Theorem 5.2 (Finite speed of propagation). Let conditions (1.2), (1.3),(1.4)and the conditions of Theorem5.1 be fulfilled. Assume that either
c0= 0, and 2< p−≤p(x, t), (5.3) or
c0>0, σ+ < p−, max 1, 2n n+ 2
< p−≤p+≤p(x, t)≤2, (5.4) and thatf = 0 in the cylinderQρ0(x0) =Bρ0(x0)×(0, T). Then every local weak solutionu(z)of equation(1.1)in Qρ0(x0), satisfying(5.1), possesses the property of finite speed of propagation: u(x, t) = 0 in x∈ Bρ(t)(x0) with 0 ≤ t ≤t∗ < T andρ(t)given by the formula
ρ1+β(t) =ρ1+β0 −CtλD1−ν(ρ0, t) (5.5) with some positive constants C, ν, and λ, β which depend on the constants in conditions(1.2),(1.3) and(1.4).
Remark 5.1. Since the functionρ(t) defined by(5.5) is monotone decreasing, the setBρ(t)(x0)is nonempty for small t.
Let us now assume that there existsρ1>0 such thatBρ1(x0)⊂Ω and that for someρ0∈(0, ρ1)
u0(x)≡0 x∈Bρ0(x0), f(x, t)≡0 in Qρ0(x0), (5.6) u022,Bρ(x0)+f22,Qρ(x0)≤ε(ρ−ρ0)1/(1+ −ν), (5.7) for all ρ ∈ [ρ0, ρ1], D(ρ1, T) < ∞, with the positive constant ν defined below, and someε > 0. This assumption means that the functions u0(x) and f(z) are sufficiently “flat” near the boundaries of their supports.
Theorem 5.3 (The waiting time effect). Let conditions (5.3) or (5.4)) and (5.6), (5.7) hold, and the conditions of Theorem 5.1 be fulfilled. Then every weak local solution u(z)of equation (1.1) possesses the waiting time property: there exists a positive constantt∗≤T such that u(x, t) = 0 inBρ0(x0)×[0, t∗].
Sketch of proof. For the sake of simplicity we consider the case (5.3) of Theorem 5.2 and assume thatf ≡0. Let us introduce the energy functions
E(ρ, t) = t 0
Bρ(x0)
|∇u|p(z)dz, b(ρ, t) =u(ã, ι)22,Bρ(x0), (5.8)
E(ρ, t) = max
τ≤tE(ρ, τ), b(ρ, t) = max
τ≤tu(ã, τ)22,Bρ(x0) (5.9) for which
Eρ= t 0
Sρ(x0)
|∇u|pdxdt, Et=
Bρ(x0)
|∇u|pdx. (5.10)
We recall that due to the regularity of weak solutions stated in Theorem 5.1 the functionsEt, Eρ andEtρare well defined in the corresponding functional spaces.
Let us consider first the cylinderBρ0(x0)×(0, T∗), ρ0>0, T∗>0 assuming that
p+−p− ≤ε(ρ0, T∗) (5.11)
with a sufficiently smallε >0. Using the interpolation inequalities (see [4], formulas (5.15), (5.22) withσ±= 2), and then following the proof of [8, Theorem 2.1, p. 133]
we estimate the right-hand sideI(ρ, t) of (5.2) as follows:
|I(ρ, t)| ≤C t 0
max
Etρ(p+−1)/p+, Etρ(p−−1)/p−
up+,Sρdt, (5.12)
up+,Sρ(x0)≤C(∇up−,Bρ(x0)+ρ−δb12)θb1−θ2
≤C
max
E1/pt +, Et1/p−
+ρ−δb12 θ
b1−θ2 ,
(5.13) where
θ=p− p+
n(p+−2) + 2
n(p−−2) + 2p− <1, δ= n(p−−2) + 2p−
2p− >1. (5.14) Substituting (5.12), (5.13) into (5.2) we arrive at the inequality
b+E≤Cb
1−θ2
t 0
max E1−
p+1
tρ , E1−
p−1
tρ max E
p+θ
t , E
p−θ
t
] +ρ−δθbθ2
dτ.
Not loosing generality we may assume thatE(ρ0, T)≤D(ρ0, T)≤1 andT ≤1.
