phase-field models with hysteresis in one-dimensional thermoviscoplasticity

PHASE-FIELD MODELS WITH HYSTERESIS INONE-DIMENSIONAL THERMOVISCOPLASTICITY∗

PAVEL KREJCI† , JURGEN SPREKELS‡ , AND ULISSE STEFANELLI§

SIAM J. MATH. ANAL. c© 2002 Society for Industrial and Applied MathematicsVol. 34, No. 2, pp. 409–434

Abstract. This paper introduces a combined one-dimensional model for thermoviscoplastic be-havior under solid-solid phase transformations that incorporates the occurrence of hysteresis effects inboth the strain-stress law and the phase transition described by the evolution of a phase-field (whichis usually closely related to an order parameter of the phase transition). Hysteresis is accounted forusing the mathematical theory of hysteresis operators developed in the past thirty years. The modelextends recent works of the first two authors on phase-field models with hysteresis to the case whenmechanical effects can no longer be ignored or even prevail. It leads to a strongly nonlinear coupledsystem of partial differential equations in which hysteresis nonlinearities occur at several places, evenunder time and space derivatives. We show the thermodynamic consistency of the model, and weprove its well-posedness.

Key words. phase-field systems, phase transitions, hysteresis operators, thermoviscoplasticity,thermodynamic consistency

AMS subject classifications. 34C55, 35K60, 47J40, 74K05, 74N30, 80A22

PII. S0036141001387604

1. Introduction and physical motivation. In this paper, we study initial-boundary value problems for systems of partial differential equations of the form

ρutt − µuxxt = σx + f(x, t) a.e. in ΩT ,(1.1)

σ = H1[ux, w] + θH2[ux, w] a.e. in ΩT ,(1.2)

(CV θ + F1[ux, w]

)t− κθxx = µu2

xt + σuxt + g(x, t, θ) a.e. in ΩT ,(1.3)

νwt = −ψ a.e. in ΩT ,(1.4)

ψ = H3[ux, w] + θH4[ux, w] a.e. in ΩT ,(1.5)

u(·, 0) = u0, ut(·, 0) = u1, θ(·, 0) = θ0, w(·, 0) = w0 a.e. in Ω,(1.6)

u(0, t) = 0, µ uxt(1, t) + σ(1, t) = 0, θx(0, t) = θx(1, t) = 0(1.7)

a.e. in (0, T ),

∗Received by the editors April 9, 2001; accepted for publication (in revised form) April 5, 2002;published electronically December 3, 2002.

http://www.siam.org/journals/sima/34-2/38760.html†Mathematical Institute, Academy of Sciences of the Czech Republic, Zitna 25, CZ–11567 Praha

1, Czech Republic ([email protected]).‡Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin,

Germany ([email protected]).§Universita degli Studi di Pavia, Dipartimento di Matematica, Via Ferrata 1, I–27100 Pavia, Italy

([email protected]).

409

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410 P. KREJCI, J. SPREKELS, AND U. STEFANELLI

where Ω := (0, 1), T > 0 denotes some final time, and where Ωt := Ω × (0, t) fort ∈ (0, T ].

The system (1.1)–(1.7) constitutes a model for the one-dimensional thermo-mechanical developments in a linearly viscous piece of wire of unit length in whicha solid-solid phase transition takes place. In this connection, the unknowns u, θ, σ,w, ψ denote displacement, absolute (Kelvin) temperature, elastoplastic stress, phasevariable (usually a so-called generalized freezing index; cf. [15]), and the thermody-namic force driving the phase transition, respectively. The positive physical constantsρ, µ, CV , κ, ν denote mass density, viscosity, specific heat, heat conductivity, and arelaxation coefficient, in that order. For the sake of notational convenience, we willalways assume without loss of generality that ρ = µ = CV = κ = ν = 1. Finally, theexpressions Hj , 1 ≤ j ≤ 4, and F1 are nonlinearities of hysteresis type (to be specifiedbelow).

The equations (1.1), (1.3), (1.4) represent the equation of motion, the balance ofinternal energy, and the phase evolution equation, in that order (see below); equation(1.2) is the constitutive law relating strain and phase variable to the elastoplasticstress, and (1.4) expresses that the phase variable evolves into the opposite directionof the thermodynamic force driving the phase transition. In addition, the boundaryconditions (1.7) indicate that the wire is thermally insulated at both ends, fixed atx = 0, and stress-free at x = 1.

The motivation to study systems of the above type is twofold. On the one hand,it is well known that for many materials the macroscopic strain-stress (ε-σ, whereε = ux is the linearized strain and u is the displacement) relations measured inuniaxial load-deformation experiments strongly depend on the absolute (Kelvin) tem-perature θ and, at the same time, exhibit a strong elastoplasticity witnessed by theoccurrence of hysteresis loops that are rate-independent, i.e., independent of the speedwith which there are traversed. Due to the hysteresis, which reflects the presence of arate-independent memory in the material, the stress-strain relation can no longer beexpressed in terms of a simple single-valued function. Among the materials showingvery strong temperature-dependent hysteretic effects are the so-called shape memoryalloys (see Figure 1 below and Chapter 5 in [2]); but even quite ordinary steels arewell known to exhibit this kind of behavior (cf. [21]), although to a smaller extent.

Fig. 1. Schematic load-deformation curves in shape memory alloys, with temperature increasingfrom left to right.

Usually the occurrence of a hysteresis in the macroscopic stress-strain relationsis accompanied (or even triggered) by changes between different configurations ofthe crystal lattice within the solid. It thus makes sense to complement macroscopicequations of thermoelastoplasticity by field equations accounting for such phase trans-

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THERMOVISCOPLASTICITY, HYSTERESIS, PHASE-FIELD MODELS 411

formations on the micro- and/or mesoscales.On the other hand, phase transition phenomena are often accompanied by macro-

scopic hysteresis effects that are caused by thermal and/or mechanical stresses actingon the micro- and/or mesoscales. It then makes sense to complement the field equa-tions describing the macroscopic phase transition by equations modeling such micro-or mesoscopic stresses.

A classical approach to such problems would be the following. One first tries toconstruct a local free energy function of the form

F (ε, w, θ) = θ(1 − log(θ)) + F1(ε, w) + θ F2(ε, w)(1.8)

in such a way that the experimentally observed ε-σ and/or w-ψ hysteresis loops areapproximately matched using the relations

σ =∂F

∂ε(ε, w, θ), ψ =

∂F

∂w(ε, w, θ),(1.9)

then determines the corresponding internal energy U and entropy S,

U(ε, w, θ) := F (ε, w, θ) − θ∂F

∂θ(ε, w, θ) = θ + F1(ε, w),

(1.10)

S(ε, w, θ) := −∂F

∂θ(ε, w, θ) = log(θ) − F2(ε, w),

and finally inserts these expressions in the governing field equations: equation ofmotion,

utt − σx = f (σ = total stress = σ + viscous stress),(1.11)

balance of internal energy,

Ut − θxx = σuxt + g (U = internal energy),(1.12)

and phase evolution equation (1.4).We then obtain (1.1), (1.3), (1.4) if we put

H1[ε, w] :=∂F1

∂ε(ε, w), H2[ε, w] :=

∂F2

∂ε(ε, w),

H3[ε, w] :=∂F1

∂w(ε, w), H4[ε, w] :=

∂F2

∂w(ε, w),(1.13)

F1[ε, w] := F1(ε, w).

In order that an ε-σ (or w-ψ, respectively) hysteresis be modeled by (1.9), F (·, w, θ)(F (ε, ·, θ), respectively) needs to be a nonconvex function within the range of inter-esting temperatures.

This approach has advantages: if the nonlinearities involved in (1.1)–(1.7) aresmooth functions, then the vast literature on one-dimensional thermoviscoelasticity(we just refer to the fundamental papers [4], [5]) can be applied to derive resultsconcerning well-posedness and asymptotic behavior. However, while this approach iscapable of correctly predicting many of the experimentally observed phenomena, italso has certain disadvantages from the phenomenological (engineering) point of view:the use of a nonconvex free energy does not guarantee that a hysteresis actually occurs;one only observes that unstable branches are traversed with a very high speed, which

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may look like a hysteresis if the speed is interpreted as infinite. For the moment, thereis no mathematical theory which would allow us to rigorously justify this singulartransition and give a correct account of the inherent memory structures that areresponsible for the complicated loopings in the interior of the external hysteresis loopsthat are observed in experiments.

To avoid these difficulties, the first two authors have recently proposed a differ-ent approach using the theory of hysteresis operators developed in the past thirtyyears (let us at least refer to the monographs [8], [20], [22], [2], [9] devoted to thissubject). In this approach, we replace the relations (1.9) by the identities (1.2),(1.5), where the expressions Hj , 1 ≤ j ≤ 4, and F1 are no longer real-valuedfunctions but hysteresis operators acting between suitable function spaces. Thisapproach has been successfully carried out for the two cases when either we haveone-dimensional thermoelastoplastic hysteresis without the order parameter evolu-tion (that is, we have (1.1)–(1.3) with no dependence on w; cf. the papers [10], [11])or we have a multidimensional phase transition without mechanical effects (that is, wehave (1.3)–(1.5) with no dependence on u, σ; see [7], [12], [13], [14], [15], [16], [17]).In this paper, we want to extend some of these results to the fully coupled problem.

