = F3(ξ, w,Dw,D2w) =F1(ξ, w,Dw,D2w)1−F2(ξ, w,Dw)

, t ≥ 0, ξ ∈ ∂Ω0, (5.13)

which, replaced in (5.10), gives the final equation for w,


wt = F(w)(ξ), t ≥ 0, ξ ∈ Ω0, (5.14)

where

F(w)(ξ) = F1(ξ, w,Dw,D2w) + F2(ξ, w,Dw)F3(ξ, w,Dw,D2w). (5.15)

Although the explicit expression of F is rather complicated, we may noteimmediately that for ξ in Ω0 \ N , that is far from the boundary, we have

F(w)(ξ) = ∆w + ∆u0.

This is because our change of coordinates reduces to the identity far from theboundary. Moreover, even near the boundary we have

F(0)(ξ) = ∆u0(ξ). (5.16)

For ξ near the boundary, the function F(v) is defined for v ∈ C2(Ω0) withsmall C1 norm. More precisely, it is defined for v ∈ C2(Ω0) such that 1 −F2(·, v(·), Dv(·)) = 0. From formulas (5.8), (5.11), (5.12), (5.13) we see thatF(v)(ξ) depends smoothly on v, Dv, D2v and their traces at the boundary.Therefore the function v → F(v) is continuously differentiable from v ∈C2(Ω0) : ‖v‖C1 ≤ r to C(Ω0), and from v ∈ C2+α(Ω0) : ‖v‖C1 ≤ r toCα(Ω0).

In the next lemma we describe the linear part of F with respect to v nearv = 0.

Lemma 5.1. Fv(0) is the sum of the Laplacian plus a nonlocal differentialoperator of order 1.

Proof — It is clear from the expression of F that the linear part of v → F(v)near v = 0 is a (possibly nonlocal) second order linear differential operator A.To identify its principal part it is sufficient to consider F1. Indeed, the prod-uct F2(ξ, v,Dv)F3(ξ, v,Dv,D2v) does not contribute to the principal part,because F2 does not depend on the second order derivatives of v, and it van-ishes at v = 0.

Using (5.11) we see that only the term L(u0 + 〈Du0, Φ〉 + w) in F1 con-tributes to the principal part of A. Its explicit expression is

N∑

i,h,j=1

∂ξj∂xi

(ξ + Φ)∂ξh∂xi

(ξ + Φ)Djh(u0 + 〈Du0, Φ〉+ v)

+N∑

i,j=1

∂2ξj∂xi∂xi

(ξ + Φ)Dj(u0 + 〈Du0, Φ〉+ v)

= L0(v, Φ) + L1(v, Φ),


where the matrix M with entries mhk = ∂ξh/∂xk is equal to (I + DΦ)−1.We recall that Φ depends on v through (5.8) and (5.2), i.e. Φ(ξ, v) =F0(ξ′, v(ξ′))ν(ξ) and Φ(ξ, 0) = 0.

At v = 0, Φ = 0, the linear part of L0 with respect to (v, Φ) is

N∑

i=1

(Diiv + Dii〈Du0, Φ〉) + lower order terms.

From the developement

(I + DΦ)−1 = I −DΦ + (DΦ)2(I + DΦ)−1 (5.17)

we get that the linear part of L1 with respect to (v, Φ) is

−N∑

i,j=1

∂2Φj∂x2i

Dju0 + lower order terms

Therefore the second order derivatives of Φ cancel in the sum; what remainsis

A = ∆ + lower order terms,

and the statement follows. Lemma 5.1 implies that equation (5.14) is parabolic near t = 0.Equation (5.14) has to be supported with an initial and a boundary con-

dition. The initial condition comes from the splitting (5.6),

w(0, ξ) = 0, ξ ∈ Ω0.

The boundary condition comes from ∂u/∂n = −1. Since

n(t, ξ + Φ) =(I +tDΦ)−1ν(ξ)|(I +tDΦ)−1ν(ξ)|

andDu(t, ξ + Φ) = (I +tDΦ)−1Du(t, ξ),

from ∂u/∂n = −1 we get

〈(I +tDΦ)−1ν, (I +tDΦ)−1D(u0 + 〈Du0, Φ〉+w〉+ |(I +tDΦ)−1ν| = 0, (5.18)

which gives a new boundary condition for w,

G(ξ, w(t, ξ), Dw(t, ξ)) = 0, t ≥ 0, ξ ∈ ∂Ω0, (5.19)

as soon as we replace Φ = w(t, ξ′)ν(ξ) in (5.18). The function G(ξ, u, p) andits first and second order derivatives with respect to u and pi, i = 1, . . . , N ,are continuous in (ξ, u, p) and C1/2+α/2 in ξ. Moreover G(ξ, 0, 0) = 0.

In the next lemma we identify the linear part of G with respect to (u, p)at (ξ, 0, 0).