Applying the integral representations t
0
Etρdt=Eρ(ρ, t),
t 0
Etdt=E(ρ, t), and using H¨older’s inequality, we derive the inequality
b+Eà
≤Ctκρ−δθmax
E1ρ−1/p+, Eρ1−1/p−
, with the exponents
à= 1− θ
p+ −1−θ
2 , κ= min 1
p+
1−θp+ p−
,1−θ
p−
.
SinceE≤1, this inequality leads to the ordinary nonlinear differential inequality for the energy functionE:
Eν ≤ b+Eν
≤Ctλ∗ρ−βEρ (5.15) with the exponents of nonlinearity
ν = p− p−−1
1 2+θ
p+−2 2p+
<1, λ= κp+
p+−1, β=δθ p+ p+−1. The function E is considered as a function of the variable ρ depending on t as a parameter. Notice that due to condition (5.11) on the oscillation of p(z) the
Parabolic Equations with Nonstandard Growth Conditions 43 inequalityà <1 with variablep(z) immediately follows ifà <1 in the special case when p+ =p−, which much easier to check. The required estimate (5.5) follows now after integration of the differential inequality in the limits (ρ, ρ0). To complete the proof, we take a ball of an arbitrary radiusρ0, take a finite covering of this ball with balls of small radiusρ such that the oscillation condition (5.11) is fulfilled, and then repeat the previous arguments in every of these balls.
In the case of Theorem 5.3 the same proceeding leads to the nonhomogeneous ordinary differential inequality
Eν≤Ctλ∗ρ−βEρ+ε[ρ−ρ0]
1−νν
+ , ρ∈(ρ0, ρ1). (5.16) The analysis of this inequality is based on [8, Ch.1, Lemma 2.4 ] (see also [4,
Lemma 5.3]).
Remark 5.2. The conclusions about the space-and-time localization properties of solutions to problem(1.1)are based on the analysis of the nonlinear ordinary dif- ferential inequalities for the energy functions. When dealing with these inequalities we always reduced them, by means of some suitable assumptions, to the nonlinear inequalities with constant exponents of nonlinearity, which are already studied(see, for instance, inequality(4.8)and its counterpart(4.9)). The study of the properties of functions satisfying the nonlinear ordinary differential inequalities with variable exponents of nonlinearity is still an open question.
References
[1] E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), pp. 121–140.
[2] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Rational Mech. Anal., 164 (2002), pp. 213–259.
[3] E. Acerbi, G. Mingione, and G.A. Seregin,Regularity results for parabolic sys- tems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 21 (2004), pp. 25–60.
[4] S. Antontsev and S. Shmarev,Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Non- linear Analysis Serie A: Theory and Methods (to appear).
[5] S. Antontsev and S. Shmarev, Existence and uniqueness of solutions of de- generate parabolic equations with variable exponents of nonlinearity, Preprint 3, Departamento de Matem´atica, Universidade da Beira Interior, 2004, 16 pp., www.ubi.pt/externos/noe.html. (To appear in “Fundamental and Applied Mathe- matics”, 2005.)
[6] ,A model porous medium equation with variables exponent of nonlinearity: ex- istence, uniqueness and localization properties of solutions, Nonlinear Analysis Serie A: Theory and Methods, 60 (2005), pp. 515–545.
[7] , On localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneracy, Siberian Math. Journal, 46(5) (2005), pp. 963–984.
[8] S. Antontsev, J.I. D´ıaz, and S. Shmarev, Energy Methods for Free Bound- ary Problems:Applications to Non-linear PDEs and Fluid Mechanics, Birkh¨auser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.
[9] S. Antontsev and V.V. Zhikov, Higher integrability for parabolic equations of p(x, t)-Laplacian type, Advances in Differential Equations, 10(9), pp. 1053–1080, (2005).
[10] L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(ã) and Wk,p(ã), Preprint 22/2002-15.07.2002, Albert-Ludwigs- University Freiburg, 2002. pp.1–13.
[11] D. Edmunds and J. R´akosnˇik,Sobolev embeddings with variable exponent, Studia Math., 143(3) (2000), pp. 267–293.
[12] P. Harjulehto and P. H¨ast¨o, An overview of variable exponent Lebesgue and Sobolev spaces, in Future trends in geometric function theory, vol. 92 of Rep. Univ.
Jyv¨askyl¨a Dep. Math. Stat., Univ. Jyv¨askyl¨a, Jyv¨askyl¨a, 2003, pp. 85–93.
