SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS IN HIGH SPACE DIMENSIONS

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS IN HIGH SPACE DIMENSIONS YIHONG DU, HIROSHI MATSUZAWA AND MAOLIN ZHOU Abstact. We conside nonlinea diffusion poblems of the fom u t = u + f(u) with Stefan type fee bounday conditions, whee the nonlinea tem f(u) is of monostable, bistable o combustion type. Such poblems aise as an altenative model (to the coesponding Cauchy poblem) to descibe the speading of a biological o chemical species, whee the fee bounday epesents the expanding font. We ae inteested in its long-time speading behavio in the adially symmetic case, whee the equation is satisfied in x < h(t), with x = h(t) the fee bounday, and lim h(t) =, lim u(t, x ) = 1. Fo the case of one space dimension (N = 1), Du and Lou 8] poved that h(t) lim = c fo some c >. Subsequently, shape estimate of the speading speed was t obtained by the authos of the cuent pape in 11], in the fom that lim h(t) c t] = Ĥ R1. In this pape, we conside the case N 2 and show that a logaithmic shifting occus, namely thee exists c > independent of N such that lim h(t) c t + (N 1)c log t] = ĥ R1. At the same time, we also obtain a athe clea desciption of the speading pofile of u(t, ). These esults contast shaply with those fo the coesponding Cauchy poblem, whee the logaithmic shifting fo the monostable case is significantly diffeent fom that fo the bistable and combustion cases. 1. Intoduction We ae inteested in obtaining exact long-time limit of the speading speed and pofile detemined by the following fee bounday poblem: u t = u + f(u), < < h(t), t >, u (1.1) (t, ) =, u(t, h(t)) =, t >, h (t) = µu (t, h(t)), t >, h() = h, u(, ) = u (), h, whee u = u + N 1 u, = h(t) is the moving bounday to be detemined, µ and h ae given positive constants. The initial function u is chosen fom { } (1.2) K (h ) := ψ C 2 (, h ]) : ψ () = ψ(h ) =, ψ() > in, h ). Fo any given h > and u K (h ), by a classical solution of (1.1) on the time-inteval, T ] we mean a pai (u(t, ), h(t)) belonging to C 1,2 (D T ) C 1 (, T ]), such that all the identities in (1.1) ae satisfied pointwisely, whee D T := { (t, ) : t (, T ],, h(t)] }. Date: Septembe 16, 213. School of Science and Technology, Univesity of New England, Amidale, NSW 2351, Austalia. (Email: ydu@tuing.une.edu.au). Numazu National College of Technology, 36 Ooka, Numazu City, Shizuoka 41-851, Japan. (Email: hmatsu@numazu-ct.ac.jp). Gaduate School of Mathematical Sciences, The Univesity of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan. (Email: zhouml@ms.u-tokyo.ac.jp). Y. Du was suppoted by the Austalian Reseach Council, and H. Matsuzawa was suppoted by the Institute of National Colleges of Technology Japan Oveseas Reseach Scholas Pogam. 1

2 Y. DU, H. MATSUZAWA AND M. ZHOU The nonlineaity f(u) is assumed to be of monostable, bistable o combustion type, whose meanings will be made pecise below. When f(u), (1.1) educes to the classical one-phase Stefan poblem, which aises in the study of melting of ice in contact with wate. Ou motivation to study the nonlinea Stefan poblem (1.1) mainly comes fom the wish to bette undestand the speading of a new species, whee u is viewed as the density of such a species, and the fee bounday epesents the speading font, beyond which the species cannot be obseved (i.e., the species has density ). The speading pocess is usually modeled by the Cauchy poblem { Ut U = f(u) fo x R N, t >, (1.3) U(, x) = U (x) fo x R N, whee U (x) is nonnegative and has nonempty compact suppot. In such a case, U(t, x) > fo all x R N once t >, but one may specify a cetain level set Γ δ (t) := {x : U(t, x) = δ} as the speading font, whee δ > is small, and Ω δ (t) := {x : U(t, x) > δ} is egaded as the ange whee the species can be obseved. A stiking featue of the long time behavio of the font Γ δ (t) is that, when speading happens (i.e., U(t, x) 1 as t ), Γ δ (t) goes to infinity at a constant asymptotic speed in all diections, namely, fo any small ϵ >, thee exists T > so that (1.4) Γ δ (t) A ϵ (t) := {x R N : (c ϵ)t x (c + ϵ)t} fo t T. The numbe c is usually called the speading speed of (1.3), and is detemined by the well-known taveling wave poblem (1.5) Q cq + f(q) =, Q > in R 1, Q( ) =, Q(+ ) = 1, Q() = 1/2. Moe pecisely, in the monostable case, c > is the minimal value of c such that (1.5) has a solution Q c (moe accuately Q c exists if and only if c c ); in the bistable and combustion cases, c is the unique value of c such that (1.5) has a solution Q c. Moeove, Q c is unique when it exists fo a given c. When U (x) is adially symmetic, then U(t, x) is adially symmetic in x fo any t >, and bette estimates of the speading speed and the pofile of U nea the font ae available, which will be ecalled biefly below. Poblem (1.1) is the spheically symmetic vesion of the geneal nonlinea Stefan poblem studied in 6] and 1], which has the fom u t u = f(u) fo x Ω(t), t >, (1.6) u = and u t = µ x u 2 fo x Γ(t), t >, u(, x) = u (x) fo x Ω, whee Ω(t) R N (N 1) is bounded by the fee bounday Γ(t) (i.e., Γ(t) = Ω(t)), with Ω() = Ω, which is a bounded domain that agees with the inteio of its closue Ω, Ω satisfies the inteio ball condition, and u C(Ω ) H 1 (Ω ) is positive in Ω and vanishes on Ω. If u (x) in (1.6) is adially symmetic, then (1.6) educes to (1.1). It follows fom 6] that (1.6) has a unique weak solution which is defined fo all t >. One of the main esults in 1] fo the geneal poblem (1.6) implies the following: Theoem A. Ω(t) is expanding in the sense that Ω Ω(t) Ω(s) if < t < s. Moeove, Ω := t> Ω(t) is eithe the entie space R N, o it is a bounded set. Futhemoe, when Ω = R N, fo all lage t, Γ(t) is a smooth closed hypesuface in R N, and thee exists a continuous function M(t) such that (1.7) Γ(t) {x : M(t) d π x M(t)}; 2 and when Ω is bounded, lim u(t, ) L (Ω(t)) =. Hee d is the diamete of Ω.

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 3 It can be shown (see 9]) that when speading happens (i.e., u(t, x) 1 as t ), thee exists c > such that M(t) (1.8) lim = c. t The numbe c is theefoe called the asymptotic speading speed of (1.6), which is detemined by the following poblem, (1.9) q cq + f(q) =, q > in (, ), q() =, q( ) = 1. The above discussion shows that when speading happens, (1.3) and (1.6) exhibit simila asymptotic behavio: Thei fonts can be appoximated by sphees, which go to infinity at some constant asymptotic speed. Moeove, by 6], if u and Ω(t) in (1.6) ae denoted by u µ and Ω µ (t), espectively, then as µ, Ω µ (t) R N ( t > ), u µ U in C (1+ν)/2,1+ν loc ((, ) R N ) ( ν (, 1)), whee U is the unique solution of (1.3) with U = u. Thus the Cauchy poblem (1.3) may be egaded as the limiting poblem of (1.6) as µ. It tuns out that undeneath these similaities, thee exist fundamental diffeences between (1.6) and (1.3). This pape is devoted to evealing these diffeences. It is ou hope that this may povide futhe insights to the undestanding of the mechanisms undelying so many diffeent speading pocesses. Fo such a pupose, we will estict to the simple spheically symmetic case (1.1), fo which we ae able to gain deepe undestanding of the speading pofile of the fee bounday model. If we take { u ( x ), x < h, (1.1) U (x) =, x h, with u given in (1.1), then the unique solution of (1.3) is adially symmetic: U = U(t, x ). Thus fo such U, (1.1) povides an altenative to (1.3) fo the desciption of the speading of a cetain species with initial density u. We will closely examine the speading behavio detemined by (1.1) and compae it with that of (1.3). While the Cauchy poblem (1.3) has been extensively studied in the past seveal decades and elatively well undestood (some elevant esults fo (1.3) will be ecalled below), the study of the nonlinea fee bounday poblem (1.1) is athe ecent. Poblem (1.1) with f(u) = au bu 2 was investigated in 5], continuing a study initiated in 7] fo the one space dimension case. A deduction of the fee bounday condition based on ecological assumptions can be found in 4], but geneally speaking, the ole of this fee bounday condition in the mechanism of speading is still pooly undestood. In 8], poblem (1.1) with a athe geneal f(u) but in one space dimension was consideed. In paticula, if f(u) is of monostable, o bistable, o combustion type, it was shown in 8] that (1.1) has a unique solution which is defined fo all t >, and as t, h(t) eithe inceases to a finite numbe h, o it inceases to +. Moeove, in the fome case, u(t, ) unifomly in, while in the latte case, u(t, ) 1 locally unifomly in, + ) (except fo a tansition case when f is of bistable o combustion type). The situation that is called the vanishing case, and is called the speading case. u and h h < + u 1 and h +