At this point we remark that in [10], [11] a more general constitutive law than(1.2) has been treated in that there the hysteresis operators could also depend on θ.However, no dependence on w was admitted in [10], [11]. It is in fact this additionaldependence that forces us to assume σ in the form (1.2) which, on the other hand,is quite typical in the Landau theory of phase transitions (cf. Chapter 4 in [2]). Theextension of the results of this paper to more general temperature-dependent hysteresisoperators seems to be a very difficult unsolved problem.

We also note that in [18] the present authors have studied a related version ofsystem (1.1)–(1.7): there, an additional curvature term was added on the left of (1.1),and the boundary conditions for u were of the form u(0, t) = u(1, t) = uxx(0, t) =uxx(1, t) = 0. Let us stress the fact that the mathematical analysis carried out in [18]differs considerably from that used in this paper. In fact, the problem investigatedhere is more difficult than that studied in [18]: if the smoothing term γuxxxx is presentin (1.1), then already the first a priori estimate (see below) yields an L∞(L2)-boundfor the strain gradient uxx and thus an L∞(L∞)-bound for ux. This means that theAndrews transformation (see (3.1) below) used in this paper to eventually bound ux

is not needed. As a consequence, the solution established in this paper enjoys lesssmoothness than that in [18]. On the other hand, our analysis does not apply tothe case when (1.1) is complemented with zero boundary conditions at both ends ofthe wire: we have to assume a stress-free regime at one of the ends in order to takeadvantage of the Andrews transformation.

Let us now recall some basic facts about the notion of hysteresis operators (fordetails, we refer to the monographs mentioned above). Let T > 0 denote some (final)time. A mapping H from the set Map[0, T ] := w : [0, T ] → R into itself is called ahysteresis operator if it is causal, that is, if for all w1, w2 ∈ Map[0, T ] and t ∈ [0, T ]we have the implication

w1(τ) = w2(τ) ∀ τ ∈ [0, t] ⇒ H[w1](t) = H[w2](t),

and if it is rate-independent, that is, if for every w ∈ Map[0, T ] and every continuousincreasing mapping α of [0, T ] onto [0, T ] we have

H[w α](t) = H[w](α(t)) ∀ t ∈ [0, T ].

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In the case of partial differential equations, when the input functions not only dependon a time variable t ∈ [0, T ] but also on a space variable x ∈ [0, 1], it is necessaryto extend the above notion. In this situation, it is natural to associate with a hys-teresis operator H defined on Map[0, T ] in the above sense an operator H acting onMap([0, 1] × [0, T ]) by simply putting

H[w](x, t) := H[w(x, ·)](t).(1.14)

It is customary to identify the operators H and H. The hysteresis operators appearingin (1.1)–(1.7) have to be understood in this way.

The advantage of this approach is that an operator equation like (1.2), (1.5),is suited much better than a relation like (1.9) to keep track of the memory effectsimprinted on the material in the past history. In fact, the output at any time t ∈ [0, T ]may depend on the whole evolution of the input in the time interval [0, t]. Observethat the rate-independence implies that the hysteresis behavior cannot be expressedin terms of an integral operator of convolution type; i.e., we are not dealing with amodel with fading memory.

Unfortunately, there are also disadvantages: the input-output behavior of hys-teresis operators usually cannot be described explicitly, and they have, as a rule, onlyvery restricted smoothness properties. In fact, nontrivial hysteresis operators are, asa rule, not differentiable, but at best only (possibly locally) Lipschitz continuous insuitable function spaces; in addition, they carry a nonlocal memory with respect totime.

Both nondifferentiability and presence of a memory are unpleasant features fromthe mathematical point of view. For instance, the classical method of deriving higherorder a priori estimates for w (namely, differentiation of (1.4) with respect to t andtesting with wt) does not immediately work, since there is no chain rule for thehysteretic nonlinearities; also, we may not simply differentiate (1.2) or (1.5) withrespect to x. These facts result in a lack of compactness and thus in difficulties inexistence proofs.

However, hysteresis operators usually dissipate energy which typically is propor-tional to the area of closed traversed loops in the hysteresis diagram. Let us explainthis fact for one fundamental hysteresis operator which plays a most prominent rolein the theory, namely the so-called stop operator or Prandtl’s normalized elastic–perfectly plastic element. To this end, let r > 0 (the yield limit) and σ0

r ∈ [−r, r] (theinitial stress) be given. For any input function ε ∈ W 1,1(0, T ), we define the outputσr ∈ W 1,1(0, T ) as the solution to the variational inequality (the index t denotes timedifferentiation)

σr(t) ∈ [−r, r] ∀ t ∈ [0, T ], σr(0) = σ0r ,(1.15)

(εt(t) − σr,t(t))(σr(t) − η) ≥ 0 ∀ η ∈ [−r, r] a.e. in (0, T ).(1.16)

In Figure 2, the typical input-output behavior is depicted.

It can easily be proved (see, for instance, [9], where also the multidimensionalcase is treated) that (1.15)–(1.16) admits a unique solution σr ∈ W 1,1(0, T ) for everyε ∈ W 1,1(0, T ) and σ0

r ∈ [−r, r]. The corresponding solution operator

sr : [−r, r] ×W 1,1(0, T ) → W 1,1(0, T ) : (σ0r , ε) → σr(1.17)

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0

r

r

r

"

Fig. 2. Prandtl’s normalized elastic–perfectly plastic element.

is just the stop operator. It has the well-known property (cf. [2], [9]) that for anyr1, r2 ∈ [0,+∞), σ0

rj ∈ [−rj ,+rj ], j = 1, 2, t ∈ [0, T ], and ε1, ε2 ∈ C[0, T ], it holdsthat

∣∣sr1 [σ0r1 , ε1](t) − sr2 [σ0

r2 , ε2](t)∣∣

(1.18)≤ |ε1(t) − ε2(t)| + max

|r1 − r2| + max

0≤τ≤t|ε1(τ) − ε2(τ)|, ∣∣σ0

r1 − σ0r2

∣∣.In addition, it holds for any ε ∈ W 1,1(0, T ) that

∥∥sr[σ0r , ε]

∥∥∞ ≤ r, sr[σ0

r , ε]t(t) = εt(t) if∣∣sr[σ0

r , ε](t)∣∣ < r,(1.19) ∣∣sr[σ0

r , ε]t∣∣2 = sr[σ0

r , ε]tεt a.e. in (0, T ).(1.20)

For each fixed initial condition σ0r , the stop operator is a hysteresis operator in

the sense of the above definition, where the value of σ0r accounts for some previous

memory. On the other hand, the value at t = 0 of the output of any rate-independentoperator can be represented as a function of the initial value of the input. For theoperators occurring in our model, we will not make any special assumptions abouttheir initial configurations, except for those that follow from the general analyticconditions imposed below in hypothesis (H4), like, e.g., Lipschitz continuous input-output relation.

We now describe the intrinsic dissipation property of the stop operator. It resultsif we insert η = 0 in (1.16). We then obtain that the energy Pr := 1

2s2r of the stop

element satisfies the inequality

d

dtPr[σ0

r , ε](t) ≤ sr[σ0r , ε](t) εt(t) a.e. in (0, T )(1.21)

for all (σ0r , ε) ∈ [−r, r] ×W 1,1(0, T ), and the difference between the right and the left

of (1.21) is the dissipated energy. Equation (1.21) can also be interpreted as a chainrule inequality for the energy operator Pr where the stop operator sr plays the roleof the “derivative” of Pr with respect to ε (only formally, since Pr is certainly notdifferentiable with respect to ε).

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Chain rule inequalities of the form (1.21) have proven to be crucial for a successfulstudy of differential equations with hysteresis (for this, see the cited literature). In thecase of system (1.1)–(1.7), an appropriate form of such a condition is to postulate theexistence of a further hysteresis operator F2 such that for any (ε, w) ∈ W 1,1(0, T ) ×W 1,1(0, T ) it holds, for a.e. t ∈ (0, T ), that

d

dtF1[ε, w](t) ≤ H1[ε, w](t) εt(t) + H3[ε, w](t)wt(t),

(1.22)d

dtF2[ε, w](t) ≤ H2[ε, w](t) εt(t) + H4[ε, w](t)wt(t).

We then can associate with system (1.1)–(1.7) free energy, entropy, and internal energyhysteresis operators by putting (compare (1.8), (1.10))

F [ε, w, θ] := θ(1 − log(θ)) + F1[ε, w] + θF2[ε, w],

S[ε, w, θ] := log(θ) −F2[ε, w],(1.23)

U [ε, w, θ] := θ + F1[ε, w],

where [ε, w, θ] ∈ Map[0, T ] × Map[0, T ] × (0,+∞). Indeed, if we associate σ and ψ asgiven by (1.2) and (1.5), respectively, with the “derivatives” of F with respect to ε andw (only formally, as they do not exist), respectively, then we arrive at system (1.1)–(1.5) as field equations. It will turn out later that the validity of (1.22) (rather, of ageneralized version thereof; see below) will guarantee the thermodynamic consistencyof the model, that is, the temperature stays positive during the evolution, and theClausius–Duhem inequality, which in view of (1.12) can be written in the form

θd

dtS[ε, w, θ] − d

dtU [ε, w, θ] ≥ −σεt a.e. in ΩT ,(1.24)

where σ = σ + εt again denotes the total stress, will be satisfied.The rest of the paper is organized as follows: In section 2, we give a detailed

statement of the mathematical problem and of the main mathematical result. Section3 brings the proof of local existence and global uniqueness, and in the concludingsection 4 we prove global existence for system (1.1)–(1.7).

