The first step of the construction of local in time solutions of the nonisothermal evolution problem consists in the analysis of the isothermal problem with ϑ = const. This has been done for the Maxwell–Boltzmann and Fermi–Dirac cases in [4] and [3, Sec. 3], respectively. In the latter publication, solutions have been constructed as limits of solutions of approximated parabolic problems where the Poisson equation has been replaced by the penalized parabolic equation
1
kϕt−∆ϕ+u= 0, k >0. (4.1) Then, the approach via energy estimates and the passage to the limitk→ ∞has led to the solution of the original parabolic-elliptic system. The next step is an analysis of the problem with a given (continuous) temperatureϑ=ϑ(t)∈(0,∞), t∈[0, T], see, e.g., [10] and for a slightly another approach [6]. The final step of the construction of solutions for the nonisothermal evolution problem is to look for a fixed point of the operator
T:ϑ−→Θ. (4.2)
Hereϑ,Θ∈C[0, T], Θ(t) is an instantaneous temperature determined by an energy relation involving both temperatures: the old oneϑ(t) (appearing implicitly inn, ϕthat solve (1.1)–(1.2)) and the new one Θ(t) (appearing in the pressure term, cf. (4.3) below). Usually, this needs Schauder type arguments, so one should prove invariance (i.e.,a priori estimates onϑ) and compactness properties (e.g., a bound on ˙Θ) of the operatorT. Here, we present thea priori estimates onϑ. The other details of the construction will appear in a more comprehensive forthcoming paper.
Evidently, assumptions guaranteeing (global in time) existence of solutions of the evolution problem are, in a sense, complementary to those implying finite time blow up forD∗= 1, e.g., assumptions on negativity ofE, cf. [8, Sec. 5].
The theorem below covers the case of three-dimensional nonisothermal Fermi–
Dirac model.
Theorem 4.1. Assume that the convex functionP ∈C1 satisfies (i) P(s)/s1+2/dε >0,
(ii) lim infs0P(s)/s >0.
Nonlinear Diffusion 115 Moreover, let the data satisfy the conditionW(0)<0 and
(iii) W(0)/M <lim infs→∞R(s)/s≡, where R(s) = sH(s)−d
2+ 1
P(s) is the entropy density in (3.2). Then the temperatureϑ for local in time solution satisfies the a priori estimatea≤ϑ≤b for some constants0< a < b <∞.
Proof. For ϑ ∈ C([0, T]; (0,∞)) we consider the map Θ := T(ϑ) as the new temperature defined for each momentt∈[0, T] by the energy relation
E=d 2
Ω
Θ1+d/2P(nΘ−d/2) dx+1 2
Ω
nϕdx. (4.3)
Thanks to the assumption (i) the operatorT is well defined by the argument in Remark 2.1. First, we prove thatT(ϑ) is bounded from above uniformly inϑ. Using the lower estimate of the pressure (ii), i.e., P(s)≥c1s for some c1>0, and the estimate of the potential energy (3.8), the upper bound for the new temperature Θ follows. Namely,
3 2E+C
2M1+ν ≥E−1 2
Ω
nϕdx= d 2
Ω
Θ1+d/2P(nΘ−d/2) dx≥d 2ΘM c1 holds, and thus we arrive at Θ≤b:= (dc1)−13E
M +CMν .
Now, we prove the second part of the claim, i.e., we estimate the temperature ϑ=T(ϑ) from below for initial data satisfying (iii), i.e., (1−δ)W(0)/M < for someδ >0. TakingK=K(δ) sufficiently large to haveR(s)/s >(1−δ)W(0)/M for alls≥K we obtain
0> W(0)≥W(t) =
{ϑd/n2≥K}
+
{ϑd/n2<K}
ϑd/2R
n ϑd/2
dx
≥ M(1−δ)W(0)/M−C(δ)|Ω|ϑd/2
with−C(δ)≤inf0≤s≤KR(s) for someC(δ)>0. This leads to the required bound forϑ:ϑ≥a >0, sinceϑd/2>−δW(0)/(C(δ)|Ω|)>0.
Observe that if = lims→∞R(s)/s, then =−d2lims→∞s1+2/d
s−2/dH(s) . Moreover, the conditionW(0)<0 is satisfied for initial data ifM2/d ϑ0 holds.