4 Y. DU, H. MATSUZAWA AND M. ZHOU When speading happens, it was shown in 8] that thee exists c > such that h(t) lim = c. t The numbe c is the same as in (1.8). These conclusions emain valid in highe space dimensions (9]). Next we will descibe the esults moe accuately. Fistly, let us ecall in detail the thee types of nonlineaities of f mentioned above: (f M ) monostable case, (f B ) bistable case, (f C ) combustion case. In the monostable case (f M ), we assume that f is C 1 and it satisfies f() = f(1) =, f () >, f (1) <, (1 u)f(u) > fo u >, u 1. A typical example is f(u) = u(1 u). In the bistable case (f B ), we assume that f is C 1 and it satisfies { f() = f(θ) = f(1) =, f(u) < in (, θ), f(u) > in (θ, 1), f(u) < in (1, ), fo some θ (, 1), f () <, f (1) < and 1 f(s)ds >. A typical example is f(u) = u(u θ)(1 u) with θ (, 1 2). In the combustion case (f C ), we assume that f is C 1 and it satisfies f(u) = in, θ], f(u) > in (θ, 1), f (1) <, f(u) < in 1, ) fo some θ (, 1), and thee exists a small δ > such that f(u) is nondeceasing in (θ, θ + δ ). The asymptotic speading speed c is detemined in the following way. Poposition 1.1 (Poposition 1.8 and Theoem 6.2 of 8]). Suppose that f is of (f M ), o (f B ), o (f C ) type. Then fo any µ > thee exists a unique c = c (µ) > and a unique solution q c to (1.9) with c = c such that q c () = c µ. We emak that this function q c is shown in 8] to satisfy q c (z) > fo z. We call q c a semi-wave with speed c, since the function v(t, x) := q c (c t x) satisfies { vt = v xx + f(v) fo t R 1, x < c t, v(t, c t) =, µv x (t, c t) = c, v(t, ) = 1. In 11], shape estimate of the speading speed in one space dimension was obtained. Moe pecisely it was shown in 11] that when speading happens fo (1.1), thee exists Ĥ R such that (1.11) (1.12) lim (h(t) c t Ĥ) =, lim h (t) = c, lim sup u(t, ) q c (h(t) ) =., h(t)] In this pape, we conside the case that the space dimension N 2, and speading happens fo (1.1), namely lim h(t) = and lim u(t, ) = 1 locally unifomly fo, ).

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 5 We will show that in such a case, we still have (1.12) and lim h (t) = c, but thee exists c > independent of N such that (1.13) lim h(t) c t + (N 1)c log t ] = ĥ R1. Moeove, the constant c is given by c = 1 ζ c, ζ = 1 + c µ 2 q c (z)2 e c z dz. The tem (N 1)c log t in (1.13) will be called a logaithmic shifting tem. Fo simplicity of notation, we will wite c N = (N 1)c. Thus fom (1.13) and (1.12) we obtain lim sup u(t, ) q c (c t c N log t + ĥ ) =., h(t)] Fo convenience of compaison, we now ecall some elevant esults fo the coesponding Cauchy poblem (1.3). The classical pape of Aonson and Weinbege 2] contains a systematic investigation of this poblem (see 1] fo the case of one space dimension). Vaious sufficient conditions fo lim U(t, x) = 1 ( speading o popagation ) and fo lim U(t, x) = ( vanishing o extinction ) ae known, and the way U(t, x) appoaches 1 as t has been used to descibe the speading of a (biological o chemical) species. In paticula, when speading happens, it was shown in 2] that, in any space dimension N 1, thee exists c > independent of N, such that, fo any small ϵ >, { lim max x (c +ϵ)t U(t, x) =, (1.14) lim max x (c ϵ)t U(t, x) 1 =. Clealy (1.4) is a consequence of (1.14) (with the same c ). The elationship between the speading speed detemined by (1.1) and that detemined by (1.3) is given by (see Theoem 6.2 of 8]) c = lim µ c (µ). Moe details on the speading behavio of the Cauchy poblem can be found, fo example, in 1, 2, 13, 14, 18, 19, 2, 24]. As we will explain below, fundamental diffeences aise between the fee bounday poblem and the Cauchy poblem when we compae thei speading pofiles closely. While the speading pofiles of all thee basic cases (f M ), (f B ) and (f C ) can be descibed in a unified fashion fo the fee bounday model (see (1.11), (1.12) and (1.13)), whee no logaithmic shifting occus in space dimension N = 1, and a synchonized logaithmic shifting happens in dimensions N 2, this is not the case fo the Cauchy poblem, whee the monostable case behaves vey diffeently fom the othe two cases: The monostable case gives ise to logaithmic shifting in all dimensions N 1, and the shifting is significantly diffeent fom the othe two cases. Moe pecisely, in one space dimension, a classical esult of Fife and McLeod 13] states that fo f of type (f B ), and fo appopiate initial function U that guaantees U(t, x) 1 as t, whee U is the unique solution to (1.3), the speading pofile of U is descibed by U(t, x) Q c (c t + x + C ) < Ke ωt fo x <, U(t, x) Q c (c t x + C + ) < Ke ωt fo x >. Hee (c, Q c ) is the unique solution of (1.5), C ± R, and K, ω ae suitable positive constants. So no logaithmic shifting occus in this case. The monostable case of (1.3) has vey diffeent behavio. Fistly we ecall that (1.5) aleady behaves diffeently in the monostable case. Secondly, a logaithmic shifting occus: When (f M ) holds

6 Y. DU, H. MATSUZAWA AND M. ZHOU and futhemoe f(u) f ()u fo u (, 1) (so f falls to the so called pulled case), thee exist constants C ± such that lim U(t, x) Q c (c t 3 ) log t x + C + =, c and max x lim max U(t, x) Q c (c t 3 ) log t + x + C =. c x Hee the logaithmic shifting tem 3 c log t is known as the logaithmic Bamson coection tem; see 3, 17, 22, 24] fo moe details. Fo space dimension N 2, if U (x) is given by (1.1) and hence the unique solution U of (1.3) is spheically symmetic (U = U(t, x )), esults in 16, 25] indicate that the Bamson coection tem fo the monostable case (with some exta conditions on f) becomes N + 2 log t (fo the pulled case of f), o N 1 log t (fo the pushed case of f), c c that is, thee exists some constant C such that fo the pulled case of f, lim sup ( U(t, x ) Q c c t N + 2 ) log t + C x =, x R N c and fo the pushed case of f, lim sup ( U(t, x ) Q c c t N 1 x R N c ) log t + C x =. In the bistable case (as well as the combustion case), the Fife-McLeod esult should be changed to (see 25]) lim sup ( U(t, x ) Q c c t N 1 ) log t + L x =, x R N c whee L is some constant. The above compaisons indicate that the singula behavio of the monostable case obseved in the Cauchy poblem does not exist anymoe in the fee bounday model, whee all thee cases behave in a athe synchonized manne. The est of the pape is oganized as follows. In section 2, we descibe how the constant c N in the logaithmic shifting tem is defined. In section 3, we estimate h(t) in seveal steps until the shap tem c N log t appeas in the uppe and lowe bounds of h(t). The main convegence esults of this pape ae poved in section 4, whee ou aguments ae based on the estimates obtained in section 3, and on a new device vey diffeent fom the enegy methods used in 11] and 13]. A key step in this eseach is to find the exact fom of the logaithmic shifting tem c N log t. This elies on the discovey that shap uppe and lowe solutions to (1.1) can be obtained by suitable petubations of h(t) = c t c N log t, u(t, ) = ϕ(µ(c c N t 1 ), h(t)), with the functions ϕ(µ, z) and µ(ξ) defined in (2.1) and (2.6), espectively. This appoach is completely diffeent fom that used fo teating the coesponding Cauchy poblem, and fom that used to handle the one space dimension case in 11]. Ou method to pove the convegence esult in section 4 also elies on innovative ideas. The method is vey poweful and should have applications elsewhee. The spiit of the method is close to those in 26] and 12].

