A MATHEMATICAL MODEL FOR CASE HARDENING OF STEEL

A MATHEMATICAL MODEL FOR CASE

HARDENING OF STEEL

ANTONIO FASANO

Dipartimento di Matematica, Universit�a degli Studi di Firenze,

Viale Morgagni 67/A, 50134 Firenze, Italyfasano@math-uni¯.it

DIETMAR HÖMBERG

Weierstrass Institute for Applied Analysis and Stochastics,

Mohrenstrasse 39, 10117 Berlin, Germany

[email protected]

LUCIA PANIZZI

Scuola Normale Superiore, Piazza dei Cavalieri 7

56126 Pisa, [email protected]

Received 18 April 2008Revised 5 January 2009

Communicated by N. Bellomo

A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the

surface layer of a low-carbon steel part at a temperature su±cient to render the steel austenitic,followed by quenching to form a martensitic microstructure. The model consists of a nonlinear

evolution equation for the temperature, coupled with a nonlinear evolution equation for the

carbon concentration, both coupled with two ordinary di®erential equations to describe the

evolution of phase fractions. We investigate questions of existence and uniqueness of a solutionand ¯nally present some numerical simulations.

Keywords: Heat treatment; phase transitions; coupled PDE.

AMS Subject Classi¯cation: 35K60, 35R05, 82B26

1. Introduction

The goal of case hardening is to create a workpiece surface which is resistant to

external stresses and abrasion, while its case is still ductile in order to reduce fatigue

e®ects. The process will be explained in detail in the next section. It exploits the

solid�solid phase transitions occurring during thermal treatment of steel and

requires a certain amount of carbon in the layer to be hardened. Accordingly, the ¯rst

Mathematical Models and Methods in Applied SciencesVol. 19, No. 11 (2009) 2101�2126

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0218202509004054

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http://dx.doi.org/10.1142/S0218202509004054

stage of case hardening is a carburization step during which the outer workpiece layer

is enriched by carbon. The second stage is a quenching step during which a hard and

wear resistant boundary layer is achieved. Sometimes, before quenching, a period of

slow di®usion is allowed. The goal of this paper is to derive and analyze a math-

ematical model capable to describe the complete process of case hardening. Con-

cerning case hardening there is mostly engineering literature available (see Refs. 15

and 16 and references therein). Carburization and quenching are usually considered

and studied separately. There are papers concerning only the carburization (see, for

example, Ref. 5) and others regarding the quenching of carburized steel (see Refs. 18

and 22). Generally, in all of the engineering papers, a lot of attention is spent to

determine the process and material parameters; nevertheless the question of ¯nding a

consistent and exhaustive database for all the parameters occurring in the whole

process seems to be still open. There is a vast literature, for instance, about the

di®usion coe±cient of carbon in iron, (see Ref. 23 and references therein), but much

less for the heat transfer coe±cient during quenching. Mathematical models for phase

transitions in steel and in their applications to heat treatments like induction

hardening have been developed and analyzed, e.g. in Refs. 7, 10, 11, 13 and 26. The

model for the phase fraction evolutions in the present paper follows the one proposed

in Ref. 10.

The main novelty of this paper is the derivation and analysis of a mathematical

model for the complete case hardening process accounting for the coupling of tem-

perature, phase transitions and carbon di®usion. This allows one to evaluate the

e®ect of additional di®usion of carbon prior to quenching, which could a®ect the ¯nal

result (see Ref. 24). Thus, from application point of view, a more accurate model

might lead to a more e±cient process guidance and reduced energy consumption.

The paper is organized as follows: in the next section we will derive the model. In

Sec. 3, we present notations and assumptions. In Secs. 4 and 5 we will prove the

existence and uniqueness of a weak solution. Section 6 is devoted to numerical

simulations; then, in Sec. 7, we collect our ¯nal considerations and remarks.

2. The Mathematical Model

To ¯x ideas we ¯rst give a sketch of the gas carburizing process. Nowadays high-

technology industry employs mostly low-carbon steels, with a carbon content around

0.2%. For this reason the enrichment of carbon in a super¯cial layer of the workpiece

may be necessary to make it resistant to fatigue. The source of carbon is a carbon-rich

furnace atmosphere produced from many gaseous components, through several

chemical reactions (see Ref. 24 for technical details). The workpiece is kept in the

furnace until the desired amount of carbon is di®used. After carburization the second

stage is quenching, a rapid coolingwhich can be performedby immersion in oil orwater.