In what follows, the norms of the standard Lebesgue spaces Lp(Ω) for 1 ≤ p ≤ ∞will be denoted by ‖ · ‖p. Finally, we shall use the usual denotations Wm,p(Ω) andHm(Ω), m ∈ N, 1 ≤ p ≤ ∞, for the standard Sobolev spaces.

2. Statement of the problem. We make the following general assumptions onthe data of the system.

(H1) u0 ∈ H2(Ω), u1 ∈ H1(Ω), θ0 ∈ H1(Ω), w0 ∈ H1(Ω), it holds that θ0(x) ≥δ > 0 for all x ∈ Ω, and the compatibility condition u0(0) = u1(0) = 0 is satisfied.

(H2) It holds that f ∈ H1(0, T ;L2(Ω)).(H3) We assume that g : Ω × (0, T ) × R → R is a Caratheodory function such

that

∃ g0 ∈ L∞(ΩT ) : θ ≤ 0 ⇒ g(x, t, θ) = g0(x, t),(2.1)

∃K1 > 0 : |g(x, t, θ1) − g(x, t, θ2)| ≤ K1|θ1 − θ2| for a.e.(2.2)

(x, t) ∈ Ω × (0, T ) and ∀θ1, θ2 ∈ R,

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g0(x, t) ≥ 0 a.e. in ΩT .(2.3)

(H4) The operators Hj , 1 ≤ j ≤ 4, and F1 are causal and map C[0, T ] × C[0, T ]continuously into C[0, T ], and W 1,1(0, T ) ×W 1,1(0, T ) into W 1,1(0, T ). In addition,the following conditions are satisfied:

(i) ∃K2 > 0 : ∀ ε, w ∈ C[0, T ] it holds that

maxj∈2,4

‖Hj [ε, w]‖∞ ≤ K2, F1[ε, w](t) ≥ −K2 ∀ t ∈ [0, T ].(2.4)

(ii) ∃K3 > 0 : ∀ ε, w ∈ W 1,1(0, T ) it holds, for a.e. t ∈ (0, T ), that

max1≤j≤4

|Hj [ε, w]t(t)| + |F1[ε, w]t(t)| ≤ K3

(|εt(t)| + |wt(t)|

).(2.5)

(iii) ∃K4 > 0 : ∀ ε1, w1, ε2, w2 ∈ C[0, T ] it holds, for every t ∈ [0, T ], that

max1≤j≤4

|Hj [ε1, w1](t) −Hj [ε2, w2](t)|(2.6)

≤ K4 max0≤r≤t

(|ε1(r) − ε2(r)| + |w1(r) − w2(r)|

),

and ∀ ε1, w1, ε2, w2 ∈ W 1,1(0, T ) it holds, for every t ∈ [0, T ], that

|F1[ε1, w1](t) −F1[ε2, w2](t)| ≤ K4

[|ε1(0) − ε2(0)| + |w1(0) − w2(0)|

(2.7)

+

∫ t

0

(|ε1,t(r) − ε2,t(r)| + |w1,t(r) − w2,t(r)|

)dr].

(H5) There exist causal operators F2 : W 1,1(0, T )×W 1,1(0, T ) → W 1,1(0, T ), G :W 1,1(0, T ) → W 1,1(0, T ), and a constant K5 > 0 such that the following conditionsare satisfied:

(i) For every ε, w ∈ W 1,1(0, T ) it holds that

F1[ε, w]t ≤ εt H1[ε, w] + G[w]t H3[ε, w] a.e. in (0, T ),(2.8)

F2[ε, w]t ≤ εt H2[ε, w] + G[w]t H4[ε, w] a.e. in (0, T ).(2.9)

(ii) For every w ∈ W 1,1(0, T ) it holds that

|G[w]t(t)|2 ≤ K5wt(t)G[w]t(t) for a.e. t ∈ (0, T ).(2.10)

Remark 1. Owing to (H4)(iii) we have, in particular, that for any ε, w ∈ H1(0, T )and t ∈ [0, T ] the following holds:

|H1[ε, w](t)| + |H3[ε, w](t)|≤ |H1[ε, w](0)| + |H3[ε, w](0)| + 2K4 max

0≤r≤t(|ε(r) − ε(0)| + |w(r) − w(0)|)

(2.11)

≤ |H1[ε, w](0)| + |H3[ε, w](0)| + 2K4

∫ t

0

(|εt(r)| + |wt(r)|

)dr

≤ |H1[ε, w](0)| + |H3[ε, w](0)| + 2K4

√t

(∫ t

0

(|εt(r)|2 + |wt(r)|2

)dr

)1/2

.

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In addition, a linear growth of H1 and H3 with respect to both ε and w is admitted,which, in particular, includes the case of simple linear elasticity. It also follows thatfor any ε, w ∈ H1(Ω;C[0, T ]) it holds, for a.e. (x, t) ∈ ΩT , that

max1≤j≤4

∣∣∣(Hj [ε, w])x(x, t)

∣∣∣ ≤ K4 max0≤r≤t

(|εx(x, r)| + |wx(x, r)|

).(2.12)

Indeed, we only have to apply (2.6) with ε1(x, t) := ε(x + h, t), ε2(x, t) := ε(x, t),w1(x, t) := w(x + h, t), w2(x, t) := w(x, t), with some h > 0, and then let h 0.Consequently, we may consider first order spatial derivatives of Hj [ε, w], and we have(Hj [ε, w])x ∈ L2(ΩT ), 1 ≤ j ≤ 4.

Remark 2. A typical example where (H4), (H5) are fulfilled is given by Prandtl–Ishlinskii operators of the form

Hj [ε, w] :=

∫ ∞

0

ϕj(r) sr[σ0,jr , ε

]dr, j = 1, 2,

(2.13)

Hj [ε, w] :=

∫ ∞

0

ϕj(r) sr[σ0,jr , w

]dr, j = 3, 4,

where σ0,jr ∈ [−r,+r], 1 ≤ j ≤ 4, are given initial values for the operators sr defined

in (1.17), and the weight functions ϕj are nonnegative on [0,+∞) and satisfy

max1≤j≤4

∫ ∞

0

(1 + r2)ϕj(r) dr < +∞.(2.14)

Indeed, defining the (energy) operators

F1[ε, w] :=1

2

∫ ∞

0

(ϕ1(r) s2

r

[σ0,1r , ε

]+ ϕ3(r) s2

r

[σ0,3r , w

])dr,

(2.15)

F2[ε, w] :=1

2

∫ ∞

0

(ϕ2(r) s2

r

[σ0,2r , ε

]+ ϕ4(r) s2

r

[σ0,4r , w

])dr,

choosing G[w] = w, and invoking the properties (1.21)–(1.24) of the stop operatorssr, we easily verify the validity of (H4), (H5). Other examples, where the dependenceon ε, w is no longer decoupled as in (2.13), can be constructed using multidimensionalstop operators as basic elements (cf. [15], [16]). For examples where the Hj are notPrandtl–Ishlinskii operators and G differs from the identity operator, we refer to [14],[15].

We can now formulate the main result of this paper.Theorem 2.1. Suppose that the hypotheses (H1) to (H5) are satisfied. Then the

system (1.1)–(1.7) admits a unique strong solution (u, θ, w) such that (1.1)–(1.5) holda.e. in ΩT , and such that

u ∈ H2(0, T ;L2(Ω)) ∩H1(0, T ;H2(Ω)), w ∈ H2(0, T ;L2(Ω)) ∩H1(0, T ;H1(Ω)),

θ ∈ H1(0, T ;L2(Ω)) ∩ L2(0, T ;H2(Ω)).(2.16)

In addition, with the finite norms β1 := ‖uxt‖L1(0,T ;L∞(Ω)) and β2 := ‖wt‖C(ΩT ) it

holds that

θ(x, t) ≥ δe−((K1+K2K5β2)t+K2β1) ∀(x, t) ∈ ΩT .(2.17)

Remark 3. We note at this point that Theorem 2.1 also implies that the secondprinciple of thermodynamics is satisfied for the system (1.1)–(1.7). Indeed, we have

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θ > 0 in ΩT , and the validity of the Clausius–Duhem inequality (1.24) follows fromthe simple calculation

θ S[ε, w, θ]t − U [ε, w, θ]t + σεt = −θF2[ε, w]t −F1[ε, w]t + σεt + ε2t

≥ −(H1[ε, w] + θH2[ε, w]

)εt −

(H3[ε, w] + θH4[ε, w]

)G[w]t + σεt + ε2

t(2.18)

≥ ε2t + wt G[w]t ≥ 0 a.e. in ΩT ,

where F , S, U are given by (1.23). We may therefore claim that our system is thermo-dynamically consistent.