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 7 2. Fomula fo c N In this section, we descibe how c N in the logaithmic shifting tem is defined, and also give a key identity (see (2.7) below) to be used in the next section. Let q c be given by Poposition 1.1 and we define ϕ(z) to be the unique solution of the following initial value poblem (2.1) ϕ + c ϕ + f(ϕ) =, ϕ() =, ϕ () = c /µ. Clealy ϕ(z) = q c ( z) fo z. To stess its dependence on µ, we wite ϕ(z) = ϕ(µ, z). Similaly we wite c = c (µ). It is easily seen that fo each given µ >, we can find ϵ > such that ϕ(µ, z) is defined fo z (, ϵ ] with ϕ z (µ, z) <, ϕ(µ, ϵ ) < η < fo µ µ /2, 2µ ] and z ϵ. Fom 8] we see that µ c (µ) is stictly inceasing. We will show below that it is a C 2 function. To this end, we need to ecall some details contained in 8]. Unde the assumptions of Poposition 1.1, it was shown in 8] that thee exists a unique c > such that fo each c, c ], the poblem (2.2) P = c f(q) P in, 1), P (1) =, P (1) < has a unique solution P c (q), which necessaily satisfies P c(1) = c c 2 4f (1), P c (q) > in (, 1). 2 Futhemoe, the following monotonicity and continuity esult holds. Lemma 2.1 (Lemma 6.1 of 8]). Fo any c 1 < c 2 c and c, c ], P c1 (q) > P c2 (q) in, 1), lim c c P c (q) = P c (q) unifomly in, 1]. Moeove, P c () = and P c (q) > in (, 1). Fom the poof of Theoem 6.2 in 8], we see that, fo µ >, c (µ) is the unique solution of P c () c µ =, c, c ]. We show below that c P c () is a C 2 function fo c (, c ). Fix c (, c ) and let h be sufficiently small so that c + h (, c ). We then conside Clealy (2.3) ˆP h (q) = 1 + The unique solution of (2.3) is given by ˆP h (q) := P c+h(q) P c (q), q, 1]. h ˆP h (q) = f(q) P c (q)p c+h (q) ˆP h (q) in, 1), ˆPh (1) =. 1 ξ q e q f(s) Pc(s)P c+h (s) ds dξ, q, 1). Let us note that fo q close to 1, f(q) is close to f (1)(q 1) and P c (q) is close to P c(1)(q 1). Hence, fo fixed q, 1), ξ f(s) q Pc(s)P e c+h (s) ds as ξ 1 unifomly in c, h.

8 Y. DU, H. MATSUZAWA AND M. ZHOU It follows that the integand function e ξ q lim ˆP h (q) = h f(s) Pc(s)P c+h (s) ds is unifomly bounded in the set {(q, ξ) : q ξ 1}. Letting h in the expession fo ˆP h (q) we obtain f(s) Pc(s) 2 ds dξ, q, 1). Theefoe (2.4) d dc P c(q) = 1 ξ q e q 1 ξ q e q f(s) Pc(s) 2 ds dξ < fo q, 1). By Lemma 2.1, we easily see fom the above identity that d dc P c(q) is continuous in c fo c (, c ). d Moeove, dc P c(1) = and the continuity of d dc P c(q) in c is unifom in q, 1]. Fom (2.4) we futhe obtain d 2 1 ξ (2.5) dc 2 P f(s) ξ ] c() = 2 e Pc(s) 2 ds f(s) d P c (s) 3 dc P c(s)ds dξ, povided that we can show the integal above is convegent. By (2.4) we can find C 1 > such that d dc P c(s) C 1 fo s, 1]. Fo ϵ (, 1) sufficiently small, thee exist C 2, C 3 > such that Hence, fo ξ 1 ϵ, 1], ξ e f(s) P c (s) 2 C 2(1 s) 1, f(s) P c (s) 3 C 3(1 s) 2 fo s 1 ϵ, 1]. f(s) ξ Pc(s) 2 ds C 1 e ξ f(s) P c (s) 3 d f(s) 1 ϵ Pc(s) 2 ds dc P c(s)ds ξ ] + 1 ϵ f(s) P c (s) 3 ds C 1 C ϵ e C ξ ξ ] 2 1 ϵ (1 s) 1 ds C ϵ + C 3 (1 s) 2 ds C ϵ (1 ξ) C 2 + (1 ξ) C 2 1 ], whee we have used C ϵ to denote vaious positive constants that depend on ϵ. Clealy this implies the convegence of the integal in the fomula fo d2 P dc 2 c () in (2.5). Moeove, by the continuous dependence of P c (q) and d dc P c(q) on c, we find fom (2.5) that d2 P dc 2 c () is continuous in c fo c (, c ). We have thus poved the following esult. 1 ϵ Lemma 2.2. The function c P c () is C 2 fo c (, c ). Define ζ(c, µ) := P c () c µ. Then c ζ(c, µ) = d dc P c() 1 µ < 1 µ <. Hence by the implicit function theoem we find that the unique solution c = c (µ) of ζ(c, µ) =, as a function of µ, is as smooth as ζ, and hence is C 2. Moeove c (µ) = µζ(c (µ), µ) c ζ(c (µ), µ) = µ 2 c (µ) c ζ(c (µ), µ) >,

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 9 and ( c ) (µ) ] = c (µ) c (µ) µ µ µ 2 = µ 2 c 1 (µ) µ c ζ 1 since c ζ > µ 1. Fom 8] we futhe have c (µ) c (µ) lim =, lim = P () >. µ µ µ µ We now fix µ > and denote c = c (µ ). Theefoe fo each ξ (, µ P ()) thee exists a unique µ = µ(ξ) such that (2.6) c (µ(ξ)) µ(ξ) < = ξ µ, ξ µ(ξ) is C 2, µ (ξ) <, µ(c ) = µ. Hee we have used the implicit function theoem and µ c (µ) µ is C 2 to conclude that ξ µ(ξ) is C 2. Let g(ξ) := c (µ(ξ)). Then g is C 2 and g (ξ) = c (µ(ξ))µ (ξ) <. The following identity will play a cucial ole in the estimates of the next section. (2.7) g(c c N t 1 ) g(c ) = g (c )(c N t 1 ) + 1 2 g (θ t )(c 2 Nt 2 ) with θ t (c c Nt 1, c ), whee c N is given by ] 1 N 1 (2.8) c N = 1 g (c ), and g (c ) can be calculated by the following fomula: Lemma 2.3. (2.9) g (c ) = c c µ 2 q c (z) 2 e c z dz. Poof. By definition, g (c ) = c (µ )µ (c ). Using c (µ(ξ)) = µ 1 ξµ(ξ), we obtain Hence By ou ealie calculation, we have Hence c (µ(ξ))µ (ξ) = µ 1 µ(ξ) + ξµ (ξ)], µ (ξ) = g (c ) = Fom (2.4) we obtain Fom 8] we know that µ (c ) = c (µ ) = c (µ ) c (µ ) µ 1 c d dc P c() = 1 c (µ ) µ 1. c µ 2 c d dc P c() µ 1. c=c µ 1 µ(ξ) c (µ(ξ)) µ 1 ξ. 1 = 1 µ 1 c c (µ ) = 1 1 d µ dc P. c() c=c 1 ξ e f(s) Pc(s) 2 ds dξ. P c (s) = P c (q c (z)) = q c(z) with s = q c (z), o equivalently z = qc 1 (s).

1 Y. DU, H. MATSUZAWA AND M. ZHOU Theefoe, making use of the change of vaiable s = q c (z), and the identity f(q c (z)) = q c (z)+cq c(z), we obtain It follows that Hence ξ f(s) P c (s) 2 ds = = d dc P c() = = = = µ c q 1 c (ξ) q 1 c (ξ) = log 1 ξ e 1 f(q c (z)) q c(z) 2 q c(z)dz q c (z) cq c(z) q c(z) dz ] (ξ)) q c() cqc 1 (ξ). q c (q 1 c f(s) Pc(s) 2 ds dξ q c(q c 1 (ξ)) q c() q c(z) q c() e cz q c(z)dz g (c ) = q c(z) 2 e cz dz. e cq 1 c (ξ) dξ c µ 2 q c (z) 2 e c z dz. 3. Shap bounds In this section we give some shap estimates fo h(t). We always assume that f satisfies the conditions of Poposition 1.1. We fix µ > and suppose that (u(t, ), h(t)) is the unique solution of (1.1) with µ = µ. Let c and c N be defined as in the pevious section (see (2.8)), and suppose that speading happens: (3.1) lim h(t) =, lim u(t, ) = 1 unifomly fo in compact subsets of, ). We make these assumptions thoughout this section. Ou aim is to show the following esult. Theoem 3.1. Thee exist positive constants C and T such that (3.2) h(t) (c t c N log t) C fo t T. Moeove, fo any c (, c ), thee exist positive constants M and σ such that (3.3) u(t, ) 1 Me σt fo t >,, ct]. These conclusions will be poved by a sequence of lemmas. 3.1. Rough bounds. We stat with some ough bounds fo u and h. Lemma 3.2. The following conclusions hold: (i) Fo any c (, c ) and δ (, f (1)), thee exist a positive constants T > and M > such that u(t, ) 1 + Me δt, h(t) ct fo t T and, h(t)].