In the boundary layer which has been enriched by carbon, the rapid cooling leads

to the growth of martensite eventually yielding the desired hard and wear resistant

layer or case, which explains why this heat treatment is called case hardening.

2102 A. Fasano, D. H€omberg & L. Panizzi

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The kinetics of the phase change can be brie°y described as follows. Depending on

temperature, two di®erent lattice structures can occur: a body-centered-cubic (b.c.c.)

and a face-centered-cubic (f.c.c.) lattice. Above a certain temperature As steel is in

the austenitic phase, a solid solution of carbon in f.c.c. iron. Below As this lattice is no

longer stable. But before the lattice can change its con¯guration to form a b.c.c.

structure, carbon atoms have to di®use, due to the higher solubility of carbon in the

f.c.c lattice. The result is pearlite, a lamellar aggregate of ferrite and cementite, soft

and ductile. Upon high cooling rate carbon has no time to di®use and is trapped,

forming a tetragonally distorted b.c.c. lattice, called martensite. Note that, depen-

ding on the cooling history and the carbon concentration, also two other phases,

ferrite and bainite, can occur.

The transformation diagrams of interest for the modelling of the phase fractions

evolution (see Eqs. (2.1a), (2.1b) below), during the cooling process, are called indeed

continuous cooling transformation (CCT) diagrams and describe the transformation

of austenite as a function of time for a continuously decreasing temperature. For

instance, in the left-hand side of Fig. 1, the CCT diagram for the steel AISI 1045 is

shown. In other words a sample is austenitized and then cooled at a predetermined

rate and the degree of transformation is measured, for example by dilatometry. The

start of transformation is de¯ned as the temperature at which 1% of the new

microstucture has formed. The transformation is completed when only 1% of the

original austenite is left.

In carburized steels the process is strongly in°uenced by the carbon content, which

varies from the carbon-enriched super¯cial layer to the core. Thus, it cannot be

described by only one continuous-cooling-transformation diagram. Figure 2 shows a

continuous cooling diagram describing, for a given austenitizing condition, the

transformation at all carbon levels in a carburized specimen. The cross sections for

¯xed carbon percentages give CCT diagrams of the type of the one plotted in Fig. 1

on the left. This ¯gure also shows that, with in¯nitely-slow cooling, the CCT diagram

Fig. 1. Equilibrium diagram of the system iron-carbon (right) as limit of the CCT-diagram with in¯nite

low cooling rate (taken from Ref. 13).

A Mathematical Model for Case Hardening of Steel 2103

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is identical with the equilibrium diagram for the chemical composition of the steel. To

avoid unnecessary technicalities for the modelling, we assume that the cooling takes

place from the high temperature phase austenite with phase fraction a to two

di®erent product phases, pearlite with fraction p and martensite with fraction m. A

more elaborate model accounting for all the phases occurring during the heat

treatment of steel can be found in Ref. 13.

The evolution of the phases p and m can be described by the following system:

_p ¼ ð1� p�mÞg1ð�; cÞ; ð2:1aÞ_m ¼ ½minfmð�; cÞ; 1� pg �m�þg2ð�; cÞ; ð2:1bÞ

pð0Þ ¼ 0; ð2:1cÞmð0Þ ¼ 0; ð2:1dÞ

where c is the concentration of carbon. Here the bracket ½ �þ denotes the positive part

function ½x�þ ¼ maxfx; 0g and the dot means the derivative with respect to t. While

the growth rate of pearlite _p is assumed to be proportional to the remaining austenite

fraction, the rate of martensite growth _m is zero if m exceeds either the non-pearlitic

fraction 1� p, or the threshold m depending on both temperature and carbon con-

centration. Indeed martensite is produced at temperatures less than a value Ms but

Fig. 2. Three-dimensional presentation of the transformation characteristic of a 14NiCr14 steel, for

continuous cooling, after austenization at 1023�K (Symbols: ZW bainite, M martensite, P pearlite, F

ferrite), (taken from Ref. 13).