Lemma 5.2. We have

G(ξ, w,Dw) = Bw −G(ξ, w,Dw)(ξ)

where B is the linear differential operator defined by

Bv =∂v

∂ν+

∂2u0

∂ν2v, (5.20)

and for every ξ ∈ ∂Ω0,

G(ξ, 0, 0) = Gu(ξ, 0, 0) = Gpi(ξ, 0, 0), i = 1, . . . , N.

Proof — Taking (5.17) into account, we rewrite the first addendum of (5.18)as

〈(I +tDΦ)−1ν, (I +tDΦ)−1D(u0 + 〈Du0, Φ〉+ w)〉 =

=∂u0

∂ν+

∂

∂ν〈Du0, Φ〉+

∂w

∂ν− 〈(tDΦ)ν,Du0〉−

−〈ν, (tDΦ)Du0〉+ R(ξ,Dw,Φ,DΦ)

where R(ξ, p, q, r) and its derivatives with respect to pi, qi, rij , i, j = 1, . . . , N ,vanish at (ξ, 0, 0, 0).

In the above sum we have

∂

∂ν〈Du0, Φ〉 =

∂2u0

∂ν2s, (tDΦ)ν = Dtangs, 〈ν, (tDΦ)Du0〉 = 0,

where the last equality follows from (DΦ)ν = 0. Replacing s = w at theboundary, we get

〈(I +tDΦ)−1ν, (I +tDΦ)−1D(u0 + 〈Du0, Φ〉+ w)〉

=∂u0

∂ν+

∂w

∂ν+

∂2u0

∂ν2w − 〈Dtangw,Dtangu0〉+ R1(ξ, w,Dw)

where R1(ξ, u, p) vanishes at (ξ, 0, 0) as well as its derivatives with respect tou, pi, i = 1, . . . , N .

On the other hand, since 〈(tDΦ)ν, ν〉 = 0, we have

|(I +tDΦ)−1ν| = 1 + R3(ξ, Φ,DΦ)

where R3(ξ, p, q) and its derivatives with respect to pi, qij , i, j = 1, . . . , N ,vanish at (ξ, 0, 0). Replacing the expression (5.2) for Φ and taking into accountthat s = w at ∂Ω0, we get

|(I +tDΦ)−1ν| = 1 + R4(ξ, w,Dw)

where R4(ξ, u, p) and its derivatives with respect to (u, pi) vanish at (ξ, 0, 0).Summing up, the statement follows.


Therefore, the final problem for the only unknown w may be rewritten as

wt = Aw + F(w), t ≥ 0, ξ ∈ Ω0,

Bw = G(w), t ≥ 0, ξ ∈ ∂Ω0,

w(0, ·) = 0, ξ ∈ Ω0,

(5.21)

where A, B are linear differential operators and F , G are nonlinear functions.Problem (5.21) is fully nonlinear because F depends on w and on its spacederivatives up to the second order, and G depends on w and on its first orderspace derivatives. Moreover, F is nonlocal in the second order derivativesof w. However, both F and G are smooth enough, and Fw, Gw vanish att = 0, w = 0. The second order linear operator A is the sum of ∆ plus anonlocal operator, acting only on w and on the first order space derivativesof w. This is crucial for our analysis: since the principal part of A coincideswith the Laplacian, the linearized problem near w0 = 0 is a good linearparabolic problem, for which optimal Holder regularity results and estimatesare available; we need such estimates to solve the nonlinear problem in astandard way, using the Contraction Theorem in a ball of the parabolic Holderspace C1+α/2,2+α([0, δ] × Ω0), with δ > 0 suitably small, or else in a ball ofC3/2+α/2,3+α([0, δ]×Ω0) if the data are more regular.

The above mentioned optimal Holder regularity results and estimates aregiven by the well known Ladyzhenskaja – Solonnikov – Ural’ceva theorem (seee.g. [16, Thm. 5.3] or [22, Thm. 5.1.20]), which we write below.

Theorem 5.3. Let α ∈ (0, 1) and T > 0. Let Ω be an open set in Rn with

uniformly C2+α boundary, and let αij , βi, γ ∈ Cα(Ω) be such that

N∑

i,j=1

αij(ξ)ηiηj ≥ m|η|2, ξ ∈ Ω,

for some m > 0. Moreover, let φi, ψ ∈ C1/2+α/2,1+α([0, T ]×∂Ω) be such that

N∑

i=1

φi(ξ)νi(ξ) = 0, ξ ∈ ∂Ω.

For every v ∈ C2(Ω) set

(Lv)(ξ) =N∑

i,j=1

αij(ξ)Dijv(ξ) +N∑

i=1

(βi(ξ)Div(ξ) + γ(ξ)v(ξ), ξ ∈ Ω,

and

(Bv)(ξ) = ψ(ξ)v(ξ) +N∑

i=1

ϕi(ξ)Div(ξ), ξ ∈ ∂Ω.