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 11 (ii) Thee exists c (, c ), δ (, f (1)), and T >, M > such that u(t, ) 1 Me δt fo, ct] and t T. Poof. (i) Conside the equation η (t) = f(η) with initial value η() = u L +1. Then η is an uppe solution of (1.1). So u(t, x) η(t) fo all t. Since f(u) < fo u > 1, η(t) is a deceasing function conveging to 1 as t. Hence thee exists T > such that η(t) < 1+ρ and η (t) = f(η) δ(1 η) fo t T. It follows that u(t, x) η(t) 1 + ρe δ(t T ) fo x h(t), t T. Next we take any c (, c ) and show that fo all lage t, h(t) ct. We constuct a lowe solution simila to the poof of Lemma 6.5 in 8]. Let us ecall that fo each c (, c ), thee exists a function q c (z) defined fo z, z c ] such that q cq + f(q) = in, z c ]; q() = q (z c ) = ; q (z) > in, z c ). Moeove, Q c := q c (z c ) < 1 and as c c, Q c 1, z c +, q c q c L (,z c ]). See page 38 of 8] fo details. We now choose c 1, c 2 (c, c ) satisfying c 1 < c 2, f(q c 2 ) >, and define We can find T 1 > such that fo t T 1. Set w(t, ) := { k(t) := z c 2 + c 2 t N 1 c 1 log t. c 1 t c 2 t N 1 c 1 q c 2 (k(t) ), q c 2 (z c 2 ), Since speading happens we can find T 2 > T 1 such that We note that Moeove, by (6.7) in 8], and k(t 1 ) h(t 2 ) log t c 2 t N 1 c 1 log t k(t), c 2 t N 1 c 1 log t. w(t 1, ) u(t 2, ) fo, k(t 1 )] w (t, ) = when c 2 t N 1 c 1 log t. k (t) = c 2 N 1 c 1 t w t w c 2 < µ(q c 2 ) () = µw (t, k(t)) = k (t)(q c 2 ) (k(t) ) (q c 2 ) (k(t) ) + N 1 (q c 2 ) (k(t) ) ( = f(q c 2 N 1 (k(t) )) + N 1 ) (q c 2 ) (k(t) ) c 1 t f(w)

12 Y. DU, H. MATSUZAWA AND M. ZHOU fo ] c 2 t N 1 c 1 log t, k(t) c 1 t, k(t)] and w t w = < f(q c 2 ) = f(w) ] fo, c 2 t N 1 c 1 log t. Since w is C 1 in, the above discussions show that (w(t T 2 + T 1, ), k(t T 2 + T 1 )) is a lowe solution of (1.1) fo t T 2. Hence thee exists some T 3 T 2 such that fo t T 3, and h(t) k(t T 2 + T 1 ) = z c 2 + c 2 (t T 2 + T 1 ) N 1 c 1 log(t T 2 + T 1 ) z c 2 + c 1 (t T 2 + T 1 ) ct u(t, ) w(t T 2 + T 1, ) fo, k(t T 1 + T 2 )], ct]. (ii) Since w(t T 2 + T 1, ) q c 2 (z c 2 ) = Q c 2 > Q c fo ct fo all t T 3, we find fom the above estimates fo u and h that (3.4) h(t) ct, u(t, ) Q c fo ct, t T 3 Since f (1) <, fo any δ (, f (1)) we can find ρ = ρ(δ) (, 1) such that f(u) δ(1 u) (u 1 ρ, 1]), f(u) δ(1 u) (u 1, 1 + ρ]). Since Q c 1 as c c, we may assume that Qc > 1 ρ. Now fo a given domain D we conside a solution ψ = ψ D to the following auxiliay poblem: ψ t ψ = δ(ψ 1), t >, x D, (3.5) ψ Q c, t >, x D, ψ Q c, t =, x D. The function Ψ = Ψ D = e δt (ψ D Q c ) satisfies Ψ t Ψ = δe δt (1 Q c ), t >, x D, (3.6) Ψ, t >, x D, Ψ, t =, x D. Take D = Q ct := {x R N ct x i ct, i = 1, 2,, N} with c = c/ N. Let G(x, t; ξ, τ) be the Geen function fo the poblem (3.6). Fom page 84 of 15] one sees that this Geen function can be expessed as follows: G(x, t; ξ, τ) = N G(x i, t; ξ i, τ) whee G is the Geen function of the one space dimension poblem: Ψ t Ψ xx = g(t, x), t >, ct x ct, Ψ, t >, x = ± ct, Ψ, t =, ct x ct. Thus Ψ Q ct (t, x) = t i=1 δe δτ (1 Q c ) Q ct G(x, t; ξ, τ)dξdτ

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 13 Fo ε (, 1), conside (t, x) R N+1 satisfying x i (1 ε) ct, i = 1, 2,, N, t ε2 c 2 T. 4 Fom the poof of Lemma 6.5 in 8] we find that fo such (t, x), thee exists T 4 T 3 such that fo T T 4, ct ct G(x i, t; ξ i, τ)dξ i 1 4 π e T/2. Hence, fo sufficiently lage T > thee exists M > such that G(x, t; ξ, τ)dξ 1 M e T/2. Q ct Fom the above estimate we obtain Ψ Q ct (t, x) δ(1 Q c ) t e δτ (1 M e T/2 )dτ = (1 Q c )(1 M e T/2 )(e δt 1) fo T T 4, x i (1 ε) ct, i = 1, 2,, N, t ε2 c 2 4 T. Since B ct Q ct B N ct B ct, using (3.4) and a simple compaison agument we obtain Hence (3.7) Fix T T 4. We have ψ Q ct (t, x) ψ BcT (t, x) u(t + T, x ) fo t, x Q ct. u(t + T, x ) ψ Q ct (t, x) fo t >, x Q ct. ψ Q ct (t, x) = Ψ Q ct (t, x)e δt + Q c 1 M e T/2 e δt fo x i ct (1 ε), i = 1, 2,, N, t ε2 c 2 ε2 c 2 4 T. Taking t = 4 T we obtain ( ε2 c 2 ) ψ Q ct 4 T, x 1 M e T/2 e ε2 c 2δT/4. We only focus on small ε > such that ε 2 c 2 δ < 2 so ( ε2 c 2 ) ψ Q ct 4 T, x 1 M e ε2 c 2δT/4 e ε2 c 2 δt/4 = 1 (M + 1)e ε2 c 2 δt/4. This holds fo x i (1 ε) ct, i = 1, 2,, N, T T 4. Thus, by (3.7), fo such T and x, we have ( ε2 c 2 ) u T + T, x 1 (M + 1)e ε2 c 2δT/4. 4 Finally, if we ewite then and t = ε2 c 2 4 T + T T = (1 + ε2 c 2 ) 1 t, 4 u(t, x ) 1 (M + 1)e δt

14 Y. DU, H. MATSUZAWA AND M. ZHOU ( ) 1 fo x i (1 ε) 1 + ε2 c 2 4 ct, i = 1, 2,, N and t T5 whee δ ( ) 1 := ε2 c 2 4 1 + ε2 c 2 4 δ and ( ) 1 T 5 = ε2 c 2 4 T 4 + T 4. This is also tue fo x (1 ε) 1 + ε2 c 2 4 ct. Since this is tue fo any c (, c / N) and fo any small ε >, the above estimate implies the conclusion in (ii). This completes the poof. Lemma 3.3. Fo any c (, c ) thee exist M >, T > and δ (, f (1)) such that u(t, ) 1 M e δ t, h(t) c t M log t fo, ct] and t T. Poof. We fist constuct a lowe solution. Define u(t, ) = (1 Me δt )q c (h(t) ), h(t) = c (t T ) + ct N 1 log t σ c T M(e δt e δt ), g(t) = ct, whee M, δ and c ae given in Lemma 3.2, σ > and T > T (T as in Lemma 3.2) will be chosen late. We will check that (u, g, h) is a lowe solution, that is, ( u t u + N 1 ) (3.8) u f(u) fo t > T, g(t) < < h(t), (3.9) u u fo t T, = g(t), (3.1) (3.11) u =, h (t) µu fo t T, = h(t), h(t ) h(t ), u(t, ) u(t, ) fo g(t ), h(t )]. We fist see that h(t ) = ct h(t ) fom Lemma 3.2. Thus we have fom Lemma 3.2. Similaly we have u(t, ) 1 Me δt u(t, ) fo g(t ), h(t )] fo t T by Lemma 3.2. Clealy u(t, h(t)) =. Next we calculate u(t, g(t)) = u(t, ct) 1 Me δt u(t, ct) = u(t, g(t)) h (t) = c N 1 ct σ M δe δt c σ δ Me δt, u (t, h(t)) = (1 Me δt )q c () = c c µ µu (t, h(t)) = c c Me δt. Hence if we take σ > so that c σ δ, then h (t) µu (t, h(t)) fo t T. Me δt It emains to pove u t ( u + N 1 ) u f(u). Put ζ = h(t). Since u t = δ Me δt q c (ζ) + (1 Me δt )h (t)q c (ζ), u = (1 Me δt )q c (ζ), u = (1 M e δt )q c (ζ),,