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complete transformation to martensite can be obtained only below some other

temperature threshold Mf . Both these temperatures depend on the local value of

carbon concentration. The quantity mð�; cÞ represents the maximum attainable

value of martensite fraction and can be de¯ned as:

mð�; cÞ ¼0 � >MsðcÞ

1 � <MfðcÞ

(

and by interpolation for intermediate temperatures. Since there is no phase transition

from pearlite to martensite, the term minfmð�; cÞ; 1� pg represents the maximal

fraction of martensite that can be reached at time t. In Fig. 3, obtained from

experimental data (see Ref. 3), we can see how the functions Ms and Mf look like.

The functions g1 and g2 are positive given functions that can be identi¯ed from the

time-temperature-transformation diagrams described before. The process of carbon

di®usion is governed by the following nonlinear parabolic equation:

@c

@t� divðð1� p�mÞDð�; cÞrcÞ ¼ 0:

The factor ð1� p�mÞ in front of the di®usion coe±cientDð�; cÞ re°ects the fact thatenrichment with carbon only takes place in the austenite phase. The di®erence in

carbon potential between the surface and the workpiece provides the driving force for

carbon di®usion into the piece. The carbon potential of the furnace atmosphere must

be greater than the carbon potential of the surface of the workpiece for carburizing to

occur. Hence we have the following boundary condition:

�ð1� p�mÞDð�; cÞ @c@�

¼ �ðc� cpÞ;

where �, the mass transfer coe±cient, controls the rate at which carbon is absorbed

by the steel during carburizing and cp is the carbon concentration in the furnace,

0

200

0

Tra

nsfo

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tem

pera

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Carbon content

Co

−200

400

0% martensite

~50%

~100%

Start of martensitetransformation

transformation

Start of martensite

End of martensite

Mass %1.60.4 0.8 1.2

Fig. 3. Level curves of function mð�; cÞ.


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usually named carbon potential of the gas. @c@� denotes the outward normal derivative.

The evolution of temperature during the entire process is described by the following

nonlinear problem

��ð�Þ @�@t

� divðkr�Þ ¼ �Lpð�Þ _pþ �Lmð�Þ _m;

�k@�

@�¼ hð�� Þ;

�ðx; 0Þ ¼ �0:

Here � is the mass density, � the speci¯c heat, k the heat conductivity of the material.

Lp and Lm denote latent heats of the austenite-pearlite and the austenite-martensite

phase changes, respectively. �� is the temperature of the coolant and �0ðxÞ is the

temperature at the beginning of the process. For simplicity � and k are taken

constant.

In the technical process, we have three di®erent time stages:

. Stage 1: carburization in a furnace, hence � 6¼ 0 and h ¼ 0.

. Stage 2: di®usion period, with � ¼ 0 and h 6¼ 0, serving as a linearized radiation

law.

. Stage 3: quenching with � ¼ 0 and h 6¼ 0.

From the mathematical point of view, without loss of generality, we will assume that

� and h are time-independent functions. Then, the mathematical result to be for-

mulated in the following section can be applied subsequently to the three process

stages, covering the complete case hardening process.

3. Assumptions and Main Result

Let � � R3 be an open bounded set with C 2-boundary @� and QT :¼ �� ð0;T Þ thecorresponding time cylinder. We use the following notations for function spaces:

. W 1;1ð0;T ;L1ð�ÞÞ ¼ fv 2 L1ð0;T ;L1ð�ÞÞ : vt 2 L1ð0;T ;L1ð�ÞÞg:

. W r;sp ðQT Þ ¼ Lpð0;T ;W r

p ð�ÞÞ \W sp ð0;T ;Lpð�ÞÞ.

For p ¼ 2 we write W r;sp ðQT Þ ¼ Hr;sðQT Þ.

. We denote by V the space H 1ð�Þ and by V � the space ðH 1ð�ÞÞ�.W ð0;T Þ ¼ fv 2 L2ð0;T ;V Þ : vt 2 L2ð0;T ;V �Þg, endowed with the norm

jjvjjWð0;T Þ ¼Z T

0

ðjjvðtÞjj2V þ jjv 0ðtÞjj 2V � Þdt� �1

2

:

Throughout the paper we will use the following assumptions:

(A1) � and k are positive constants.

(A2) � 2 CðRÞ and there exist positive constants �0; �1 such that 0 < �0 ��ð�Þ � �1. Lp;Lm 2 L1ðRÞ and they are Lipschitz-continuous.