Then for every δ ∈ (0, T ] and for every f1 ∈ Cα/2,α([0, δ] × Ω), g1 ∈C1/2+α/2,1+α([0, δ] ×∂Ω) satisfying the compatibility condition

g1(0, ·) = 0 in ∂Ω, (5.22)

the problem

vt = Lv + f1, 0 ≤ t ≤ δ, ξ ∈ Ω,

Bv = g1, t ≥ 0, ξ ∈ ∂Ω,

v(0, x) = 0, ξ ∈ Ω,

(5.23)

has a unique solution v ∈ C1+α/2,2+α([0, δ]×Ω). Moreover there exists C > 0independent of δ such that

‖v‖C1+α/2,2+α([0,δ]×Ω) ≤ C(‖f1‖Cα/2,α([0,δ]×Ω) + ‖g1‖C1/2+α/2,1+α([0,δ]×∂Ω)).(5.24)

Looking at the expression of F , we see that it depends on third orderderivatives of the initial datum u0. Moreover both the boundary operator Band the nonlinear function G depend on second order derivatives of u0.

Using theorem 5.3 it is possible to solve locally problem (5.21). To be moreprecise, first one extends the result of theorem 5.3 to the case where L is theLaplacian plus a nonlocal first order perturbation, and then one solves (5.21)by a fixed point theorem in the metric space Y = u ∈ C1+α/2,2+α([0, δ]×Ω0) :u(0, ·) = 0, ‖u‖C1+α/2,2+α ≤ R.

Some comments about the application fixed point theorem are in order.For every u ∈ Y , the function F(u) is well defined if δ and R are suitably cho-sen. Note that F(u) depends on u0 through its derivatives up to the thirdorder. Since u0 ∈ C3+α(Ω0), then F(u) ∈ Cα/2,α([0, δ] × Ω0). Similarly,G(u) depends on u0 through its derivatives up to the second order, so thatG(u) ∈ C1/2+α/2,1+α([0, δ] × ∂Ω0); the coefficients of the boundary opera-tor B depend on the second order derivatives of u0 and hence they are inC1/2+α/2,1+α([0, δ]× ∂Ω0). Therefore it is reasonable to look for a local solu-tion to (5.21) as a fixed point of the operator Γ defined by Γu = w, w beingthe solution of

wt = Aw + F(u), t ≥ 0, ξ ∈ Ω0,

Bw = G(u), t ≥ 0, ξ ∈ ∂Ω0,

w(0, ·) = 0, ξ ∈ Ω0,

This procedure works, and it gives the following result.

Theorem 5.4. There is R0 > 0 such that for every R ≥ R0 and for ev-ery sufficiently small δ > 0 problem (5.21) has a unique solution in the ballB(0, R) ⊂ C1+α/2,2+α([0, δ]×Ω0).


Once we have locally solved (5.21) we come back to the original problem(1.2) using (5.9) to define ∂Ωt. Note that s has the same regularity of w, i.e.it is in C1+α/2,2+α([0, δ]× ∂Ω0).

Then we define u through (5.6), where Φ is given by (5.2). Again, u hasthe same regularity of w. As a last step we define u through the change ofcoordinates, u(t, x) = u(t, ξ) where x = ξ + Φ(t, ξ).

It is clear now why we lose regularity: not only because of the change ofcoordinates, but mainly because w has the same space regularity of Du0.

However, it is easy to see that uniqueness of w implies uniqueness of (s, u),and hence uniqueness of the regular solution to the original problem (1.2).Moreover, using standard techniques of fully nonlinear problems, it is possibleto prove that t → w(t, ·) is smooth with values in C2(Ω0) in the interval (0, δ)if the data are smooth. It follows that both the free boundary and u aresmooth with respect to time for t in (0, δ).

By a covering argument we get eventually that the free boundary of anysufficiently smooth solution defined in an interval [0, a] is C∞ in time in (0, a].

6 Special Geometries

6.1 Radially Symmetric Solutions

The paper [13] deals with a detailed study of radially symmetric solutions to(1.1), of the type Ωt = B(0, s(t)), u(t, x) = v(t, |x|). The initial set is the ballB0 = B(0, s(0)), and the initial function u0 has positive values in the interiorof B0. We may assume that u0 is defined in the whole R

N , and that it hasnegative values outside B(0, s0).

Problem (1.1) is transformed to a fixed boundary elliptic-parabolic one-dimensional problem, as follows.

Assume we have a radially symmetric solution in [0, T ], such that u haspositive values in B(0, s(t)), and let R > s(t) for 0 ≤ t ≤ T . We extendcontinuously u by a harmonic function outside B(0, s(t)). Then, still denotingby u the extension, u satisfies

(c(u)(t, x))t = ∆u(t, x), t ≥ 0, |x| = s(t),

wherec(u) = max0, u.