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 15 we have fo t T and ( ct, h(t)), ( u t u + N 1 ) u f(u) = δ Me δt q c (ζ) + (1 Me δt )h (t)q c (ζ) (1 Me δt )q c (ζ) + N 1 (1 Me δt )q c (ζ) f((1 Me δt )q c (ζ)) ( = δ Me δt q c (ζ) + (1 Me δt ) c N 1 ) δt σ M δe q c ct (ζ) (1 Me δt )q c (ζ) + N 1 (1 Me δt )q c (ζ) f((1 Me δt )q c (ζ)) = δ Me δt q c (ζ) + (1 Me δt )(c q c (ζ) q c (ζ)) ( σ M δe δt (1 Me δt )q c (ζ) + (1 Me N 1 δt ) f((1 Me δt )q c (ζ)) δ Me δt q c (ζ) σ M δe δt (1 Me δt )q c (ζ) + (1 Me δt )f(q c (ζ)) f((1 Me δt )q c (ζ)). N 1 ) q c ct (ζ) Let us conside the tem (1 Me δt )f(q c (ζ)) f((1 Me δt )q c (ζ)), which is of the fom The mean value theoem implies that (1 + ξ)f(u) f((1 + ξ)u). ξf(u) + f(u) f((1 + ξ)u) = ξf(u) ξf (u + θ ξ,u ξu)u fo some θ ξ,u (, 1). Since < δ < f (1), we can find an η > such that { δ f (u) fo 1 η u 1 + η, (3.12) f(u) fo 1 η u 1. Since q c (ζ) 1 as ζ, thee exists ζ η > such that q c (ζ) 1 η/2 fo ζ ζ η. We may assume that Me δt η/2 fo t T. Fo ζ ζ η, we have ( u t u + N 1 ) u f(u) δ Me δt q c (ζ) σ M δe δt (1 Me δt )q c (ζ) Me { δt f(q c (ζ)) f ( q c (ζ) θ ζ,t Me δt q c (ζ) ) } q c (ζ) = Me δt f(q c (ζ)) σ M δe δt (1 Me δt )q c (ζ) + Me { δt f ( q c (ζ) θ ζ,t Me δt q c (ζ) ) + δ } q c (ζ), fo some θ ζ,t (, 1). Hee we have use the fact that q c (ζ) θ ζ,t Me δt q c (ζ) q c (ζ) Me δt q c (ζ) 1 η and hence f ( q c (ζ) θ ζ,t Me δt q c (ζ) ) + δ.

16 Y. DU, H. MATSUZAWA AND M. ZHOU Fo ζ ζ η, we have ( u t u + N 1 ) u f(u) δ Me δt q c (ζ) σ M δe δt (1 Me δt )q c (ζ) Me { δt f(q c (ζ)) f ( q c (ζ) θ ζ,t Me δt q c (ζ) ) } q c (ζ) = Me δt f(q c (ζ)) σ M δe δt (1 Me δt )q c (ζ) + Me { δt f ( q c (ζ) θ ζ,t Me δt q c (ζ) ) + δ } q c (ζ) δt Me min f(s) σ M δe δt (1 Me δt )q c s 1 (ζ) + Me { δt max f (s) + δ } s 1 = Me { δt min f(s) + max f (s) + δ σ δ(1 Me } δt )q c s 1 s 1 (ζ), fo sufficiently lage σ > and all lage t. Finally we note that we can take T > T so lage that the above holds and ct h(t) fo t T. Thus we have shown that (3.8)-(3.11) hold and (u, g, h) is a lowe solution of (1.1). It follows that Hence u(t, ) u(t, ), h(t) h(t) fo t T and g(t), h(t)]. u(t, ) (1 Me δt )q c (h(t) ) q c (h(t) ) Me δt fo t T and ct h(t). Fo any c (, c ) and any κ (, c c), thee exists T > such that fo t T and, ct], we have h(t) (c c)t N 1 c log t T + ct σ M κt. Since thee exist C > and β > such that q c satisfies q c (z) 1 Ce βz fo z, we thus obtain (3.13) u(t, ) 1 Ce βκt Me δt = 1 M e δ t fo t T and ct, ct], whee δ = min{βκ, δ}. Moeove, if M > (N 1)/ c, then h(t) h(t) = c t N 1 log t c C c t M log t fo all lage t. Thus combined with (3.14) and Lemma 3.2, we find that u(t, ) 1 M e δ t, h(t) c t M log t fo t T and, ct] povided that M and T ae chosen lage enough. This completes the poof of Lemma 3.3. Clealy (3.3) follows diectly fom Lemmas 3.2 and 3.3. Let us note that fom the poof of Lemma 3.3, we have, fo t T and ct, c t M log t], u(t, ) (1 Me δt )q c (c t M log t ).

Since we immediately obtain SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 17 q c (z) 1 M 1 e δ 1z fo z, ) and some M 1, δ 1 >, (3.14) u(t, ) (1 Me δt )(1 M 1 e δ 1(c t M log t ) ) fo t T and ct, c t M log t]. 3.2. Shap bounds. We now make use of the ough bounds fo u and h to obtain shap bounds fo h. We fist impove the estimate fo h(t) in Lemma 3.3. Lemma 3.4. Thee exist C > and T > such that whee c N is given by (2.8). h(t) c t c N log t C fo t T, Poof. With B > a constant to be detemined, and ϕ(z) = ϕ(µ, z) given in (2.1), we set k(t) = c t c N log t + Bt 1 log t, ( v(t, ) = ϕ µ(c c N t 1 ), k(t) ) t 2 log t. We have v(t, k(t)) = t 2 log t < fo t > 1, and v(t, k(t) t 1 ) = ϕ ( µ(c c N t 1 ), t 1) t 2 log t = ϕ (µ, )t 1 + o(t 1 ) > fo all lage t. Moeove, v (t, ) = ϕ (µ(c c Nt 1 ), k(t) ) < fo all t > and (, k(t)]. Theefoe, thee exists a unique k(t) ( k(t) t 1, k(t)) such that v(t, k(t)) = fo all lage t. By the implicit function theoem we know that t k(t) is smooth, and by the mean value theoem we obtain ϕ (µ, ) + o(1) ] k(t) k(t) ] = t 2 log t. Since ϕ (µ, ) = c /µ, we thus obtain (3.15) k(t) k(t) = µ ] c + o(1) t 2 log t fo all lage t. Using v t (t, k(t)) + v (t, k(t))k (t) = we obtain It follows that ϕ µ µ c N t 2 + ϕ k (t) k (t) ] + 1 + o(1) ] 2t 3 log t =. k (t) = k (t) + O(t 2 ) = c c N t 1 Bt 2 log t + O(t 2 ) fo all lage t. We want to show that thee exist positive constants M and T such that (v(t, ), k(t)) satisfies, fo t T and k(t) M log t k(t), (3.16) v(t, k(t)) =, k (t) µ v (t, k(t)), (3.17) v(t, k(t) M log t) u(t + s, k(t + s) M log(t + s)), s >, (3.18) v t v N 1 v f(v).