The proof of Theorem 2.1 will be given in the following sections. During itscourse, we will make repeated use of Young’s inequality,

ab ≤ δ

2a2 +

1

2δb2 ∀ a, b ∈ R, δ > 0,(2.19)

of the elementary inequality,

|z(t)|2 ≤ 2|z(0)|2 + 2t

∫ t

0

z2t (r) dr ∀ t ∈ (0, T ) ∀ z ∈ H1(0, T ),(2.20)

and of the one-dimensional Gagliardo–Nirenberg inequality,

‖w‖p ≤ K0

(‖w‖1−ω

q ‖wx‖ωr + ‖w‖q)

∀w ∈ W 1,r(Ω) ∩ Lq(Ω),(2.21)

where K0 > 0 is a constant depending only on p, q, r, and where

1 ≤ r ≤ +∞, 1 ≤ q ≤ p ≤ +∞, ω

(1

q− 1

r+ 1

)=

1

q− 1

p.(2.22)

3. Local existence. We rewrite the system (1.1)–(1.7), using the transforma-tion due to Andrews [1],

ux = p + q, where p(x, t) :=

∫ x

1

ut(ξ, t) dξ.(3.1)

We easily find that (1.1)–(1.6) is equivalent to the system

pt − pxx = σ +

∫ x

1

f(ξ, t) dξ,(3.2)

p(1, t) = px(0, t) = 0, p(x, 0) =

∫ x

1

u1(ξ) dξ,(3.3)

σ = H1[p + q, w] + θH2[p + q, w],(3.4)

qt = −σ −∫ x

1

f(ξ, t) dξ,(3.5)

q(x, 0) = u′0(x) −

∫ x

1

u1(ξ) dξ,(3.6)

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(θ + F1[p + q, w]

)t− θxx = p2

xx + σpxx + g(x, t, θ),(3.7)

wt = −ψ,(3.8)

ψ = H3[p + q, w] + θH4[p + q, w],(3.9)

θ(x, 0) = θ0(x), w(x, 0) = w0(x), θx(0, t) = θx(1, t) = 0.(3.10)

Let V0 := z ∈ H1(Ω); z(1) = 0, and let V ∗0 denote its dual space. We are going

to show the following result.Theorem 3.1. Suppose that the hypotheses (H1) to (H4) are fulfilled. Then

there is some τ > 0 such that the initial-boundary value problem (3.2)–(3.10) admitsa unique solution quadruple (p, q, θ, w) on Ω × [0, τ ] satisfying

p ∈ H2(0, τ ;V ∗0 ) ∩H1(0, τ ;H1(Ω)) ∩ L2(0, τ ;H3(Ω)),(3.11)

q, w ∈ H2(0, τ ;L2(Ω)) ∩H1(0, τ ;H1(Ω)) ∩ C1([0, τ ];C(Ω)),(3.12)

θ ∈ H1(0, τ ;L2(Ω)) ∩ L2(0, τ ;H2(Ω)) ∩ C(Ωτ ),(3.13)

θ(x, t) ≥ δ

2> 0 for every (x, t) ∈ Ω × [0, τ ].(3.14)

Proof of Theorem 3.1. We divide the proof of Theorem 3.1 into several steps,each formulated as a separate lemma. The existence part of the proof is based on thefollowing special case of the Schauder–Tikhonov fixed point principle (cf., for instance,Theorem 3.6.1 in [6]).

Lemma 3.2. Let the operator T map the nonempty, closed, convex, and weaklycompact subset M of the separable Hilbert space X into itself, and suppose that T isweakly sequentially continuous on M; that is, it holds that T (vn) → T (v) weakly inX whenever vn → v weakly in X for some sequence vn ⊂ M. Then T has a fixedpoint in M.

We aim to apply Lemma 3.2 to the following setting. Consider for τ ∈ (0, T ] theseparable Hilbert spaces

Pτ := H2(0, τ ;V ∗0 ) ∩H1(0, τ ;H1(Ω)) ∩ L2(0, τ ;H3(Ω)),

Qτ := H2(0, τ ;L2(Ω)) ∩H1(0, τ ;H1(Ω)),

Zτ := H1(0, τ ;L2(Ω)) ∩ L2(0, τ ;H2(Ω)),(3.15)

Wτ := H2(0, τ ;L2(Ω)) ∩H1(0, τ ;H1(Ω)),

Xτ := Pτ ×Qτ × Zτ ×Wτ

and introduce the sets

Mτ :=

(p, q, θ, w) ∈ Xτ ; (3.3), (3.6), (3.10) hold, pt + qt = pxx a.e. in Ωτ ,

∫ τ

0

∫Ω

(θ2t + θ2

xx

)dx dt + max

0≤t≤τ

∫Ω

|θx(x, t)|2 dx ≤ M1,(3.16)

max(x,t)∈Ωτ

|θ(x, t)| ≤ M2,(3.17)

max0≤t≤τ

∫Ω

(|qt(x, t)|2 + |wt(x, t)|2

)dx ≤ M3,(3.18)

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∫ τ

0

∫Ω

(q2xt + w2

xt

)dx dt ≤ M4,(3.19)

∫ τ

0

∫Ω

(p2t + p2

xx

)dt + max

0≤t≤τ

∫Ω

|px(x, t)|2 dx ≤ M5,(3.20)

max0≤t≤τ

∫Ω

(|qx(x, t)|2 + |wx(x, t)|2

)dx ≤ M6,(3.21)

‖pt‖2H1(0,τ ;V ∗

0 ) +

∫ τ

0

∫Ω

p2xt dx dt + max

0≤t≤τ

∫Ω

|pt(x, t)|2 dx ≤ M7,(3.22)

max0≤t≤τ

∫Ω

|pxx(x, t)|2 dx ≤ M8,(3.23)

∫ τ

0

∫Ω

p2xxx dx dt ≤ M9,(3.24)

∫ τ

0

∫Ω

(q2tt + w2

tt) dx dt ≤ M10,(3.25)

min(x,t)∈Ωτ

θ(x, t) ≥ δ

2> 0

,(3.26)

where the positive constants Mi, i = 1, . . . , 10, will have to be specified later. Ob-viously, Mτ is a nonempty, closed, convex, and bounded (hence weakly compact)subset of the separable Hilbert space Xτ .

Next, we introduce the operator T on Mτ by T (p, q, θ, w) := (p, q, θ, w), wherefor (p, q, θ, w) ∈ Mτ the quadruple (p, q, θ, w) is the unique solution to the linearinitial-boundary value problem

pt − pxx = σ +

∫ x

1

f(ξ, t) dξ,(3.27)

p(1, t) = px(0, t) = 0, p(x, 0) =

∫ x

1

u1(ξ) dξ,(3.28)

σ = H1[p + q, w] + θH2[p + q, w],(3.29)

qt = −σ −∫ x

1

f(ξ, t) dξ,(3.30)

q(x, 0) = u′0(x) −

∫ x

1

u1(ξ) dξ,(3.31)

θt − θxx = −F1[p + q, w]t + p2xx + σpxx + g(x, t, θ),(3.32)

wt = −ψ,(3.33)

ψ = H3[p + q, w] + θH4[p + q, w],(3.34)

θ(x, 0) = θ0(x), w(x, 0) = w0(x), θx(0, t) = θx(1, t) = 0.(3.35)

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We have the following result.Lemma 3.3. There exist τ ∈ (0, T ] and positive constants Mi, i = 1, . . . , 10, such

that T (Mτ ) ⊂ Mτ for any τ ∈ (0, τ ].Proof. Let τ ∈ (0, T ] be given. Without loss of generality, we may assume that

τ ≤ 1. We have pt + qt = pxx a.e. in Ωτ , and we infer from the general hypothesesthat pt, pxx, qt, wt, θt, θxx ∈ L2(Ωτ ). Therefore, θ ∈ Zτ . Also, since (p, q, θ, w) ∈ Mτ ,it follows from Remark 1 that the right-hand sides of both (3.30) and (3.33) belongto H1(Ωτ ) so that q ∈ Qτ and w ∈ Wτ .

Next, we consider the parabolic initial-boundary value problem

zt − zxx = v := σt +

∫ x

1

ft(ξ, t) dξ,(3.36)

zx(0, t) = z(1, t) = 0, z(x, 0) = u′1(x) + σ(0) +

∫ x

1

f(ξ, 0) dξ.(3.37)

Since z(·, 0) ∈ L2(Ω), and since the right-hand side v of (3.36) belongs to L2(Ωτ ),it follows from general linear parabolic theory (cf. Lions and Magenes [19]) that(3.36)–(3.37) admits a unique weak solution z ∈ L2(0, τ ;H1(Ω)) ∩ H1(0, τ ;V ∗

0 ) ∩C([0, τ ];L2(Ω)), and there is some constant C > 0, not depending on τ ∈ (0, T ], suchthat

‖z‖2H1(0,τ ;V ∗

0 ) +

∫ τ

0

∫Ω

z2x dx dt + max

0≤t≤τ

∫Ω

|z(x, t)|2 dx(3.38)

≤ C

(∫Ω

|z(x, 0)|2 dx +

∫ τ

0

∫Ω

v2 dx dt

).

Invoking the compatibility condition u1(0) = 0, we easily verify that

p(x, t) =

∫ x

1

u1(ξ) dξ +

∫ t

0

z(x, r) dr,(3.39)

so that p ∈ H2(0, τ ;V ∗0 ) ∩ H1(0, τ ;H1(Ω)) ∩ C1([0, τ ];L2(Ω)), and (3.38) holds for

z = pt. Hence, using (3.27), we can conclude that pxxx ∈ L2(Ωτ ), and also pxx ∈C([0, τ ];L2(Ω)). In conclusion, p ∈ Pτ , and we have shown that T (Mτ ) ⊂ Xτ .

Now let (p, q, θ, w) = T (p, q, θ, w) for some (p, q, θ, w) ∈ Mτ , where the constantsM1, . . . ,M10 and τ ∈ (0, T ] are assumed to be fixed. We are going to derive a numberof estimates for (p, q, θ, w) in terms of M1, . . . ,M10 and of the data of the system. Inwhat follows, we denote by Ci, i ∈ N ∪ 0, positive constants which may depend onthe given data u0, u1, θ0, w0, f , g0, and on the constants Ki, 0 ≤ i ≤ 4, but neitheron τ nor on M1, . . . ,M10.