For |x| > s(t), u(t, ·) is harmonic, so that (rN−1ur)r = 0 and thereforeur = −(s(t)/r)N−1. Since R > s(t) for each t ∈ [0, T ], this gives a Neumannboundary condition at ∂B(0, R), i.e.

ur(t, x) = −(

s(t)R

)N−1

, |x| = R.


On the other hand, since u vanishes at |x| = s(t), from ur = −(s(t)/r)N−1

we get also (if N > 2)

u(t, x) =∫ r

s(t)

(s(t)ρ

)N−1

dρ =s(t)

N − 2

((s(t)r

)N−2

− 1), |x| = r > s(t).

This gives a Dirichlet boundary condition at ∂B(0, R),

u(t, x) =s(t)

N − 2

((s(t)R

)N−2

− 1), |x| = R.

Eliminating s(t) we get a nonlinear boundary condition for u,

u(t, R) = − R

N − 2((−ur(t, R))1/(N−1) + ur(t, R)) = G(ur(t, R)).

To write it in the usual form ur = F (u) we note that G is not invertible, it hasa minimum at aN = −(N − 1)(N−1)/(2−N) ∈ (−1, 0), with G(aN ) = −bNR.Since we are interested in the free boundary conditions (u, ur) = (0,−1) wetake R close to s0 and we want (u(t, R), ur(t, R)) to be close to (0,−1) for tsmall; so we invert G in [−1, aN ] and we denote by F : [−bNR, 0] → [−1, aN ]its inverse. We extend F setting F (u) = F (−bNR) = aN for u < −bNR.Then our problem is rewritten as an elliptic-parabolic problem with nonlinearNeumann boundary condition,

(c(u)t) = ∆u, (t, x) ∈ QT = (0, T ]×B(0, R),

ur(t, R) = F (u(t, R)), 0 < t ≤ T,

c(u(0, x)) = v0(x), |x| ≤ R,

(6.1)

with v0 = c(u0).A function u ∈ L2(0, T ;H1(B(0, R)) is called a weak solution of (6.1) in

[0, T ] if c(u) ∈ C(QT ) and∫ ∫

QT

(−ϕtc(u) + 〈Du,Dϕ〉dx dt +∫

B(0,R)

ϕ(T, x)c(u(T, x))dx

=∫

B(0,R)

ϕ(0, x)v0(x)dx +∫ T

0

∫

∂B(0,R)

ϕ(t, x)F (u(t, R))dx dt,

for every ϕ ∈ H1(QT ). The existence result of [13] is the following.

Theorem 6.1. Let R > s0 > 0, and let v0 be continuous and radially sym-metric in B(0, R), positive for |x| < s0, zero for |x| ≥ s0. Then for R closeto s0 and T small there exists a unique radially symmetric solution of (6.1),u ∈ L∞(0, T ;H1(B(0, R))) with ∆u ∈ L2(QT ). Moreover u satisfies the freeboundary conditions almost everywhere.


In [13] other properties of the solution are shown. If the initial datumsatisfies also the conditions of [9], then the solution of (6.1) given by theorem6.1 coincides with the solution obtained by the approximation procedure of[9]. If the initial datum is C2 and satisfies the natural compatibility conditionsu0(x) = 0, u0(x) = −1 at |x| = s0, then the solution is analytic for t > 0.Moreover, the behavior near the extinction time is studied.

6.2 Solutions Monotonic Along Some Direction

Another example without loss of regularity was considered in [1], where a two-phase version of (1.2) was studied in a strip for C2+α initial data satisfyingsuitable monotonicity conditions: u0 is assumed to be strictly monotonic inthe direction orthogonal to the strip. This allows to take u as new independentvariable for small t, an old trick already used by Meirmanov for the Stefanproblem and also for a two-dimensional version of problem (1.2), see [23].

To fix the ideas, let us consider a modification of problem (1.1), as follows.Denote by (x, y) the coordinates in R

N , with x ∈ R, y ∈ RN−1. Fix b ∈ R

and look for Ωt of the type

Ωt = x ∈ RN : s(t, y) < x < b.

Here the free boundary Γt is not the whole boundary of Ωt but it is onlythe graph of the unknown function s(t, ·). On the fixed boundary x = b theother unknown u is prescribed, say u = c > 0. The other conditions of (1.1)remain unchanged. Therefore we have the problem

ut = ∆u, t > 0, s(t, y) ≤ x ≤ b, y ∈ RN−1,

u = 0, ∂u∂ν = −1, t > 0, x = s(t, y), y ∈ R

N−1,

u = c, t > 0, x = b,

(6.2)

supported by the initial conditions

s(0, y) = s0(y), y ∈ RN−1; u(0, x, y) = u0(x, y), y ∈ R

N−1, s0(y) ≤ x ≤ b.(6.3)