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(A3) �� is a positive constant. h 2 L1ð@�Þ with hðxÞ 0 a.e. in @�. We assume that

�0 2 H 1ð�Þ and c0 2 L2ð�Þ.(A4) g1; g2 are Lipschitz-continuous in both variables, moreover there are positive

constants �1; �2 such that 0 � g1ð�; cÞ � �1; 0 � g2ð�; cÞ � �2; 8 �; c 2 R.

(A5) m is Lipschitz-continuous satisfying mð�; cÞ 2 ½0; 1� for every �; c 2 R.

(A6) Dð�; cÞ is Lipschitz in both arguments and there are constants �3; �4 such that

0 < �3 � Dð�; cÞ � �4; 8 �; c 2 R.

(A7) cp is a positive constant. � 2 L1ð@�Þ with � 0 a.e. in @�.

Summarizing the model equations of Sec. 2, we consider the following boundary value

problem:

��ð�Þ @�@t

� divðkr�Þ ¼ �Lpð�Þpt þ �Lmð�Þmt in QT ; ð3:1aÞ

@c

@t� divðð1� p�mÞDð�; cÞrcÞ ¼ 0 in QT ; ð3:1bÞ

pt ¼ ð1� p�mÞg1ð�; cÞ in QT ; ð3:1cÞmt ¼ ½minfmð�; cÞ; 1� pg �m�þg2ð�; cÞ in QT ; ð3:1dÞ

�k@�

@�¼ hð�� Þ on @�� ð0;T Þ; ð3:1eÞ

�ð1� p�mÞDð�; cÞ @c@�

¼ �ðc� cpÞ on @�� ð0;T Þ; ð3:1fÞ

�ðx; 0Þ ¼ �0 in �; ð3:1gÞcðx; 0Þ ¼ c0 in �; ð3:1hÞ

pð0Þ ¼ 0 in �; ð3:1iÞmð0Þ ¼ 0 in �: ð3:1jÞ

We are going to prove that, under the hypothesis above, the considered problem

has a weak solution.

Theorem 3.1. (Existence of a weak solution) Assume (A1)�(A7), then there exists

a weak solution ð�; c; p;mÞ to problem (3.1a)�(3.1j) such that � 2 H 2;1ðQT Þ,c 2 W ð0;T Þ; p;m 2 W 1;1ð0;T ;L1ð�ÞÞ.

With slightly stronger assumptions on the data, we can also prove uniqueness.

Theorem 3.2. (Uniqueness) Suppose that (A1)�(A7) are satis¯ed. Assume

moreover that � is constant, D ¼ Dð�Þ;h; � 2 W 15 ð@�Þ; �0; c0 2 W 2

5 ð�Þ. Then the

solution to (3.1a)�(3.1j) is unique.

Remark 3.1. The regularity assumptions on the boundary and initial values in the

uniqueness theorem could be weakened; to avoid unnecessary technicalities we

assumed �� and cp to be constants, but they could in fact be functions of space and

time.


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4. Proof of Theorem 3.1

The proof is carried out using a nested ¯xed point argument. We divide the proof in

three steps. The ¯rst is a preliminary lemma concerning the ODE system (3.1c)�(3.1d) only, for � and c prescribed. The second step is the coupling of the ODE system

and the temperature equation, which gives a solution p;m; � depending on c and the

third is the further coupling with the equation for c.

We begin with considering the initial value problem

zt ¼ fðz; �; cÞ in QT ; ð4:1aÞ

zð0Þ ¼ 0 in �; ð4:1bÞ

where z ¼ ðp;mÞT and f ¼ ðf1; f2ÞT denotes the right-hand side of (3.1c) and (3.1d).

Lemma 4.1. Under the assumptions (A4), (A5) the following statements are valid:

(a) For every �; c 2 L2ðQT Þ problems (4.1a)–(4.1b) has a unique solution z such that

p 0;m 0 and

jjjzjjjW 1;1ð0;T ;L1ð�ÞÞ � M

for a constant M independent of � and c. Moreover, there exists a constant cTsuch that

0 � pðx; tÞ þmðx; tÞ � cT < 1 for a:e: ðx; tÞ in QT :

(b) There are constants M1;M2 > 0 such that for every �1; �2; c1; c2 2 LpðQT Þ, foralmost all t 2 ð0;T Þ and all p 2 we have

jjjz1ðtÞ � z2ðtÞjjjpW 1;pð�Þ � M1

Z t

0

jj�1 � �2 jjpLpð�ÞdsþM2

Z t

0

jjc1 � c2 jjpLpð�Þds;

ð4:2Þ

where pi;mi is the solution corresponding to ð�i; ciÞ and j � j is the Euclidean

norm in R2.