18 Y. DU, H. MATSUZAWA AND M. ZHOU Moeove, we will show that the above inequalities imply (3.19) k(t) h(t + T 1 ), v(t, ) u(t + T 1, ) fo (k(t) M log t, k(t)) and t T. Clealy the equied estimate fo h(t) follows diectly fom (3.19) and (3.15). By the definition of k(t), we have v(t, k(t)) =. We now calculate v (t, k(t)) = ϕ (µ(c c N t 1 ), k(t) k(t)) = ϕ (µ(c c N t 1 ), ) + ϕ (µ, ) + o(1) ] k(t) k(t) ] = 1 µ (c c N t 1 ) + ϕ (µ, ) + o(1) ] µ c Using ϕ (µ, ) + c ϕ (µ, ) + f(ϕ(µ, )) = and f(ϕ(µ, )) = f() =, we obtain It follows that ϕ (µ, ) = c ϕ (µ, ) = c 2 µ. µ v (t, k(t)) = c c N t 1 + µ c t 2 log t + o(t 2 log t) > c c N t 1 Bt 2 log t + O(t 2 ) = k (t) fo all lage t. ] + o(1) t 2 log t. Hence (3.16) holds. Since c t M log t k(t) M log t ] = (c N + M M ) log t + o(1) > (M/2) log t fo all lage t, povided that M > 2M, we obtain fom (3.14) that u(t, k(t) M log t) (1 Me ( ) δt ) 1 M 1 t δ 1M/2 > 1 t 2 fo all lage t, povided that M > 4/δ 1. We now fix M such that M > max{2m, 4/δ 1 }. Thus u(t + s, k(t + s) M log(t + s)) > 1 (t + s) 2 > 1 t 2 log t > v(t, k(t) M log t) fo all lage t and evey s >. This poves (3.17). Next we show (3.18). We have, with ξ = c c Nt 1, and Hence, v t =ϕ µ (µ(ξ), k(t))µ (ξ)c N t 2 ϕ (µ(ξ), k(t)) k (t) + 2t 3 log t t 3 =O(t 2 ) + ϕ ( c + c N t 1 + Bt 2 log t Bt 2), v (t, ) = ϕ (µ(ξ), k(t)), v (t, ) = ϕ (µ(ξ), k(t)). v t v N 1 v f(v) = O(t 2 ) + ϕ c + c N t 1 + Bt 2 log t Bt 2 N 1 ] ϕ f ( ϕ t 2 log t ) = O(t 2 ) + ϕ g(ξ) g(c ) + c N t 1 + Bt 2 log t Bt 2 N 1 ] g(ξ)ϕ ϕ f ( ϕ t 2 log t ) = O(t 2 ) + ϕ J + f(ϕ) f ( ϕ t 2 log t ),

whee SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 19 J := g(ξ) g(c ) + c N t 1 + Bt 2 log t Bt 2 N 1. Fo k(t) M log t, k(t)], we have k(t) M log t = k(t) M log t + O(t 2 log t) = c t (c N + M) log t + Bt 1 log t + O(t 2 log t) c t M 2 log t fo all lage t, whee M 2 = c N + M. It follows that, fo such, N 1 N 1 c t M 2 log t = N 1 c t + (N 1)M 2 log t c 2 t 2 1 + o(1) ]. Theefoe J g (c )c N t 1 + c N t 1 N 1 c t 1 + B (N 1)M ] 2 c 2 t 2 log t + o(t 2 log t) = B (N 1)M ] 2 c 2 + o(1) t 2 log t > fo all lage t, povided that B is lage enough. We now fix ϵ > small so that f (u) σ < fo u 1 2ϵ, 1 + 2ϵ ]. Then when ϕ(µ(ξ), k(t)) 1 ϵ, 1] we have fo all lage t. Hence in such a case, f(ϕ) f(ϕ t 2 log t) σ t 2 log t O(t 2 ) + ϕ J + f(ϕ) f(ϕ t 2 log t) O(t 2 ) σ t 2 log t < fo all lage t. If ϕ(µ(ξ), k(t)), 1 ϵ ], then we can find σ 1 > such that ϕ σ 1, and hence ϕ J σ 1 B (N 1)M ] 2 c 2 + o(1) t 2 log t. On the othe hand, thee exists σ 2 > such that Thus in this case we have f(ϕ) f(ϕ t 2 log t) σ 2 t 2 log t. O(t 2 ) + ϕ J + f(ϕ) f(ϕ t 2 log t) σ 1 B (N 1)M ] 2 + o(1) t 2 log t + σ 2 t 2 log t + O(t 2 ) < c 2 fo all lage t, povided that B is lage enough. This poves (3.18). We ae now eady to show (3.19). Since as t, h(t) and u(t, ) 1 locally unifomly in, ), we can find T > T such that h(t ) > k(t ), u(t, ) > v(t, ) fo k(t ) M log T, k(t )],

2 Y. DU, H. MATSUZAWA AND M. ZHOU whee T > is a constant such that (3.16), (3.17) and (3.18) hold fo t T. We may now use a compaison agument to conclude that h(t + t) k(t + t), u(t + t, ) v(t + t, ) fo t >, k(t + t) M log(t + t), k(t + t)], which is equivalent to (3.19) with T 1 = T T. Lemma 3.5. Thee exist C > and T > such that whee c N is given by (2.8). h(t) c t c N log t + C fo t T, Poof. With B > and C > constants to be detemined, and ϕ(z) = ϕ(µ, z) given in (2.1), we set ˆk(t) = c t c N log t Bt 1 log t + C, ( v(t, ) = ϕ µ(c c N t 1 ), ˆk(t) ) + t 2 log t. We have v(t, ˆk(t)) = t 2 log t > fo t > 1, and v(t, ˆk(t) + t 1 ) = ϕ ( µ(c c N t 1 ), t 1) + t 2 log t = ϕ (µ, ) + o(1) ] t 1 < fo all lage t. Moeove, v (t, ) = ϕ (µ(c c Nt 1 ), ˆk(t) ) < fo all t > and (, ˆk(t)]. Theefoe, thee exists a unique k(t) (ˆk(t), ˆk(t) + t 1 ) such that v(t, k(t)) = fo all lage t. By the implicit function theoem we know that t k(t) is smooth, and by the mean value theoem we obtain ϕ (µ, ) + o(1) ] k(t) ˆk(t) ] = t 2 log t. Since ϕ (µ, ) = c /µ, we thus obtain (3.2) k(t) ˆk(t) = µ Using v t (t, k(t)) + v (t, k(t))k (t) = we obtain c ] + o(1) t 2 log t fo all lage t. ϕ µ µ c N t 2 + ϕ k (t) ˆk (t) ] 1 + o(1) ] 2t 3 log t =. It follows that k (t) = ˆk (t) + O(t 2 ) = c c N t 1 + Bt 2 log t + O(t 2 ) fo all lage t. We want to show that, by choosing B and C popely, thee exists a positive constant T such that (v(t, ), k(t)) satisfies, fo t T and 1 k(t), (3.21) v(t, k(t)) =, k (t) µ v (t, k(t)), (3.22) v(t, 1) u(t, 1), (3.23) v t v N 1 v f(v), and (3.24) k(t ) h(t ), v(t, ) u(t, ) fo 1, h(t )]. If these inequalities ae poved, then we can apply a compaison agument to conclude that (3.25) k(t) h(t), v(t, ) u(t, ) fo 1, h(t)] and t T.

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 21 Clealy the equied estimate fo h(t) follows diectly fom (3.25) and (3.2). By the definition of k(t), we have v(t, k(t)) =. We now calculate It follows that v (t, k(t)) = ϕ (µ(c c N t 1 ), k(t) ˆk(t)) = ϕ (µ(c c N t 1 ), ) + ϕ (µ, ) + o(1) ] k(t) ˆk(t) ] Hence (3.21) holds. Since = 1 µ (c c N t 1 ) + ϕ (µ, ) + o(1) ] µ c = 1 µ (c c N t 1 ) + c t 2 log t + o(t 2 log t). µ v (t, k(t)) = c c N t 1 µ c t 2 log t + o(t 2 log t) < c c N t 1 + Bt 2 log t + O(t 2 ) = k (t) fo all lage t. ] + o(1) t 2 log t v(t, 1) = ϕ ( µ(c c N t 1 ), 1 ˆk(t) ) + t 2 log t 1 M 1 e δ 11 ˆk(t)] + t 2 log t 1 + t 2 fo all lage t, and by Lemma 3.2, u(t, 1) 1 + Me δt fo all t >, we find that u(t, 1) < v(t, 1) fo all lage t. This poves (3.22). Next we show (3.23). We have, with ξ = c c Nt 1, and Hence, v t =ϕ µ (µ(ξ), ˆk(t))µ (ξ)c N t 2 ϕ (µ(ξ), ˆk(t))ˆk (t) 2t 3 log t + t 3 =O(t 2 ) + ϕ ( c + c N t 1 Bt 2 log t + Bt 2), v (t, ) = ϕ (µ(ξ), ˆk(t)), v (t, ) = ϕ (µ(ξ), ˆk(t)). v t v N 1 v f(v) = O(t 2 ) + ϕ c + c N t 1 Bt 2 log t + Bt 2 N 1 ] ϕ f ( ϕ + t 2 log t ) = O(t 2 ) + ϕ g(ξ) g(c ) + c N t 1 Bt 2 log t + Bt 2 N 1 ] g(ξ)ϕ ϕ f ( ϕ + t 2 log t ) = O(t 2 ) + ϕ Ĵ + f(ϕ) f ( ϕ + t 2 log t ), whee Ĵ := g(ξ) g(c ) + c N t 1 Bt 2 log t + Bt 2 N 1. Fo 1, k(t)], we have N 1 N 1 k(t) = N 1 c t = N 1 ˆk(t) + o(t 1 ) + (N 1)c N log t c 2 t 2 1 + o(1) ].