Therefore, Ω0 = (x, y) ∈ RN : s0(y) < x < b. The initial function s0

has values in [ε, b − ε] for some ε > 0, and belongs to C2+α(RN−1) for someα ∈ (0, 1). The initial datum u0 is in C2+α(Ω0), and the natural compatibilityconditions

u0(s0(y), y) = 0, u0(b, y) = c,∂u

∂ν(s0(y), y) = −1 (6.4)

are satisfied.The main assumption is that u0 is increasing in the variable x; more pre-

cisely


Dxu0 ≥ ε, (x, y) ∈ Ω0. (6.5)

Since we look for a regular solution, for small t > 0 the unknown u(t, ·) has tobe increasing with respect to x. Therefore, it can be taken as a new coordinateξ. Setting for fixed t

Φ(x, y) = (ξ, y) = (u(t, (x, y)), y),

the change of coordinates Φ transforms Ωt into the strip (0, c)×RN−1, indepen-

dent of time. The inverse change of coordinates is denoted by (x, y) = Φ(t, ξ)where

(x, y) = Φ−1(ξ, y) = (f(t, (ξ, y)), y),

and the function f is our new unknown. Deducing a problem for f is easy:observing that

Dxu =1

Dξf, Dyiu = −Dyif

Dξf, i = 1, . . . , N − 1, Dtu = −Dtf

Dξf,

∆u = −∆yf

Dξf+ 2

〈Dyfξ, Dyf〉(Dξf)2

− (|Dyf |2 + 1)Dξξf

(Dξf)3,

we get

ft = ∆yf +Dξξf

(Dξf)2(|Dyf |2 + 1)− 2

〈Dyfξ, Dyf〉Dξf

,

t ≥ 0, ξ ∈ [0, c], y ∈ RN−1,

Dξf(t, 0, y) =√

1 + |Dyf |2, y ∈ RN−1,

y(t, c, y) = b, y ∈ RN−1,

f(0, ξ, y) = f0(ξ, y), ξ ∈ [0, c], y ∈ RN−1.

(6.6)

This is a quasilinear parabolic initial-boundary value problem with smoothnonlinearities, which is shown to have a unique regular (local in time) solutionby classical methods of parabolic equations. Indeed it is standard to set it as afixed point problem in the parabolic Holder space C1+α/2,2+α([0, T ]× ([0, c]×RN−1)), with T small, using again theorem 5.3 as a main tool.

Once a local solution y ∈ C1+α/2,2+α([0, T ] × ([0, c] × RN−1)) exists, it

is easy to come back to the original problem (6.2) by the inverse change ofcoordinates Φ−1.

6.3 Solutions Close to Self-similar Solutions, Travelling Waves etc.

If the initial datum (Ω0, u0) is close to the initial datum (Ω,U) of a smoothgiven solution, the method of section 5 gives existence and uniqueness of alocal classical solution without loss of regularity.


First of all, the free boundary problem is transformed into a fixed boundaryproblem in Ω by the changement of coordinates defined in (5.3), with Ω0

replaced by Ω. Therefore, the unknown s is now the signed distance from ∂Ω.Then the splitting (5.6) is replaced by

u(t, ξ) = U(ξ) + 〈DU(ξ), Φ(t, ξ)〉 + w(t, ξ), ξ ∈ Ω. (6.7)

This gives again s(t, ξ) = w(t, ξ) for ξ ∈ ∂Ω. The final problem for w hasw0 = u0−U −〈DU,Φ(0, ξ)〉 which in general is not zero, but it is small. Fullynonlinear parabolic problems of the type (5.21) may be studied as well withnonzero initial data.

To fix the ideas, we consider the case where (Ω0, u0) is close to the initialdatum of the self-similar solution (B(0, r

√T − t), f(|x|/

√T − t)) introduced

in §3. It is convenient to transform the problem to selfsimilar variables

x =x

(T − t)12, t = − log(T−t), u(x, t) =

u(x, t)(T − t)

12, Ωt = x : x ∈ Ωt. (6.8)

Omitting the hats, we arrive at

ut = ∆u− 12〈x,Du〉+

12u, x ∈ Ωt, (6.9)

with boundary conditions

u = 0,∂u

∂n= −1 on ∂Ωt. (6.10)

The selfsimilar solution (3.1) is transformed by (6.8) into a stationary solution

U(x) = f(|x|), Ω = Br = x ∈ Rn : |x| < r, (6.11)

of (6.9), (6.10).Stability of stationary solutions to free boundary problems in bounded

domains has been studied in [3]. The change of coordinates (t, x) → (t, ξ)which transforms the unknown domain Ωt into the fixed ball Ω is describedin sect. 5. Setting u(t, ξ) = u(t, x) and defining w by the splitting (5.7) wearrive at a final equation for w,

wt = ∆w − 12 〈ξ,Dw〉+ w

2 + φ(w,Dw,D2w), t ≥ 0, ξ ∈ Ω,

∂w

∂n+(

N − 1r

− r

2

)w = ψ(w,Dw), t ≥ 0, ξ ∈ ∂Ω,

w(0, ξ) = w0(ξ), ξ ∈ Ω,

(6.12)

where φ and ψ are smooth and quadratic near w = 0.This problem is similar to (5.21). The differences are due to the fact that

we linearize around a stationary solution instead of around the initial datum.