Proof of Lemma 4.1. In order to prove (a) it is convenient to rewrite problem

(4.1a)�(4.1b) as:

zt ¼ F ðz; tÞ in ð0;T Þ ð4:3aÞ

zð0Þ ¼ 0 ð4:3bÞ

with F ðz; �Þ ¼ fðz; �ð�Þ; cð�ÞÞ.First of all we are going to show that the hypothesis of the existence Car-

ath�eodory's theorem are satis¯ed:

(i) t 7! F ðz; tÞ is measurable on ð0;T Þ for each z 2 ½0; 1� � ½0; 1�;

z 7! F ðz; tÞ is continuous on ½0; 1� � ½0; 1� for almost all t 2 ð0;T Þ:


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These conditions follow from the de¯nition of F as a consequence of the

measurability of � and c on ð0;T Þ and of the fact that g1ð�; cÞ; g2ð�; cÞ are

Lipschitz-continuous in both variables.

(ii) Using assumptions (A4), (A5) we have

jF1ðz; tÞj � j1� p�mjg1ð�; cÞ � �1 on ½0; 1� � ½0; 1� � ð0;T Þ;jF2ðz; tÞj � jm �mjg2ð�; cÞ � �2 on ½0; 1� � ½0; 1� � ð0;T Þ:

According to Carath�eodory's theorem (cf. e.g. p. 1044 of Ref. 27) (4.3a)–(4.3b)

has a solution on some time interval ð0;TþÞ.

Next we are going to show that the solution is unique. To this end we have to

prove that there holds

jF ðz1; tÞ � F ðz2; tÞj � Ljz1 � z2j 8 ðz1; tÞ; ðz1; tÞ 2 ½0; 1� � ½0; 1� � ð0;T Þ: ð4:4Þ

Indeed, according to the de¯nition of F:

jF ðz1; tÞ � F ðz2; tÞj2 ¼ jð1� p1 �m1Þg1ðtÞ � ð1� p2 �m2Þg1ðtÞj2

þ j½minfmðtÞ; 1� p1g �m1�þg2ðtÞ

� ½minfmðtÞ; 1� p2g �m2�þg2ðtÞj2:

Thanks to the boundedness of g1 and g2, we obtain

jð1� p1 �m1Þg1ðtÞ � ð1� p2 �m2Þg1ðtÞj � �1ðjp1 � p2j þ jm2 �m1jÞ

and

j½minfmðtÞ; 1� p1g �m1�þg2ðtÞ � ½minfmðtÞ; 1� p2g �m2�þg2ðtÞj

� �2jminfmðtÞ; 1� p1g �m1 �minfmðtÞ; 1� p2g þm2j:

We shall now distinguish some cases.

If either minfmðtÞ; 1� pig ¼ 1� pi or minfmðtÞ; 1� pig ¼ mðtÞ, for i ¼ 1; 2,

(4.4) immediately follows.

If minfmðtÞ; 1� p1g ¼ 1� p1 and minfmðtÞ; 1� p2g ¼ mðtÞ (the same holds for

inverted indices), we have

�2jminfmðtÞ; 1� p1g �m1 �minfmðtÞ; 1� p2g þm2j

� �2ðjm1 �m2j þ j1� p1 �mðtÞjÞ � �2 jm1 �m2j þ jp1 � p2jð Þ: ð4:5Þ

Thus, there exists a positive constant L such that

jF ðz1; tÞ � F ðz2; tÞj � Ljz1 � z2j:

Hence we have proved uniqueness of z on ð0;TþÞ:Now, we de¯ne T� as the maximal time such that the solution to (4.3a,b) exists and

Z < 1� � on ð0;T�Þ, where Z ¼ pþm.

The last step in order to prove point (a) of the lemma is to show that for any T > 0

there exists an � such that jZj � 1� � in ½0;T �.


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This will be done by means of a classical comparison criterium for ODE (see for

instance Ref. 17, Chap. I, Prop. 3.1).