At first, we conclude from (2.11), (2.4), and (2.5) that∫Ω

(|σ|2 + |ψ|2

)(x, t) dx

≤ 2

∫Ω

(H2

1[p + q, w] + H23[p + q, w] + θ2(H2

2[p + q, w] + H24[p + q, w])

)(x, t) dx

(3.40)

≤ 4K22M

22 + C0

[1 + t

∫ t

0

∫Ω

(p2t + q2

t + w2t

)dx dr

]

≤ 4K22M

22 + C0

(1 + t(M3 + M7)

)∀t ∈ [0, τ ],

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|σt| + |ψt| ≤ 2K3(1 + M2)(|pt| + |qt| + |wt|) + 2K2|θt| a.e. in Ωτ .(3.41)

Also, it follows from (2.12) that for 1 ≤ i ≤ 4 and a.e. (x, t) ∈ Ωτ we have∣∣∣(Hi[p + q, w])x(x, t)

∣∣∣ ≤ K4 max0≤r≤t

(|px(x, r)| + |qx(x, r)| + |wx(x, r)|

).(3.42)

In addition, owing to (2.5),

|F1[p + q, w]t| ≤ K3(|pt| + |qt| + |wt|) a.e. in Ωτ ,(3.43)

as well as, by virtue of (H3),

|g(x, t, θ(x, t))| ≤ g0(x, t) + K1M2.(3.44)

Now multiply (3.32) first by θt and then by −θxx; add the resulting equationsand integrate over Ω × [0, t] for any t ∈ (0, τ ]. Using Young’s inequality and invoking(3.40), (3.43), and (3.44), we find that∫ t

0

∫Ω

(θ2t + θ2

xx

)dx dr +

∫Ω

|θx(x, t)|2 dx

≤ C1

[1 + tM2

2 +

∫ t

0

∫Ω

(p2t + q2

t + w2t

)dx dr(3.45)

+

∫ t

0

∫Ω

p4xx dx dr +

∫ t

0

∫Ω

σ2 p2xx dx dr

].

Invoking the Gagliardo–Nirenberg inequality (2.21) for p = +∞, q = r = 2, ω = 1/2,we infer that∫ t

0

‖pxx(·, r)‖2∞ dr ≤ 2K2

0

(∫ t

0

∫Ω

p2xx dx dr

+ max0≤r≤t

‖pxx(·, r)‖2

∫ t

0

‖pxxx(·, r)‖2 dr

)(3.46)

≤ 2K20

(tM8 +

√M8

√t√

M9

)≤ 2K2

0

√t(M8 +

√M8M9

).

It follows that ∫ t

0

∫Ω

p4xx dx dr ≤ max

0≤r≤t‖pxx(·, r)‖2

2

∫ t

0

‖pxx(·, r)‖2∞ dr

(3.47)≤ 2K2

0

√t(M2

8 + M3/28

√M9

).

In addition, by (3.40),∫ t

0

∫Ω

σ2p2xx dx dr ≤ max

0≤r≤t‖σ(·, r)‖2

2

∫ t

0

‖pxx(·, r)‖2∞ dr

(3.48)≤ 2K2

0

√t(M8 +

√M8M9

)(4K2

2M22 + C0(1 + t(M3 + M7))

).

In conclusion, we have shown the estimate∫ τ

0

∫Ω

(θ2t + θ2

xx

)dx dr + max

0≤t≤τ

∫Ω

|θx(x, t)|2 dx

≤ C2

[1 +

√τ(M2

2 + M3 + M7 + M28 + M

3/28 M

1/29(3.49)

+(M8 + M

1/28 M

1/29

)(1 + M2

2 + M3 + M7

))].

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Next, we consider (3.30) and (3.33). By the general hypotheses and (3.40), we have

max0≤t≤τ

∫Ω

(|qt(x, t)|2 + |wt(x, t)|2

)dx ≤ C3

(1 + M2

2 + t(M3 + M7)).(3.50)

Now differentiate (3.30) and (3.33) with respect to x. Then, by (3.42),

|σx(x, t)|+|ψx(x, t)| ≤ 2K2|θx(x, t)|+2K4(1+M2) max0≤r≤t

((|px|+|qx|+|wx|)(x, r)

)(3.51)for a.e. (x, t) ∈ Ωτ . Therefore, using (2.20), we can conclude that∫ τ

0

∫Ω

(q2xt + w2

xt

)dx dt ≤ C4

(1 +

∫ τ

0

∫Ω

(|σx|2 + |ψx|2

)dx dt

)

≤ C5

[1 + M2

2 +

∫ τ

0

∫Ω

|θx|2 dx dt + (1 + M22 )τ

∫ τ

0

∫Ω

(p2xt + q2

xt + w2xt

)dx dt

]

≤ C5

(1 + M2

2 + τ(M1 + (1 + M2

2 )(M4 + M7)))

.

(3.52)

But then also

max0≤t≤τ

∫Ω

(|qx(x, t)|2 + |wx(x, t)|2

)dx

(3.53)≤ C6

(1 + M2

2 + τ(M1 + (1 + M2

2 )(M4 + M7)))

.

Next, we consider the linear parabolic system (3.27)–(3.28). Standard parabolicestimates, using the general hypotheses and (3.40), yield that∫ τ

0

∫Ω

(p2t + p2

xx

)dx dt ≤ C7

(1 +

∫ τ

0

∫Ω

|σ|2 dx dt)

(3.54)≤ C8

(1 + τ(M2

2 + M3 + M7)).

Moreover, since (3.38) is valid for z = pt, we can infer from (H2) and from (3.41) that

‖p‖2H2(0,τ ;V ∗

0 ) +

∫ τ

0

∫Ω

p2xt dx dt + max

0≤t≤τ

∫Ω

|pt(x, t)|2 dx

≤ C9

(1 +

∫ τ

0

∫Ω

(θ2t + (1 + M2

2 )(p2t + q2

t + w2t

))dx dt

)(3.55)

≤ C9

(1 + M1 + τ(1 + M2

2 )(M3 + M7)).

But then we obtain from (3.27), also using (3.29) and (H2), that

max0≤t≤τ

∫Ω

|pxx(x, t)|2 dx ≤ 2 max0≤t≤τ

∫Ω

|pt(x, t)|2 dx + 2C10

(1 + max

0≤t≤τ

∫Ω

|σ(x, t)|2 dx)

≤ C11

(1 + M1 + M2

2 + τ(1 + M22 )(M3 + M7)

).(3.56)

In addition, employing (3.51) and (3.55), and arguing as in the derivation of (3.52),we can deduce the estimate∫ τ

0

∫Ω

p2xxx dx dt ≤ 2

∫ τ

0

∫Ω

p2xt dx dt + 2

∫ τ

0

∫Ω

|σx + f |2 dx dt

≤ C12

(1 + M1 + M2

2 + τ(M1 + (1 + M22 )(M3 + M4 + M7))

).(3.57)

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Next, we differentiate (3.30) and (3.33), respectively, with respect to t, and invoke(H2) and (3.41) to obtain the bound

∫ τ

0

∫Ω

(q2tt + w2

tt) dx dt ≤ C13(1 + (1 + M22 )(M3 + M5) + M1).(3.58)

Finally, since Ω is one-dimensional, H1(0, τ ;L2(Ω)) ∩ L2(0, τ ;H2(Ω)) is continuouslyimbedded in C(Ωτ ). Hence, there is some C14 > 0 such that

max(x,t)∈Ωτ

|θ(x, t)| ≤ C14

(1 +

√M1

),(3.59)

where M1 is equal to the expression on the right-hand side of (3.49).Now, we can define the constants M1 . . . ,M10. We make the choices

M1 := 2C2, M2 := C14

(1 +

√M1

), M3 := 2C3 + C3M

22 ,

M4 := 2C5 + C5M22 , M5 := 2C8, M6 := 2C6 + C6M

22 ,

(3.60)

M7 := 2C9 + C9M1, M8 := 2C11 + C11(M1 + M22 ),

M9 := 2C12 + C12(M1 + M22 ), M10 := C13(1 + (1 + M2

2 )(M3 + M5) + M1).

It then follows from the estimates (3.49)–(3.50), (3.52)–(3.57), and (3.59) that thereexists some τ0 ∈ (0, T ] such that the inequalities (3.16)–(3.27) are fulfilled for anyτ ∈ (0, τ0]. In particular, (3.16) implies that the assumptions of Lemma 3.2.2 in [2]

are satisfied. Therefore, we have θ ∈ C12 ,

16 (Ωτ0), and there is some constant C15 > 0,

depending only on M1 and τ0, such that for every (x, t), (y, s) ∈ Ωτ0 it holds that

|θ(x, t) − θ(y, s)| ≤ C15

(|t− s|1/6 + |x− y|1/2

).(3.61)

Consequently, for sufficiently small τ ∈ (0, τ0],

θ(x, t) ≥ θ0(x) − |θ(x, t) − θ0(x)| ≥ δ − C15t1/6 ≥ δ

2(3.62)

for all (x, t) ∈ Ω × [0, τ ]. With this, the proof of the lemma is complete.Lemma 3.4. The operator T is weakly sequentially continuous in Mτ .Proof. Suppose a sequence (pn, qn, θn, wn) ⊂ Mτ is given such that

(pn, qn, θn, wn) → (p, q, θ, w) weakly in Xτ as n → ∞.(3.63)

Since Mτ is weakly closed, it holds that (p, q, θ, w) ∈ Mτ . Now, let

(pn, qn, θn, wn) := T (pn, qn, θn, wn), n ∈ N, (p, q, θ, w) := T (p, q, θ, w).(3.64)