It comes out that the differential equation for w is a bit simpler, and the initialdatum is not zero.

However, the technique to find a local solution is the same. We give herethe proof of a slightly more general theorem, taken from the paper [3]. It givesexistence of the solution for arbitrarily long time intervals, provided the initialdatum is small enough. This may be seen as continuous dependence of thesolution on the initial datum at w0 = 0.

Theorem 6.2. Let Ω be a bounded open set in RN with C2+α boundary,

0 < α < 1.Let L = aijDij + biDi+ c be a uniformly elliptic operator with coefficients

aij, bi, c in Cα(Ω) and let B = βiDi + γ be a nontangential operator withcoefficients βi, γ in C1+α(∂Ω).

Let F : B(0, R) ⊂ C2(Ω) → C(Ω), G : B(0, R) ⊂ C1(Ω) → C(∂Ω)be smooth functions with smooth restrictions to B(0, R) ⊂ C2+α(Ω) and toB(0, R) ⊂ C2+α(Ω) with values in Cα(Ω) and in C1+α(∂Ω), respectively.Assume moreover

F (0) = 0, F ′(0) = 0; G(0) = 0, G′(0) = 0.

Let w0 ∈ C2+α(Ω) satisfy the compatibility condition

Bw0 = G(w0(·))(ξ), ξ ∈ ∂Ω. (6.13)

Then for every T > 0 there are r, ρ > 0 such that the problem

wt = Lw + F (w), t ≥ 0, ξ ∈ Ω,

Bw = G(w), t ≥ 0, ξ ∈ ∂Ω,

w(0, ξ) = w0(ξ), ξ ∈ Ω,

(6.14)

has a solution w ∈ C1+α/2,2+α([0, T ]×Ω) if ‖w0‖C2+α(Ω) ≤ ρ. Moreover w isthe unique solution in B(0, r) ⊂ C1+α/2,2+α([0, T ]×Ω).

Proof — Let 0 < r ≤ R, and set

K(r) = sup‖F ′(ϕ)‖L(C2+α(Ω),Cα(Ω)) : ϕ ∈ B(0, r) ⊂ C2+α(Ω),

H(r) = sup‖G′(ϕ)‖L(C2+α(Ω),C1+α(∂Ω)) : ϕ ∈ B(0, r) ⊂ C2+α(Ω).

Since F ′(0) = 0 and G′(0) = 0, K(r) and H(r) go to 0 as r → 0. Let L > 0be such that, for all ϕ, ψ ∈ B(0, r) ⊂ C2(Ω) with small r,

‖F ′(ϕ) − F ′(ψ)‖L(C2(Ω),C(Ω)) ≤ L‖ϕ− ψ‖C2(Ω),

‖G′(ϕ)−G′(ψ)‖L(C1(Ω),C(∂Ω)) ≤ L‖ϕ− ψ‖C1(Ω).


For every 0 ≤ s ≤ t ≤ T and for every w ∈ B(0, r) ⊂ C1+α/2,2+α([0, T ]× Ω)with r so small that K(r), H(r) < ∞, we have

‖F (w(·, t))‖Cα(Ω) ≤ K(r)‖w(·, t)‖C2+α(Ω), ‖F (w(·, t))− F (w(·, s))‖C(Ω) ≤

Lr‖w(·, t) − w(·, s)‖C2(Ω) ≤ Lr|t− s|α/2‖w‖C1+α/2,2+α([0,T ]×Ω),

and similarly

‖G(w(·, t))‖C1+α(∂Ω) ≤ H(r)‖w(·, t)‖C2+α(Ω) ≤ H(r)‖w‖C1+α/2,2+α([0,T ]×Ω),

‖G(w(·, t)) −G(w(·, s))‖C(∂Ω) ≤

Lr‖w(·, t)− w(·, s)‖C1(Ω) ≤ Lr|t− s|1/2+α/2‖w‖C1+α/2,2+α([0,T ]×Ω).

Therefore, (ξ, t) → F (w(·, t))(ξ) belongs to Cα/2,α([0, T ] × Ω), (ξ, t) →G(w(·, t))(ξ) belongs to C1/2+α/2,1+α([0, T ]× ∂Ω) and

‖F (w)‖Cα,α/2([0,T ]×Ω) ≤ (K(r) + Lr)‖w‖C1+α/2,2+α([0,T ]×Ω),

‖G(w)‖C1/2+α/2,1+α([0,T ]×∂Ω) ≤ Cα(2H(r) + Lr)‖w‖C1+α/2,2+α([0,T ]×Ω).