22 Y. DU, H. MATSUZAWA AND M. ZHOU Theefoe, fo such, Ĵ g (c )c N t 1 + c N t 1 N 1 c t 1 B + (N 1)c ] N c 2 t 2 log t + o(t 2 log t) = B + (N 1)c ] N c 2 + o(1) t 2 log t < fo all lage t. We now fix ϵ > small so that f (u) σ < fo u 1 2ϵ, 1+2ϵ ]. Then fo ϕ(µ(ξ), ˆk(t)) 1 ϵ, 1] we have f(ϕ) f(ϕ + t 2 log t) σ t 2 log t fo all lage t. Hence in such a case, O(t 2 ) + ϕ Ĵ + f(ϕ) f(ϕ + t 2 log t) O(t 2 ) + σ t 2 log t > fo all lage t. If ϕ(µ(ξ), ˆk(t)), 1 ϵ ], then we can find σ 1 > such that ϕ σ 1, and hence ϕ Ĵ σ 1 B + (N 1)c ] N c 2 + o(1) t 2 log t. On the othe hand, thee exists σ 2 > such that Thus in this case we have f(ϕ) f(ϕ + t 2 log t) σ 2 t 2 log t. O(t 2 ) + ϕ Ĵ + f(ϕ) f(ϕ + t 2 log t) σ 1 B + (N 1)c ] N + o(1) t 2 log t σ 2 t 2 log t + O(t 2 ) > c 2 fo all lage t, povided that B is lage enough. This poves (3.23). Finally we show that (3.24) holds if C is chosen suitably. Indeed, we set Then C = h(t ) c T + c N log T + 2T. k(t ) = ˆk(T ) + o(t 1 ) = h(t ) BT 1 log T + 2T + o(t 1 ) > h(t ) + T fo T lage enough. By enlaging T if necessay we have, fo 1, h(t )], while Theefoe v(t, ) v(t, h(t )) = ϕ(µ(c c N T 1 ), h(t ) ˆk(T )) + T 2 log T ϕ(µ(c c N T 1 ), T ) + T 2 log T 1 M 1 e δ 1T + T 2 log T > 1 + T 2, u(t, ) 1 + Me δt. v(t, ) u(t, ) fo 1, h(t )] povided that T is lage enough. This poves (3.24). The poof of the lemma is now complete.

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 23 4. Convegence Thoughout this section we assume that (u, h) is the unique solution of (1.1) with µ = µ >, and speading happens: As t, h(t) and u(t, ) 1 fo in compact subsets of, ). We will pove the following convegence esult. Theoem 4.1. Thee exists a constant ĥ R1 such that { h(t) c t c N log t ]} = and lim ĥ, lim h (t) = c lim u(t, ) q c (h(t) ) L (,h(t)]) =. Again we will pove this theoem by a seies of lemmas. By Lemmas 3.4 and 3.5 we know that thee exist C, T > such that We now denote and define Clealy Moeove, and (v, g) satisfies v t v C h(t) c t c N log t] C fo t T. k(t) = c t c N log t 2C v(t, ) = u(t, + k(t)), g(t) = h(t) k(t), t T. C g(t) 3C fo t T. u = v, u = v, u t = v t (c c N t 1 )v, ] c c Nt 1 + N 1 +k(t) v = f(v), k(t) < g(t), t > T, v(t, g(t)) =, g (t) = µ v (t, g(t)) c + c Nt 1, t > T. 4.1. Limit along a subsequence of t n. Let t n be an abitay sequence satisfying t n > T fo evey n 1. Define Lemma 4.2. Subject to a subsequence, k n (t) = k(t + t n ), v n (t, ) = v(t + t n, ), g n (t) = g(t + t n ). g n G in C 1+ α 2 loc (R 1 ) and v n V 1+α C 2,1+α loc (D n ) whee α (, 1), D n = {(t, ) D : g n (t)}, D = {(t, ) : < G(t), t R 1 }, and (V (t, ), G(t)) satisfies { Vt V c V = f(v ), (t, ) D, (4.1), V (t, G(t)) =, G (t) = µ V (t, G(t)) c, t R1. Poof. By 9] thee exists C > such that < h (t) C fo all t >. It follows that Define c < g n(t) C fo t + t n lage and evey n 1. s = g n (t), w n(t, s) = v n (t, ).

24 Y. DU, H. MATSUZAWA AND M. ZHOU Then (w n (t, s), g n (t)) satisfies (4.2) (w n ) t (w n) ss g n (t) 2 fo k n(t) g n (t) s < 1, t > T t n, and sg n(t) + c c N (t + t n ) 1 + (4.3) w n (t, 1) = fo t > T t n, (4.4) g n(t) = µ (w n ) s (t, 1) g n (t) ] N 1 (wn ) s g n (t)s + k n (t) g n (t) = f(w n) c + c N (t + t n ) 1 fo t > T t n. Fo any given R > and T R 1, using the patial inteio-bounday L p estimates (see Theoem 7.15 in 23]) to (4.2) and (4.3) ove T 1, T + 1] R 1, 1], we obtain, fo any p > 1, w n W 1,2 p (T,T +1] R,1]) C R fo all lage n, whee C R is a constant depending on R and p but independent of n and T. Theefoe, fo any α (, 1), we can choose p > 1 lage enough and use the Sobolev embedding theoem (see 21]) to obtain (4.5) w n 1+α C C 2,1+α R fo all lage n, (T, ) R,1]) whee C R is a constant depending on R and α but independent of n and T. Fom (4.4) and (4.5) we deduce g n C 1+ α 2 (T, )) C 1 fo all lage n, with C 1 a constant independent of T and n. Hence by passing to a subsequence we may assume that, as n, w n W in C α+1 2,1+α loc (R 1 (, 1]), g n G in C 1+ α 2 loc (R 1 ), whee α (, α ). Moeove, using (4.2),(4.3) and (4.4), we find that (W, G) satisfies in the W 1,2 p sense (and hence classical sense by standad egulaity theoy), Define V (t, ) = W (t, W t Wss G(t) 2 (sg (t) + c ) Ws G(t) = f(w ), s (, 1], t R1, W (t, 1) =, G (t) = µ W s (t,1) G(t) c, t R1. G(t) ). We easily see that (V, G) satisfies (4.1) and lim v n V n C 1+α 2,1+α loc (D n) 4.2. Detemine the limit pai (V, G). We show by a sequence of lemmas that G(t) G is a constant, and hence V (t, ) = ϕ( G ). Since C g(t) 3C fo t T, we have C G(t) 3C fo t R 1. By the poof of Lemma 3.5, we have, fo 1 k(t + t n ), g(t + t n )] and t + t n lage, Letting n we obtain =. v n (t, ) ϕ ( µ(c c N (t + t n ) 1 ), 3C ) + (t + t n ) 2 log(t + t n ). V (t, ) ϕ(µ, 3C) fo all t R 1, < G(t).

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 25 Define Then and Lemma 4.3. R = inf { R : V (t, ) ϕ(µ, R) fo all (t, ) D }. R = sup t R 1 G(t). V (t, ) ϕ(µ, R ) fo all (t, ) D C inf G(t) sup G(t) R 3C. t R1 t R 1 Poof. Othewise we have R > sup t R 1 G(t). We ae going to deive a contadiction. Choose δ > such that G(t) R δ fo all t R 1. We deive a contadiction in thee steps. To simplify notations we will wite ϕ() instead of ϕ(µ, ). Step 1. V (t, ) < ϕ( R ) fo all t R 1 and G(t). Othewise thee exists (t, ) D such that V (t, ) = ϕ( R ) ϕ( δ) >. Hence necessaily < G(t ). Since V (t, ) ϕ( R ) in D, and ϕ( R ) satisfies the fist equation in (4.1), we can apply the stong maximum pinciple to conclude that V (t, ) ϕ( R ) in D := {(t, ) : < G(t), t t }, which clealy contadicts with the assumption that G(t) R δ. Step 2. M := inf t R 1 ϕ( R ) V (t, ) ] > fo (, R δ]. Hee we assume that V (t, ) = fo > G(t). Othewise thee exists (, R δ] such that M =, since the definition of R implies M fo all R δ. By Step 1 we know that M is not achieved at any finite t. Theefoe thee exists s n R 1 with s n such that Define ϕ( R ) = lim n V (s n, ). (V n (t, ), G n (t)) = (V (t + s n, ), G(t + s n )). Then the same agument used in the poof of Lemma 4.2 shows that, by passing to a subsequence, (V n, G n ) (Ṽ, G) with (Ṽ, G) satisfying { Ṽt Ṽ c Ṽ = f(ṽ ), < < G(t), t R 1, (4.6) Moeove, Ṽ (t, G(t)) =, t R 1. (4.7) Ṽ (t, ) ϕ( R ), G(t) R δ, Ṽ (, ) = ϕ( R ) >. Since ϕ( R ) satisfies (4.6) with G(t) eplaced by R, we can apply the stong maximum pinciple to conclude, fom (4.7), that Ṽ (t, ) ϕ( R ) fo t, G(t), which is clealy impossible. Step 3. Reaching a contadiction. Choose ϵ > small and R < lage negative such that Then choose ϵ (, ϵ ) such that ϕ( R ) 1 ϵ fo R, f (u) < fo u 1 2ϵ, 1 + 2ϵ ]. ϕ(r R + ϵ) ϕ(r R ) M R, ϕ( R + ϵ) 1 2ϵ fo R.