We have to show that

(pn, qn, θn, wn) → (p, q, θ, w) weakly in Xτ as n → ∞.(3.65)

Clearly, as (pn, qn, θn, wn) ∈ Mτ for all n ∈ N, we have, on a subsequence which isagain indexed by n,

(pn, qn, θn, wn) → (p, q, θ, w) weakly in Xτ as n → ∞(3.66)

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for some (p, q, θ, w) ∈ Mτ . It remains to show that (p, q, θ, w) = (p, q, θ, w). Theuniqueness of the limit point then entails that (3.66), and thus (3.65) holds for theentire sequence. To this end, note that we have the convergences

θn,t → θt, θn,xx → θxx, pn,t → pt, pn,xx → pxx, pn,xxx → pxxx, pn,xt → pxt,

wn,t → wt, wn,xt → wxt, qn,t → qt, qn,xt → qxt, all weakly in L2(Ωτ ).(3.67)

By compact imbedding, we may also assume that

θn → θ, pn,x → px, both uniformly in Ωτ ,(3.68)

pn → p, qn → q, wn → w, all strongly in L2(Ω;C[0, τ ]).(3.69)

But then, owing to (H4), it follows that

σn → σ, ψn → ψ, F1[pn + qn, wn] → F1[p + q, w],(3.70)

all strongly in L2(Ω;C[0, τ ]),

where σn, ψn have obvious meaning. Also,

g(·, ·, θn) → g(·, ·, θ) strongly in L∞(Ωτ ),(3.71)

σnpn,xx → σpxx weakly in L2(Ωτ ).(3.72)

In addition, owing to (3.18), (3.22), and (3.43), the sequence F1[pn + qn, wn]t isbounded in L∞(0, τ ;L2(Ω)), so that we may assume that

F1[pn + qn, wn]t → y weakly-star in L∞(0, τ ;L2(Ω))(3.73)

for some y ∈ L∞(0, τ ;L2(Ω)). But then it follows from (3.70) that y = F1[p + q, w]t.Finally, we conclude from (3.67) and (3.68) that

p2n,xx → p2

xx weakly in L2(Ωτ ).(3.74)

Indeed, we have for any test function η ∈ C∞0 (Ωτ ) that

limn→∞

∫ τ

0

∫Ω

p2n,xxη dx dt

= − limn→∞

∫ τ

0

∫Ω

(pn,xxxpn,xη + pn,xxpn,xηx

)dx dt(3.75)

= −∫ τ

0

∫Ω

(pxxxpxη + pxxpxηx

)dx dt =

∫ τ

0

∫Ω

p2xxη dx dt.

Since C∞0 (Ωτ ) is a dense subset of L2(Ωτ ), and since p2

n,xx is bounded in L2(Ωτ )(cf. (3.47)), we conclude (3.74) from the Banach–Steinhaus theorem.

Now observe that (3.66) implies, in particular, the convergences

θn,t → θt, θn,xx → θxx, pn,t → pt, pn,xx → pxx, qn,t → qt, wn,t → wt,(3.76)

all weakly in L2(Ωτ ). Combining all the above convergences, and letting n → ∞,

we finally can infer that (p, q, θ, w) = T (p, q, θ, w). This concludes the proof of thelemma.

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By virtue of Lemmas 3.3 and 3.4, we deduce from Lemma 3.2 that T has a fixedpoint in Mτ which then is a solution to the system (3.2)–(3.10). To conclude theproof of Theorem 3.1, we still need to show the uniqueness. We achieve this throughthe following result, which even shows global uniqueness.

Lemma 3.5. Let the assumptions of Theorem 3.1 be fulfilled, and let τ ∈ (0, T ]be arbitrary. Then the system (3.2)–(3.10) has at most one solution in Xτ .

Proof. Suppose that (pi, qi, θi, wi) ∈ Xτ , i = 1, 2, are two solutions to (3.2)–(3.10)on Ωτ for some τ ∈ (0, T ]. Let p := p1 − p2, q := q1 − q2, θ := θ1 − θ2, w := w1 −w2,and put σi := H1[pi+qi, wi]+θi H2[pi+qi, wi], ψi := H3[pi+qi, wi]+θi H4[pi+qi, wi]for i = 1, 2. Then it holds that

pt − pxx = σ1 − σ2,(3.77)

qt = σ2 − σ1,(3.78)

θt − θxx = −F1[p1 + q1, w1]t + F1[p2 + q2, w2]t + p21,xx − p2

2,xx(3.79)

+σ1p1,xx − σ2 p2,xx + g(x, t, θ1) − g(x, t, θ2),

wt = ψ1 − ψ2,(3.80)

with corresponding zero initial and boundary conditions. Owing to (H4)(iii), we havefor every (x, t) ∈ Ωτ

max|σ1(x, t) − σ2(x, t)|, |ψ1(x, t) − ψ2(x, t)|≤ C1

(|θ(x, t)| + max

0≤r≤t(|p(x, r)| + |q(x, r)| + |w(x, r)|)

)(3.81)

≤ C1

(|θ(x, t)| +

∫ t

0

(|pt(x, r)| + |qt(x, r)| + |wt(x, r)|

)dr

),

where by Ci, i ∈ N, we denote positive constants that depend only on the data ofthe system and on the Xτ -norms of (pi, qi, θi, wi), i = 1, 2. Hence, we may multiply(3.77) by pt, and by −pxx, respectively, (3.78) by qt, and (3.80) by wt, respectively,add the four resulting equations, integrate over space and time, and apply Young’sinequality appropriately to arrive at the estimate

∫ t

0

∫Ω

(p2t + p2

xx + q2t + w2

t

)dx ds ≤ C2

∫ t

0

∫ s

0

∫Ω

(p2t + q2

t + w2t

)dx dr ds

+C3

∫ t

0

∫Ω

θ2 dx ds.(3.82)

Next, we integrate (3.79) over [0, s] for some s > 0. We obtain

θ −∫ s

0

θxx dr = −F1[p1 + q1, w1] + F1[p2 + q2, w2] +

∫ s

0

(p21,xx − p2

2,xx

)dr

+

∫ s

0

(σ1p1,xx − σ2p2,xx

)dr +

∫ s

0

(g(x, r, θ1) − g(x, r, θ2)

)dr.(3.83)

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Our aim is to multiply (3.83) by θ, and integrate over Ω × [0, t] for some t ∈ [0, τ ].In what follows, we first estimate the terms resulting from the right-hand side indi-vidually. To this end, let γ > 0 (to be specified later). First, we note that for a.e.(x, t) ∈ Ωτ it holds that

|F1[p1+q1, w1](x, t)−F1[p2+q2, w2](x, t)| ≤ C4

∫ t

0

(|pt|+|qt|+|wt|

)(x, r) dr,(3.84)

so that, using Young’s inequality,∫ t

0

∫Ω

|θ(x, s)| |F1[p1 + q1, w1](x, s) −F1[p2 + q2, w2](x, s)| dx ds

(3.85)

≤ γ

2

∫ t

0

∫Ω

θ2 dx ds +C5

2γ

∫ t

0

∫ s

0

∫Ω

(p2t + q2

t + w2t

)dx dr ds.

Moreover, owing to (H3), we have

∫ t

0

∫Ω

|θ(x, t)|∫ s

0

|g(x, r, θ1(x, r)) − g(x, r, θ2(x, r)| dr dx ds

≤ C6

∫ t

0

∫Ω

|θ(x, t)|∫ s

0

|θ(x, r)| dr dx ds(3.86)

≤ γ

2

∫ t

0

∫Ω

θ2 dx ds +C6

2γ

∫ t

0

∫ s

0

∫Ω

θ2 dx dr ds.

Next, we estimate∫ t

0

∫Ω

|θ(x, s)|∫ s

0

|σ1p1,xx − σ2p2,xx|(x, r) dr dx ds

≤∫ t

0

∫Ω

|θ(x, s)|∫ s

0

|σ2(x, r)| |pxx(x, r)| dr dx ds

(3.87)

+

∫ t

0

∫Ω

|θ(x, s)|∫ s

0

(|p1,xx| |σ1 − σ2|

)(x, r) dr dx ds

=: B1 + B2.

By the boundedness of σ2,

B1 ≤ γ

2

∫ t

0

∫Ω

θ2 dx ds +C7

2 γ

∫ t

0

∫ s

0

∫Ω

p2xx dx dr ds.(3.88)

Next, we employ the Gagliardo–Nirenberg inequality (2.21) with p = +∞, q = r = 2,ω = 1/2, to conclude that, for i = 1, 2 and every x ∈ Ω,∫ s

0

|pi,xx(x, r)|2 dr ≤∫ s

0

‖pi,xx(·, r)‖2∞ dr

≤ 2K20

∫ s

0

(‖pi,xx(·, r)‖2

2 + ‖pi,xx(·, r)‖2‖pi,xxx(·, r)‖2

)dr

(3.89)

≤ C8 max0≤r≤s

‖pi,xx(·, r)‖22 + C9 max

0≤r≤s‖pi,xx(·, r)‖2

(∫ s

0

∫Ω

|pi,xxx|2 dx dr

)1/2

≤ C10.

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Hence, we have

∫ s

0

(|σ1 − σ2| |p1,xx|

)(x, r) dr ≤

(∫ s

0

|(σ1 − σ2)(x, r)|2 dr)1/2(∫ s

0

|p1,xx(x, r)|2 dr)1/2

≤√

C10

(∫ s

0

|(σ1 − σ2)(x, r)|2 dr)1/2

,(3.90)

so that, by virtue of (3.81) and of Young’s inequality,

B2 ≤ γ

2

∫ t

0

∫Ω

θ2 dx ds +C11

2γ

∫ t

0

∫ s

0

∫Ω

|σ1 − σ2|2 dx dr ds

(3.91)

≤ γ

2

∫ t

0

∫Ω

θ2 dx ds +C12

2γ

∫ t

0

∫ s

0

∫Ω

(θ2 + p2

t + q2t + w2

t

)dx dr ds.