So, if ‖w0‖C2+α(Ω) is small enough, we define a nonlinear map

dom (Γ ) := w ∈ B(0, r) ⊂ C1+α/2,2+α([0, T ]×Ω) : w(·, 0) = w0Γ : dom (Γ ) → C1+α/2,2+α([0, T ]×Ω),

by Γw = v, where v is the solution of

vt(t, ξ) = Lv + F (w(t, ·))(ξ), 0 ≤ t ≤ T, ξ ∈ Ω,

Bv = G(w(t, ·))(ξ), 0 ≤ t ≤ T, ξ ∈ ∂Ω,

v(0, ξ) = w0(ξ), ξ ∈ Ω.

Actually, thanks to the compatibility condition Bw0 = G(w0) and the regular-ity of F (w) and G(w), the range of Γ is contained in C1+α/2,2+α([0, T ]×Ω).Moreover, by theorem 5.3 there is C = C(T ) > 0, independent of r, such that

‖v‖C1+α/2,2+α([0,T ]×Ω) ≤

C(‖w0‖C2+α(Ω) + ‖F (w)‖Cα/2,α([0,T ]×Ω) + ‖G(w)‖C1/2+α/2,1+α([0,T ]×∂Ω)),

so that

‖Γ (w)‖C1+α/2,2+α([0,T ]×Ω) ≤

C(‖w0‖C2+α(Ω) + (K(r) + Lr + Cα(2H(r) + Lr))‖w‖C1+α/2,2+α([0,T ]×Ω)).


Therefore, if r is so small that

C(K(r) + Lr + Cα(2H(r) + Lr)) ≤ 1/2, (6.15)

and w0 is so small that

‖w0‖C2+α(Ω) ≤ Cr/2,

Γ maps the ball B(0, r) into itself. Let us check that Γ is a 1/2-contraction.Let w1, w2 ∈ B(0, r), wi(·, 0) = w0. Writing wi(·, t) = wi(t), i = 1, 2, we have

‖Γw1 − Γw2‖C1+α/2,2+α([0,T ]×Ω) ≤

≤ C(‖F (w1)− F (w2)‖Cα/2,α([0,T ]×Ω)+

+‖G(w1)−G(w2)‖C1/2+α/2,1+α([0,T ]×∂Ω)),

and, arguing as above, for 0 ≤ t ≤ T ,

‖F (w1(t))− F (w2(t))‖Cα(Ω) ≤

K(r)‖w1(t)− w2(t)‖C2+α(Ω) ≤ K(r)‖w1 − w2‖C1+α/2,2+α([0,T ]×Ω),

‖G(w1(t))−G(w2(t))‖C1+α(∂Ω) ≤

H(r)‖w1(t)− w2(t)‖C2+α(Ω) ≤ H(r)‖w1 − w2‖C1+α/2,2+α([0,T ]×Ω),

while for 0 ≤ s ≤ t ≤ T

‖F (w1(t))− F (w2(t)) − F (w1(s)) − F (w2(s))‖C(Ω) =

=∥∥∥∥∫ 1

0

F ′(σw1(t) + (1 − σ)w2(t)(w1(·, t)− w2(t))−

−F ′(σw1(s) + (1− σ)w2(s))(w1(s)− w2(s))dσ∥∥∥∥C(Ω)

≤

≤∫ 1

0

‖(F ′(σw1(t) + (1− σ)w2(t)− F ′(σw1(s)+

+(1− σ)w2(s))(w1(t)− w2(t))dσ‖C(Ω)+

+∫ 1

0

‖F ′(σw1(s))(w1(t)− w2(t)− w1(s) + w2(s))‖C(Ω) ≤


≤ L

2(‖w1(t)− w1(s)‖C2(Ω)+

+‖w2(t)− w2(s)‖C2(Ω))‖w1(t)− w2(t)‖C2(Ω)+

+Lr‖w1(t)− w2(t)− w1(s) + w2(s))‖C2(Ω) ≤

≤ 2Lr(t− s)α/2‖w1 − w2‖C1+α/2,2+α([0,T ]×Ω),

and similarly

‖G(w1(·, t))−G(w2(·, t))−G(w1(·, s))−G(w2(·, s))‖C(∂Ω)

≤ 2Lr(t− s)1/2+α/2‖w1 − w2‖C1+α/2,2+α([0,T ]×Ω).

Therefore,

‖Γw1 − Γw2‖C1+α/2,2+α([0,T ]×Ω)

≤ C(K(r) + Lr + Cα(2H(r) + Lr))‖w1 − w2‖C1+α/2,2+α([0,T ]×Ω)

≤ 12‖w1 − w2‖C1+α/2,2+α([0,T ]×Ω),

the last inequality being a consequence of (6.15). The statement follows. Local existence and uniqueness for initial data close to travelling wave

solutions may be studied similarly. See [4] for the free boundary heat equation,[7] for a two phase Burgers equation, [18] for a one phase - two phase systemfrom combustion theory.

References

1. D. Andreucci, R. Gianni: Classical solutions to a multidimensional freeboundary problem arising in combustion theory, Comm. P.D.E’s 19 (1994), 803-826.