[13] E. Henriques and J.M. Urbano, Intrinsic scaling for PDE’s with an expo- nential nonlinearity, Preprint 04-33, Universidade de Coimbra, Departamento de Matem´atica, 2004.
[14] O. Kov´aˇcik and J. R´akosn´ık,On spacesLp(x) andWk,p(x), Czechoslovak Math.
J., 41 (1991), pp. 592–618.
[15] O. A. Ladyˇzenskaja, V.A. Solonnikov, and N.N. Ural’tseva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R.I., 1967. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23.
[16] J. Musielak,Orlicz spaces and modular spaces, vol. 1034 of Lecture Notes in Math- ematics, Springer-Verlag, Berlin, 1983.
S. Antontsev and S. Shmarev Departamento de Matem´atica
Universidade da Beira Interior, Portugal e-mail:anton@ubi.pt
S. Shmarev
Departam´ento de Matematicas Universidad de Oviedo, Espa˜na
e-mail:shmarev@orion.ciencias.uniovi.es
International Series of Numerical Mathematics, Vol. 154, 45–54 c 2006 Birkh¨auser Verlag Basel/Switzerland
Parabolic Systems with the
Unknown Dependent Constraints Arising in Phase Transitions
Masayasu Aso, Michel Fr´ emond and Nobuyuki Kenmochi
Abstract. We consider a system of nonlinear parabolic PDEs which includes a constraint on the time-derivative depending on the unknowns. This system is a mathematical model for irreversible phase transitions. In our phase transition model, the constraint p := p(θ, w) is a function of the temperature θ and the order parameter (state variable) w and it is imposed on the velocity of the order parameter, for instance, in such a way that p(θ, w) ≤ wt ≤ p(θ, w) + (a positive constant). We give an existence result of the problem.
Mathematics Subject Classification (2000).Primary 35K45; Secondary 35K50.
Keywords.System of nonlinear parabolic PDEs, irreversible phase transitions.
1. Introduction
The irreversible phase change is very often observed in solid-liquid systems, for instance, in the solidification process of eggs; in fact, once eggs are solidified in high temperature, their states never return to raw ones even if they are put in cold water. It is called that the phase transition is irreversible.
In this paper we consider the following system:
θt+wt−ν∆θ=h(x, t) in Q:= Ω×(0, T), (1.1) wt+α(wt−p(θ, w))−κ∆w+β(w)f(θ, w) inQ, (1.2)
∂θ
∂n= ∂w
∂n = 0 on Σ := Γ×(0, T), (1.3) θ(ã,0) =θ0, w(ã,0) =w0on Ω, (1.4) where Ω is a bounded domain inR3with smooth boundary Γ, 0< T <∞;ν and κare positive constants;αandβ are maximal monotone graphs in RìR;p(ã,ã) is a function ofC2-class onRìRandf(ã,ã) is a Lipschitz continuous function on R×R. Moreoverhis a function onQ, andθ0 and w0 are functions on Ω, which are prescribed as the data. We denote by (P) the system of (1.1)–(1.4).
In this paper, we suppose forα,β,p(ã,ã) andf(ã,ã) that
(A) 0∈α(0),D(α) = [0, N0] or [0, N0) for some finite positive numberN0, where D(α) is the domain ofα.
(B) β is the subdifferential of the indicator function of the interval (−∞,1], namely
β(r) =
⎧⎨
⎩
∅ forr >1, [0,∞) forr= 1, {0} forr <1.
(C) All of the first- and second-order partial derivatives ofpare continuous and bounded onR×R, p≥0 onR×Rand
p(θ, w) = 0, ∀θ∈R, ∀w∈Rwithw≥1; (1.5) note here that 1 = supD(β).
(D) f is Lipschitz continuous and bounded onR×R.
We note here that the term α(wt−p(θ, w)) requires automatically 0 ≤ p(θ, w) ≤ wt ≤ p(θ, w) +N0, which is a velocity constraint depending on the unknownsθ and w; in the context of irreversible phase transition in solid-liquid systems, the unknownθ is temperature andw, 0≤w≤1, is the volume fraction of solid in the system under consideration. In this paper we give an existence result for problem (P). We refer for related works on irreversible phase transition, for instance, to [7, 9] in the case of prescribed constraints.