Finally, using (3.89), we estimate

∫ t

0

∫Ω

|θ(x, s)|∫ s

0

(p21,xx − p2

2,xx

)(x, r) dr dx ds

≤∫ t

0

∫Ω

|θ(x, s)|∫ s

0

(|pxx| |p1,xx + p2,xx|

)(x, r) dr dx ds

≤∫ t

0

∫Ω

|θ(x, s)|(∫ s

0

|pxx(x, r)|2dr)1/2(∫ s

0

|(p1,xx + p2,xx)(x, r)|2dr)1/2

dx ds(3.92)

≤ C13

∫ t

0

∫Ω

|θ(x, s)|(∫ s

0

|pxx(x, r)|2 dr)1/2

dx ds

≤ γ

2

∫ t

0

∫Ω

θ2 dx ds +C14

2γ

∫ t

0

∫ s

0

∫Ω

p2xx dx dr ds.

Now, we multiply (3.83) by θ and integrate over Ω×[0, t] for some t ∈ [0, τ ]. Combiningthe estimates (3.85)–(3.92) and choosing γ > 0 appropriately small, we obtain theinequality

∫ t

0

∫Ω

θ2 dx ds ≤ C15

∫ t

0

∫ s

0

∫Ω

(θ2 + p2

xx + p2t + q2

t + w2t

)dx dr ds.(3.93)

Consequently, combining inequalities (3.82) and (3.93), we have finally shown that

∫ t

0

∫Ω

(θ2 + p2

t + p2xx + q2

t + w2t

)dx ds

(3.94)

≤ C16

∫ t

0

∫ s

0

∫Ω

(θ2 + p2

t + p2xx + q2

t + w2t

)dx dr ds,

whence, by Gronwall’s lemma, pt = qt = wt = θ = 0 a.e. in Ωτ , so that the assertionfollows. With this, the proof of Theorem 3.1 is complete.

4. Global existence. Suppose now that the hypotheses (H1) to (H5) hold sothat (3.2)–(3.11) has a unique solution (p, q, θ, w) on Ωτ which satisfies (3.11)–(3.14).Using the compatibility condition u0(0) = 0, we then easily verify that (u, θ, w), where

u(x, t) =

∫ x

0

(p(ξ, t) + q(ξ, t)

)dξ,(4.1)

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solves (1.1)–(1.7) on Ωτ and satisfies (2.16). Now let τ ∈ (0, T ] be arbitrary such that(u, θ, w) can be extended to a solution of (1.1)–(1.7) on Ωτ and satisfies θ(x, t) ≥ θfor some θ > 0, as well as the smoothness properties (2.16). Owing to the globaluniqueness result of Lemma 3.5, this solution is unique. We are now going to derivea number of global a priori estimates. To this end, we denote by Ci, i ∈ N, positiveconstants which may depend on the given data of system (1.1)–(1.7), but neither onτ nor on the lower bound θ for the temperature. For notational convenience, we putε := ux.

First estimate. We multiply (1.1) by ut, add the result to (1.3), and thenintegrate over Ωt, t ∈ (0, τ ], and by parts. In light of (H1), we have

1

2

∫Ω

u2t (x, t) dx +

∫Ω

(θ(x, t) + F1[ε, w](x, t)

)dx

≤ C1 +

∫ t

0

∫Ω

g(x, r, θ(x, r)) dx dr +

∫ t

0

∫Ω

fut dx dr.(4.2)

Invoking (2.2), (2.4), (H2), (H4)(i), the positivity of θ, and Gronwall’s lemma, we findthat

max0≤t≤τ

(‖θ(·, t)‖1 + ‖ut(·, t)‖2

)≤ C2.(4.3)

Second estimate. We multiply (1.3) by −θ−1 and integrate over Ωt, t ∈ (0, τ ].(Note that θ−1 is actually bounded, since θ ≥ θ > 0.) It follows that

∫ t

0

∫Ω

(θ2x

θ2+

ε2t

θ

)dx dr ≤ C3 +

∫ t

0

∫Ω

1

θ

(F1[ε, w]t − σεt − g(x, r, θ)

)dx dr

+

∫Ω

log(θ(x, t)) dx.(4.4)

In light of (4.3) and of the elementary inequality log(θ) ≤ θ for θ > 0, the secondintegral on the right-hand side is bounded. Also, we obtain from (1.2), (1.4), (H3),(H5), and Young’s inequality that a.e. in Ωτ it holds that

F1[ε, w]t − σεt − g(x, r, θ) ≤ H3[ε, w]G[w]t − θH2[ε, w] εt − g0(x, r) + K1θ

≤ −(θH4[ε, w] + wt)G[w]t + K2θ|εt| + K1θ ≤ K1θ + K2θ|εt| +K5

4K2

2θ2.

(4.5)

Therefore, using (4.3), we find from Young’s inequality that

∫ t

0

∫Ω

1

θ

(F1[ε, w]t − σεt − g(x, r, θ)

)dx dr ≤ C4

(1 +

∫ t

0

∫Ω

|εt| dx dr)

≤ C5 +1

2

∫ t

0

∫Ω

ε2t

θdx dr.(4.6)

In conclusion, we have shown the estimate

∫ τ

0

∫Ω

(θ2x

θ2+

ε2t

θ

)dx dt ≤ C6.(4.7)

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But then, using Schwarz’s inequality for a fixed t ∈ (0, τ),

∫Ω

∣∣∣(√θ)x

∣∣∣ (x, t) dx =

∫Ω

∣∣∣∣∣θx(x, t)

2√

θ(x, t)

∣∣∣∣∣ dx(4.8)

≤ 1

2

(∫Ω

θ(x, t) dx

)1/2 (∫Ω

θ2x(x, t)

θ2(x, t)dx

)1/2

,

whence, using the Gagliardo–Nirenberg inequality (2.21) with w =√θ, p = +∞,

q = 2, r = 1, and ω = 1, and invoking (4.3), we obtain∫ τ

0

‖θ(·, t)‖∞ dt ≤ C7 + C8

∫ τ

0

∫Ω

θ2x

θ2dx dt ≤ C9.(4.9)

Hence, ∫ τ

0

∫Ω

θ2 dx dt ≤ max0≤t≤τ

‖θ(·, t)‖1

∫ τ

0

‖θ(·, t)‖∞ dt ≤ C10.(4.10)

Third estimate. We now exploit the decomposition (3.1). We have ux = ε =p + q, where, owing to (4.3), ‖p‖L∞(Ωτ ) ≤ C11. Therefore, invoking (H4) and (2.11),

it holds that for every (x, t) ∈ Ωτ ,

|σ(x, t)| + |ψ(x, t)| ≤ C12 + 2K2 θ(x, t) + 2K4 max0≤r≤t

(|q(x, r)| + |w(x, r)|

).(4.11)

Now multiply (3.5) by q, and (3.8) by w, respectively, add the resulting equations,and integrate over [0, t], where t ∈ (0, τ ]. Using Young’s inequality and invoking (4.9),we obtain from (4.11) the estimate

1

2

(q2(x, t) + w2(x, t)

)

≤ C13

[1 + max

0≤r≤t

(|q(x, r)| + |w(x, r)|

)(1 +

∫ t

0

(|q(x, r)| + |w(x, r)|

)dr

)](4.12)

≤ C14 +1

4max0≤r≤t

(q2(x, r) + w2(x, r)

)+ C15

∫ t

0

(q2(x, r) + w2(x, r)

)dr.

Taking the maximum with respect to t on both sides, we obtain from Gronwall’slemma that

‖q‖L∞(Ωτ ) + ‖w‖L∞(Ωτ ) ≤ C16,(4.13)

whence also

‖ε‖L∞(Ωτ ) + maxj∈1,3

‖Hj [ε, w]‖L∞(Ωτ ) ≤ C17,(4.14)

and we obtain from (3.5) and (3.8), using (4.9)–(4.11), that

‖qt‖L1(0,τ ;L∞(Ω))∩L2(Ωτ )∩L∞(0,τ ;L1(Ω))(4.15)

+ ‖wt‖L1(0,τ ;L∞(Ω))∩L2(Ωτ )∩L∞(0,τ ;L1(Ω)) ≤ C18.

Moreover, (4.10) and (4.14) imply that the right-hand side of (3.2) is bounded inL2(Ωτ ); hence, using standard parabolic estimates, we can conclude that

‖p‖H1(0,τ ;L2(Ω))∩L2(0,τ ;H2(Ω))∩C([0,τ ];H1(Ω)) ≤ C19,(4.16)

which yields, in particular, that uxt = pxx is bounded in L2(Ωτ ).