2. O. Baconneau, A. Lunardi: Smooth solutions to a class of free boundaryparabolic problems, preprint Dipart. Mat. Univ. Parma 253 (2001) (submitted).

3. C.M. Brauner, J. Hulshof, A. Lunardi: A general approach to stability infree boundary problems, J. Diff. Eqns. 164 (2000), 16-48.

4. C.M. Brauner, J. Hulshof, A. Lunardi: A critical case of stability in a freeboundary problem, J. Evol. Eqns. 1 (2001), 85-113.

5. C.M. Brauner, A. Lunardi: Instabilities in a combustion model with freeboundary in R2, Arch. Rat. Mech. Anal. 154 (2000), 157-182.

6. C.M. Brauner, A. Lunardi, Cl. Schmidt-Laine: Une nouvelle formulationde modeles de fronts en problemes completement non lineaires, C.R.A.S. Paris311 (1990), Serie I, 597-602.


7. C.M. Brauner, A. Lunardi, Cl. Schmidt-Laine: Stability analysis in a mul-tidimensional interface problem, Nonlinear Analysis T.M.A. 44 (2001), 263-280.

8. J.D. Buckmaster, G.S.S. Ludford: Theory of Laminar Flames, CambridgeUniversity Press, Cambridge, 1982.

9. L.A. Caffarelli, J.L. Vazquez: A free boundary problem for the heat equationarising in flame propagation, T.A.M.S. 347 (1995), 411-441.

10. W. Desch, W. Schappacher: Some Perturbation Results for Analytic Semi-groups, Math. Ann. 281 (1988), 157-162.

11. J. Escher, G. Simonett: Maximal regularity for a free boundary problem,NoDEA 2 (1995), 463-510.

12. J. Escher, G. Simonett: Classical solutions of multidimensional Hele-Shawmodels, SIAM J. Math. Anal. 28 (1997), 1028-1047.

13. V.A. Galaktionov, J. Hulshof, J.L. Vazquez: Extinction and focusing be-haviour of spherical and annular flames described by a free boundary problem,J. Math. Pures Appl. 76 (1997), 563-608.

14. D. Hilhorst, J. Hulshof: An elliptic-parabolic problem in combustion theory:convergence to travelling waves, Nonlinear Analysis TMA 17 (1991), 519-546.

15. D. Hilhorst, J. Hulshof: A free boundary focusing problem, Proc. AMS 121,1193-1202 (1994).

16. O.A. Ladyzhenskaja, V.A. Solonnikov, N.N. Ural’ceva: Linear and quasi-linear equations of parabolic type, Nauka, Moskow 1967 (Russian). Englishtransl.: Transl. Math. Monographs, AMS, Providence, 1968.

17. C. Lederman, J.L. Vazquez, N. Wolanski: Uniqueness of solution to a freeboundary problem from combustion, TAMS 353 (2001), 655-692.

18. L. Lorenzi: A free boundary problem stemmed from Combustion Theory. Part I:Existence, uniqueness and regularity results, preprint Dipart. Mat. Univ. Parma245 (2000), to appear in J. Math. Anal. Appl.

19. L. Lorenzi: A free boundary problem stemmed from Combustion Theory. PartII: Stability, instability and bifurcation results., preprint Dipart. Mat. Univ.Parma 246 (2000), to appear in J. Math. Anal. Appl.

20. L. Lorenzi, A. Lunardi: Stability in a two-dimensional free boundary com-bustion model, preprint Dipart. Mat. Univ. Parma 271 (2001), to appear inNonlinear Analysis TMA.

21. A. Lunardi: An introduction to geometric theory of fully nonlinear parabolicequations, in “Qualitative aspects and applications of nonlinear evolution equa-tions”, T.T. Li, P. de Mottoni Eds., World Scientific, Singapore (1991),107-131.

22. A. Lunardi: Analytic semigroup and optimal regularity in parabolic problems,Birkhauser Verlag, Basel, 1995.

23. A.M. Meirmanov: On a problem with free boundary for parabolic equations,Math. USSR Sbornik 43 (1982), 473-484.

24. M. Primicerio: Diffusion problems with a free boundary, Boll. U.M.I. A (5) 18(1981), 11-68.

25. E. Sinestrari: On the abstract Cauchy problem in spaces of continuous func-tions, J. Math. Anal. Appl. 107 (1985), 16-66.

26. J. L. Vazquez: The Free Boundary Problem for the Heat Equation with fixedGradient Condition, Proc. Int. Conf. “Free Boundary Problems, Theory andApplications”, Zakopane, Poland, Pitman Research Notes in Mathematics 363,pp. 277-302, 1995.

27. T.D. Ventsel: A free-boundary value problem for the heat equation, Dokl. Akad.Nauk SSSR 131 (1960), 1000-1003; English transl. Soviet Math. Dokl. 1 (1960).

[lecture notes in mathematics] functional analytic methods for evolution equations volume 1855 || an...

Documents