Acknowledgment
The preparation of this paper was partially supported by the KBN (MNI) grant 2/P03A/002/24.
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Piotr Biler and Robert Sta´nczy
Mathematical Institute, University of Wroclaw pl. Grunwaldzki 2/4, PL-50-384 Wroclaw, Poland
e-mail:biler@math.uni.wroc.pl, stanczr@math.uni.wroc.pl
International Series of Numerical Mathematics, Vol. 154, 117–124 c 2006 Birkh¨auser Verlag Basel/Switzerland
Existence, Uniqueness and an Explicit Solution for a One-Phase Stefan Problem for
a Non-classical Heat Equation
Adriana C. Briozzo and Domingo A. Tarzia
Abstract. Existence and uniqueness, local in time, of the solution of a one- phase Stefan problem for a non-classical heat equation for a semi-infinite material is obtained by using the Friedman-Rubinstein integral representa- tion method through an equivalent system of two Volterra integral equations.
Moreover, an explicit solution of a similarity type is presented for a non- classical heat source depending on time and heat flux on the fixed facex= 0.
Mathematics Subject Classification (2000).Primary 35R35, 80A22, 35C05; Sec- ondary 35K20, 35K55, 45G15, 35C15.
Keywords.Stefan problem, Non-classical heat equation, Free boundary prob- lems, Similarity solution, Nonlinear heat sources, System of Volterra integral equations.
1. Introduction
The one-phase Stefan problem for a semi-infinite material is a free boundary prob- lem for the classical heat equation which requires the determination of the tem- perature distributionuof the liquid phase (melting problem) or of the solid phase (solidification problem), and the evolution of the free boundaryx=s(t). Phase- change problems appear frequently in industrial processes and other problems of technological interest [2, 4, 6, 9, 12]. A large bibliography on the subject was given in [20].
Non-classical heat conduction problem for a semi-infinite material was studied in [3, 5, 10, 22, 23], e.g., problems of the type
ut−uxx=−F(ux(0, t)), x >0, t >0, u(0, t) = 0, t >0
u(x,0) =h(x), x >0
(1.1)
It was supported by CONICET PIP No. 3579 and ANPCYT PICT No. 03-11165.
where h(x), x > 0, andF(V), V ∈ R, are continuous functions. The functionF, henceforth referred as control function, is assumed to fulfill the following condition
(H1) F(0) = 0.
As it was observed in [22, 23] the heat fluxw(x, t) =ux(x, t) for problem (1.1) satisfies a classical heat conduction problem with a nonlinear convective condition atx= 0,which can be written in the form
⎧⎨
⎩
wt−wxx= 0, x >0, t >0, wx(0, t) =F(w(0, t)), t >0, w(x,0) =h(x)≥0, x >0.
(1.2) The literature concerning problem (1.2) has constantly increased from the appearance of the papers [13, 15, 17]. In [21] a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material was presented. The free boundary problem consists in determining the temperatureu=u(x, t) and the free boundaryx=s(t) with a control functionF which depends on the evolution of the heat flux at the boundaryx= 0,satisfying the following conditions
⎧⎪
⎪⎨
⎪⎪
⎩
ut−uxx=−F(ux(0, t)), 0< x < s(t),0< t < T , u(0, t) =f(t)≥0, 0< t < T ,
u(s(t), t) = 0, ux(s(t), t) =−s. (t) , 0< t < T ,
u(x,0) =h(x)≥0, 0≤x≤b=s(0) (b >0).
(1.3)
In Section 2 we present a result on the local existence and uniqueness in time of the solution of the one-phase Stefan problem (1.3) for a non-classical heat equation with temperature boundary condition at the fixed face x = 0. First, we prove that the free boundary problem (1.3) is equivalent to a system of two Volterra integral equations (2.4)–(2.5) [8, 14] following the Friedman-Rubinstein’s method given in [7, 18](see also [19]). Then, we prove that the problem (2.4)–(2.5) has a unique local solution in time by using the Banach contraction theorem.
In Section 3 we show an explicit solution of a similarity type for a one-phase Stefan problem for a non classical control functionF which depends on time and heat flux on the fixed facex= 0.