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Fourth estimate. In the next step, we perform a classical estimate (cf. [5]);namely, we multiply (1.3) by −θ−1/3 and integrate over Ωt for t ∈ (0, τ ]. It thenfollows from (4.5) that∫ t

0

∫Ω

(θ−4/3θ2

x + θ−1/3ε2t

)dx dr ≤ C20

(1 +

∫Ω

θ2/3(x, t) dx +

∫ t

0

∫Ω

θ2/3 dx dr

)

+C21

(∫ t

0

∫Ω

θ2/3|εt| dx dr +

∫ t

0

∫Ω

θ5/3 dx dr

).(4.17)

Owing to (4.3) and (4.10), the first, second, and fourth integrals on the right of (4.17)are bounded, and the remaining expression is estimated as follows:∫ t

0

∫Ω

θ2/3|εt| dx dr =

∫ t

0

∫Ω

θ5/6θ−1/6|εt| dx dr

(4.18)

≤ 1

2

∫ t

0

∫Ω

θ−1/3ε2t dx dr +

1

2

∫ t

0

∫Ω

θ5/3 dx dr.

Since the second summand on the right of (4.18) is again bounded, we have shownthe estimate ∫ τ

0

∫Ω

(θ−4/3θ2

x + θ−1/3ε2t

)dx dt ≤ C22.(4.19)

But then θ1/3 is bounded in L∞(0, τ ;L3(Ω)) ∩ L2(0, τ ;H1(Ω)), and the Gagliardo–Nirenberg inequality (2.21), with p = +∞, r = 2, q = 3, and ω = 2/5, yields that∫ τ

0

‖θ(·, t)‖5/3∞ dt ≤ C23,(4.20)

whence, using (4.3) once more, we obtain∫ τ

0

∫Ω

θ8/3 dx dt ≤ C24.(4.21)

Thus, the right-hand sides of (3.2), (3.5), and (3.8), respectively, are bounded inL8/3(Ωτ ), and we can infer from standard parabolic estimates, using εt = pxx, that

‖pt‖L8/3(Ωτ ) + ‖εt‖L8/3(Ωτ ) + ‖qt‖L8/3(Ωτ ) + ‖wt‖L8/3(Ωτ ) ≤ C25.(4.22)

Fifth estimate. We now turn our attention to (1.3). By virtue of (4.21) and(4.22), and invoking (2.5), we easily verify that F1[ε, w]t and the right-hand side of(1.3) are bounded in L4/3(Ωτ ). Therefore, multiplying (1.3) by θ, integrating over Ωτ

for t ∈ (0, τ ], and applying Young’s inequality and (4.3), we see that for any γ > 0 itholds that

‖θ(·, t)‖22 +

∫ t

0

∫Ω

θ2x dx dr ≤ C26

(1 + γ−1

)+ γ

∫ t

0

∫Ω

θ4 dx dr

(4.23)

≤ C27

(1 + γ−1 + γ

∫ t

0

‖θ(·, r)‖3∞ dr

).

Now we use (4.3) and the Gagliardo–Nirenberg inequality (2.21) with p = +∞, q = 1,r = 2, and ω = 2/3 in order to obtain that

∫ t

0

‖θ(·, r)‖3∞ dr ≤ C28

(1 +

∫ t

0

∫Ω

θ2x dx dr

).(4.24)

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Hence, choosing γ > 0 small enough, we have shown the estimate

‖θ‖L∞(0,τ ;L2(Ω))∩L2(0,τ ;H1(Ω)) ≤ C29,(4.25)

whence, using interpolation once more, we obtain

‖θ‖L6(Ωτ ) ≤ C30.(4.26)

Thus, just as in the derivation of (4.22), we have

‖pt‖L6(Ωτ ) + ‖εt‖L6(Ωτ ) + ‖qt‖L6(Ωτ ) + ‖wt‖L6(Ωτ ) ≤ C31.(4.27)

But then F1[ε, w]t and the right-hand side of (1.3) are bounded in L2(Ωτ ), and stan-dard parabolic estimates, also using the continuous imbeddings H1(0, τ ;L2(Ω)) ∩L2(0, τ ;H2(Ω)) → C([0, τ ];H1(Ω)) → C(Ωτ ), yield that

‖θ‖H1(0,τ ;L2(Ω))∩L2(0,τ ;H2(Ω))∩C([0,τ ];H1(Ω))∩C(Ωτ ) ≤ C32.(4.28)

This implies, in particular, that σt and ψt are bounded in L2(Ωτ ), so that

‖qtt‖L2(Ωτ ) + ‖wtt‖L2(Ωτ ) ≤ C33.(4.29)

In addition, we may argue as in the derivation of (3.38) to conclude that also

‖p‖H2(0,τ ;V ∗0 )∩H1(0,τ ;H1(Ω)) ≤ C34.(4.30)

Sixth estimate. In light of the above estimates, we have, for a.e. (x, t) ∈ Ωτ ,

|σx(x, t)| + |ψx(x, t)| ≤ C35

(|θx(x, t)| + max

0≤r≤t((|px| + |qx| + |wx|)(x, r))

)(4.31)

≤ C36

(1 + |θx(x, t)| +

∫ t

0

(|pxt(x, r)| + |qxt(x, r)| + |wxt(x, r)|

)dr

).

Hence, differentiating (3.5) and (3.8), respectively, with respect to x, and invokingthe estimates (4.28) and (4.30), we easily derive the estimate

‖qxt‖L2(Ωt) + ‖wxt‖L2(Ωt) ≤ C37

(1 +

∫ t

0

(‖qxt‖L2(Ωr) + ‖wxt‖L2(Ωr)

)dr

),(4.32)

whence, using Gronwall’s lemma, we have

‖qxt‖L2(Ωτ ) + ‖wxt‖L2(Ωτ ) ≤ C38.(4.33)

Finally, we conclude from the above estimates that also

‖pxxx‖L2(Ωτ ) ≤ C39.(4.34)

In conclusion, combining all previously shown estimates, we have shown that

‖(p, q, θ, w)‖Xτ ≤ C40,(4.35)

where Xτ is the space introduced in (3.15).

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Conclusion of the proof of Theorem 2.1. So far we have shown that (4.35)holds as long as there is some θ > 0 such that θ ≥ θ on Ωτ . To conclude the proof ofthe assertion, we still have to prove the validity of (2.17). To this end, first observethat we have shown above that εt = pxx is bounded in L6(Ωτ ) ∩ L2(0, τ ;H1(Ω)). Inparticular, there is some β1 > 0, independent of τ , such that

∫ t

0

‖εt(·, r)‖∞ dr ≤ β1 ∀t ∈ (0, τ ].(4.36)

Besides, there is some β2 > 0, independent of τ , such that

max(x,t)∈Ωτ

|wt(x, t)| ≤ β2.(4.37)

Now test (1.3) with an arbitrary function z ∈ H1(Ωτ ) satisfying z ≤ 0 a.e. in Ωτ . Inview of (2.10), (4.5), and (4.37) it follows, for a.e. t ∈ (0, T ), that∫

Ω

(zθt + zxθx

)(x, t) dx ≤

∫Ω

|z(x, t)|(

(F [ε, w]t − σεt − g(·, ·, θ))(x, t))dx

≤∫

Ω

|z(x, t)|(

(−θH4[ε, w] − wt)G[w]t + K2θ|εt| + K1θ)

(x, t) dx

(4.38)

≤(K1 + K2K5β2 + K2‖εt(·, t)‖∞

)∫Ω

|z(x, t)| θ(x, t) dx

≤ ϕ(t)

∫Ω

|z(x, t)| θ(x, t) dx,

where ϕ(t) := (K1 + K2K5β2 + K2‖εt(·, t)‖∞) is by (4.36) bounded in L1(0, τ) by aconstant which does not depend on τ ∈ (0, T ]. Now, put

z(x, t) := −(δ exp

(−∫ t

0

ϕ(s) ds

)− θ(x, t)

)+

for (x, t) ∈ Ωτ .(4.39)

Then it follows from inequality (4.38) that

∫Ω

(z

(z + δ exp

(−∫ t

0

ϕ(s) ds

))t

+ z2x

)(x, t) dx

(4.40)

≤ ϕ(t)

∫Ω

|z|(|z| + δ exp

(−∫ t

0

ϕ(s) ds

))(x, t) dx.

This yields, in particular,

1

2

d

dt

∫Ω

z2(x, t) dx +

∫Ω

z2x(x, t) dx ≤ ϕ(t)

∫Ω

z2(x, t) dx.(4.41)

Therefore, by Gronwall’s lemma, z = 0, and thus

θ(x, t) ≥ δe−((K1+K2K5β2)t+K2β1) ∀(x, t) ∈ Ωτ .(4.42)

Therefore, we can claim that τ = T , and the assertion of Theorem 2.1 is completelyproved.

Remark 4. It does not present any major difficulties to extend the above proofto the more general case when H3 and H4 are vector hysteresis operators and, ac-cordingly, (1.4) is a vector differential equation (then, of course, the hypotheses (H4)and (H5) have to be appropriately modified).

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Remark 5. It is easy to see that the solution (u, θ, w) depends Lipschitz continu-ously on the data of the system. Indeed, a closer look at the proof of Lemma 3.5 revealsthat L2(Ω)-variations of u0, u1, θ0, w0 and L2(ΩT )-variations of f lead to Lipschitzvariations of (p, q, θ, w) in the norm of the space (H1(0, T ;L2(Ω))∩L2(0, T ;H2(Ω)))×H1(0, T ;L2(Ω)) × L2(ΩT ) ×H1(0, T ;L2(Ω)). A similar result holds for variations ofg. As the line of arguments should be clear, we leave the explicit formulation and theproof of the corresponding result to the reader.

Remark 6. It seems natural to discuss the asymptotic behavior of system (1.1)–(1.7) as µ 0. As our method of proof strongly relies on the presence of the viscousterm −µuxxt in (1.1), our analysis does not cover this problem, which seems to bevery difficult.

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phase-field models with hysteresis in one-dimensional thermoviscoplasticity

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