26 Y. DU, H. MATSUZAWA AND M. ZHOU We conside the auxiliay poblem V t V c V = f(v ), t >, < R, (4.8) V (t, R ) = ϕ(r R + ϵ), t >, V (, ) = 1, < R. Since the initial function is an uppe solution of the coesponding stationay poblem of (4.8), its unique solution V (t, ) is deceasing in t. Clealy V (t, ) := ϕ( R + ϵ) is a lowe solution of (4.8). It follows fom the compaison pinciple that Hence Moeove, V satisfies 1 V (t, ) ϕ( R + ϵ) fo all t >, < R. V () := lim V (t, ) ϕ( R + ϵ), < R. (4.9) V c V = f(v ) in (, R ), V ( ) = 1, V (R ) = ϕ(r R + ϵ). Wite ψ() = ϕ( R + ϵ). We notice that ψ() also satisfies (4.9). Moeove 1 2ϵ ψ() V () 1 fo (, R ]. Hence W () := V () ψ() and thee exists c() < such that f(v ()) f(ψ()) = c()w () in (, R ]. Theefoe W c W = c()w in (, R ), W (R ) =, and by the maximum pinciple we deduce, fo any R < R, W () W (R) fo R, R ]. Letting R we deduce W () in (, R ]. It follows that W. Hence V () ψ() = ϕ( R + ϵ). We now look at V (t, ), which satisfies the fist equation in (4.8), and fo any t R 1, V (t, ) 1, V (t, R ) ϕ(r R ) M R ϕ(r R + ϵ). Theefoe we can use the compaison pinciple to deduce that V (s + t, ) V (t, ) fo all t >, < R, s R 1. O equivalently V (t, ) V (t s, ) fo all t > s, < R, s R 1. Letting s we obtain (4.1) V (t, ) V () = ϕ( R + ϵ) fo all < R, t R 1. By Step 2 and the continuity of M in, we have If ϵ 1 (, ϵ] is small enough we have M σ > fo R, R δ]. ϕ( R + ϵ 1 ) ϕ( R ) σ fo R, R δ], and hence V (t, ) ϕ( R + ϵ 1 ) σ M fo R, R δ], t R 1. Theefoe we can combine with (4.1) to obtain V (t, ) ϕ( R + ϵ 1 ) fo (, R δ], t R 1, fo all small ϵ 1 (, ϵ), which contadicts the definition of R. The poof is now complete.

SPREADING SPEED AND PROFILE FOR NONLINEAR STEFAN PROBLEMS 27 Lemma 4.4. Thee exists a sequence {s n } R 1 such that G(t + s n ) R, V (t + s n, ) ϕ( R ) as n unifomly fo (t, ) in compact subsets of R 1 (, R ]. Poof. Thee ae two possibilities: (i) R = sup t R 1 G(t) is achieved at some finite t = s, (ii) R > G(t) fo all t R 1 and G(s n ) R along some unbounded sequence s n. In case (i), necessaily G (s ) =. Since V (t, ) ϕ( R ) fo G(t) and t R 1, with V (s, G(s )) = ϕ(g(s ) R ) = ϕ() =, we can apply the stong maximum pinciple and the Hopf bounday lemma to conclude that V (s, G(s )) > ϕ () unless V (t, ) ϕ( R ) in D = {(t, ) : G(t), t s }. On the othe hand, we have V (s, G(s )) = µ 1 G (s ) + c ] = µ 1 c = ϕ (). Hence we must have V (t, ) ϕ( R ) and G(t) R in D. Using the uniqueness of (4.1) with a given initial value, we conclude that V (t, ) ϕ( R ) fo all G(t) and t R 1. Thus the conclusion of the lemma holds by taking s n s. In case (ii), we conside the sequence V n (t, ) = V (t + s n, ), G n (t) = G(t + s n ). By the same easoning as in the poof of Lemma 4.2, we can show that, by passing to a subsequence, V n Ṽ in C 1+α 2,1+α loc (D), G n G in C 1 loc (R1 ) and (Ṽ, G) satisfies (4.1), whee D := {(t, ) : < G(t), t R 1 }. Moeove, G(t) R, G() = R. Hence we ae back to case (i) and thus Ṽ (t, ) ϕ( R ) in D, and G R. The conclusion of the lemma now follows easily. By the poof of Lemma 3.4, we have v n (t, ) ϕ ( µ(c c N (t + t n ) 1 ), C ) (t + t n ) 2 log(t + t n ) fo k(t + t n ) k(t + t n ) M log(t + t n ), k(t + t n ) k(t + t n )] and t + t n lage. Letting n we obtain V (t, ) ϕ(µ, C) fo all t R 1, < G(t). Define R = sup { R : V (t, ) ϕ(µ, R) fo all (t, ) D }. Then V (t, ) ϕ(µ, R ) fo all (t, ) D and C R inf G(t) sup G(t) R 3C. t R1 t R 1 Lemma 4.5. R = inf t R 1 G(t), and thee exists a sequence { s n } R 1 such that G(t + s n ) R, V (t + s n, ) ϕ( R ) as n unifomly fo (t, ) in compact subsets of R 1 (, R ]. Poof. The poof uses simila aguments to those used to pove Lemmas 4.3 and 4.4, and we omit the details.

28 Y. DU, H. MATSUZAWA AND M. ZHOU Lemma 4.6. R = R and hence G(t) G is a constant, which implies V (t, ) = ϕ( G ). Poof. Ague indiectly we assume that R < R. Set ϵ = (R R )/4. We show next that thee exists T ϵ > such that (4.11) G(t) R ϵ and G(t) R ϵ fo t T ϵ, which implies R R 2ϵ. This contadiction would complete the poof. To pove (4.11), we use Lemmas 4.4 and 4.5, and a modification of the agument in section 3.3 of 11]. Indeed, by using Lemma 4.4 and constucting a suitable lowe solution we can show that thee exists n 1 = n 1 (ϵ) lage such that G(t) R ϵ fo all t s n1. Similaly we can use Lemma 4.5 and constuct a suitable uppe solution to show that G(t) R ϵ fo all t s n2 with n 2 = n 2 (ϵ) lage enough. Hence (4.11) holds fo t T := max{s n1, s n2 }. Fo completeness, the detailed constuctions of the above mentioned uppe and lowe solutions ae given in the Appendix at the end of the pape. 4.3. Convegence of h and u. Lemma 4.7. Thee exist a constant C > and a function ξ C 1 (R 1 +) such that ξ(t) C fo all t >, { lim h(t) c t c N log t + ξ(t) ]} =, lim ξ (t) =, and lim u(t, ) q c (h(t) ) L (,h(t)]) =. Poof. By Lemmas 4.2 and 4.6, we find that fo any sequence t n, by passing to a subsequence, h(t + t n ) k(t + t n ) G in C 1+ α 2 loc (R 1 ). Hence h (t + t n ) c in Cα/2 loc (R1 ). We now define U(t, ) = u(t, + h(t)) fo t >, h(t), ], and It is easily checked that (4.12) (U n ) t h n(t) + U n (t, ) = U(t + t n, ), h n (t) = h(t + t n ). ] N 1 +h n (t) (U n ) (U n ) = f(u n ), t > t n, ( h n (t), ], U n (t, ) =, (U n ) (t, ) = h n(t)/µ, t > t n. By the same easoning as in the poof of Lemma 4.2, we can use the paabolic egulaity to (4.12) plus Sobolev embedding to conclude that, by passing to a futhe subsequence, as n, U n U in C 1+α 2,1+α loc (R 1 (, ]), and U satisfies, in view of h n(t) c, { Ut c U U = f(u), t R 1, (, ], U(t, ) =, U (t, ) = c /µ, t R 1. This is equivalent to (4.1) with V = U and G =. Hence we may epeat the agument in Lemmas 4.2-4.5 to conclude that U(t, ) ϕ(µ, ) fo (t, ) R 1 (, ]. Thus we have poved that, as n, u(t + t n, + h(t + t n )) q c ( ) in C 1+α 2,1+α loc (R 1 (, ]).