Z ¼ pþm satis¯es, on ½0;T�Þ:

_ZðtÞ ¼ ð1� ZðtÞÞg1ðtÞ þ ½minfmðtÞ; 1� pðtÞg �mðtÞ�þg2ðtÞ

� gðt;ZðtÞÞ :¼ ð1� ZðtÞÞðg1ðtÞ þ g2ðtÞÞ;

Zð0Þ ¼ 0:

Now, if we consider on ½0;T � the auxiliary problem:

V:ðtÞ ¼ ð1� V ðtÞÞðg1ðtÞ þ g2ðtÞÞ ¼ gðt;V ðtÞÞ;

V ð0Þ ¼ 0;

the solution is given by

V ðtÞ ¼ 1� e�R t

0ðg1þg2ÞðsÞds 8 t 2 ½0;T �

and we immediately have that there exists a constant CT > 0 such that:

0 � V ðtÞ � CT < 1 on ½0;T Þ:

Notice that gðt;V ðtÞÞ ¼ ð1� V ðtÞÞðg1ðtÞ þ g2ðtÞÞ is Lipschitz-continuous on ½0;T Þwith respect to V.

Thus, choosing � ¼ 1� CT , we have

ZðtÞ � V ðtÞ � 1� � on ½0;T �:

Since T� was chosen maximally such that ZðtÞ � 1� � on ½0;T��, it follows that

T� T .

(b) Let us consider again the equation zt ¼ fðz; �; cÞ. Let zi be the solution to

(4.1a)�(4.1b), corresponding to �i; ci; i ¼ 1; 2. Denoting z ¼ z1 � z2, subtracting the

equations and taking the scalar product with the function jzjp�2z, we obtain:

1

p

Z�

jzðtÞjpdx ¼Z t

0

Z�

ðfðz1; �1; c1Þ � fðz1; �2; c2ÞÞ � zjzjp�2dxds: ð4:6Þ

Invoking (A4), f is Lipschitz-continuous in all variables, thus, proceeding from (4.6),

the conclusion follows through standard application of Young's inequality and

Gronwall lemma. The proof is thus complete.

Next, we de¯ne

Bð�; cÞ :¼ �Lpð�Þ _pþ �Lmð�Þ _m; ð4:7Þ

where ðp;mÞ depends on �; c as characterized by the previous lemma.


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Lemma 4.2. Suppose that (A2), (A4) hold. Then the operator B de¯ned by (4.7) has

the following properties

(a) There exists a constant �B independent of �, c such that, for all � 2 L2ðQT Þ; c 2L2ðQT Þ there holds

jjBð�; cÞjjL1ðQT Þ � �B:

(b) Given c 2 L2ðQT Þ, let �k � L2ðQT Þ be any sequence converging strongly in

L2ðQT Þ to � 2 L2ðQT Þ. Then for every p 2 ½1;1Þ, we have

Bð�k; cÞ ! Bð�; cÞ strongly in LpðQT Þ: ð4:8Þ

(c) There are constants K1;K2 > 0 such that for all �1; �2; c1; c2 2 L2ðQT Þ and for

almost all x 2 � and every t 2 ð0;T ÞZ t

0

jBð�1ðx; sÞ; c1ðx; sÞÞ � Bð�2ðx; sÞ; c2ðx; sÞÞj2ds

� K1

Z t

0

j�1ðx; sÞ � �2ðx; sÞj2 dsþK2

Z t

0

jc1ðx; sÞ � c2ðx; sÞj2ds:

Proof of Lemma 4.2. (a) follows directly from assumptions (A2), (A4), (A5) and

Lemma 4.1(a).

(b) We have

_p�;c ¼ ð1� p�mÞg1ð�; cÞ; ð4:9Þ_m�;c ¼ ½minfmð�; cÞ; 1� pg �m�þg2ð�; cÞ: ð4:10Þ

Let x 2 �nN, with N � � of zero measure and consider z ¼ ðp;mÞ. By Lemma

4.1(a), jjz�kjjW 1;1ð0;T ;L1ð�ÞÞ � M 8 k, thus jjz�kjjW 1;pð0;T ;L1ð�ÞÞ � M 8 k; 8 p <1. Thus,

there exists a subsequence, f�k 0 g, and some z such that

z�k 0 ðx; �Þ ! zðx; �Þ weakly-star in W 1;1ð0;T Þ; for a:e: x 2 �:

Thus, we have

_z�k 0 ðx; �Þ ! z:ðx; �Þ weakly in Lpð0;T Þ 8 p <1; for a:e: x 2 �: ð4:11Þ

z� 0kðx; �Þ ! zðx; �Þ strongly in C½0;T �; for a:e: x 2 �: ð4:12Þ

Since the solution to (4.3a)�(4.3b) is unique we have zðx; �Þ ¼ z�ðx; �Þ and the con-

vergence holds for the whole sequence, hence we can conclude that z�kðx; tÞ ! z�ðx; tÞpointwise in Q. Since �k ! � strongly in L2ðQT Þ, using assumption (A4), possibly

extracting a subsequence, we have

�Lpð�k 0 Þ _p�k 0 ;c þ �Lmð�k 0 Þ _m�k 0 ;c ! �Lpð�Þ _p� þ �Lmð�Þ _m� a:e: in QT : ð4:13Þ

But, applying Lebesgue theorem, we get

Bð�k 0 ; cÞ ! Bð�; cÞ strongly in LpðQT Þ:Since the limit does not depend on the extracted subsequence the convergence holds

for the whole sequence f�kg, hence we obtain (4.8).

(c) follows directly from assumption (A2) and Lemma 4.1(b).


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Lemma 4.3. Let c 2 L2ð0;T ;L2ð�ÞÞ. Then there exists a unique �ðcÞ 2 H 2;1ðQT Þand a unique zðcÞ ¼ ðpðcÞ; mðcÞÞ 2 W 1;1ð0; T ; L1ð�ÞÞ �W 1;1ð0;T ; L1ð�ÞÞ,satisfying

��ð�Þ @�@t

� divðkr�Þ ¼ Bð�; cÞ in QT ; ð4:14aÞ

�k@�

@�¼ hð�� Þ on @�� ð0;T Þ; ð4:14bÞ

�ðx; 0Þ ¼ �0 in �; ð4:14cÞzt ¼ fðz; �; cÞ in QT ; ð4:14dÞ

zð0Þ ¼ 0 in �; ð4:14eÞ

where f is de¯ned as in (4.1a). Moreover, there exist �1; �2 > 0 such that

jj�1 � �2 jj2L 2ð0;t;L 2ð�ÞÞ � �1

Z t

0

jjc1 � c2 jj 2L2ð0;s;L 2ð�ÞÞds ð4:15Þ

and

jjjz1 � z2jjj2L 2ð0;t;L 2ð�ÞÞ � �2

Z t

0

jjc1 � c2 jj2L 2ð0;s;L 2ð�ÞÞds; ð4:16Þ

where ð�i; ciÞ is the solution corresponding to ci; i ¼ 1; 2.

Proof of Lemma 4.3. Existence. We introduce the operator

P : L2ðQT Þ ! L2ðQT Þ;

� ¼ P �;

by demanding � to be the solution of the linear parabolic problem

��ð�Þ @�@t

� k4� ¼ Bð�; cÞ in QT ; ð4:17aÞ

�k@�

@�¼ hð�� Þ on @�� ð0;T Þ; ð4:17bÞ

�ðx; 0Þ ¼ �0 in �: ð4:17cÞ

According to classical results about parabolic equations, problems (4.17a)�(4.17c)

has a unique strong solution � 2 H 2;1ðQT Þ (see, for instance, Ref. 19), therefore the

operator P is well-de¯ned.

Moreover, thanks to Lemma 4.2(a), there exists a constant M > 0, independent of

�, such that:

jj�jjH 2;1ðQT Þ � M : ð4:18Þ

We shall now show the continuity of the operator P.


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Let ð�nÞ � L2ðQT Þ with �n ! � strongly in L2ðQT Þ. De¯ning �n ¼ P �n, in view of

(4.18), jj�n jjH 2;1ðQT Þ � M. Thus, we can ¯nd a subsequence �n 0 such that

�n 0 ! � weakly in H 2;1ðQT Þ; strongly in L2ðQT Þ; ð4:19aÞ�n 0 ! � a:e: in QT : ð4:19bÞ

Testing Eq. (4.17a), written for the index n 0, by 2 L2ð0; t;H 1ð�ÞÞ, we getZ t

0

Z�

��ð�n 0 Þ�n 0;s dx dsþ k

Z t

0

Z�

r�n 0rdx ds

þZ t

0

Z@�

hðÞð�n 0 � ��Þ d ds�Z t

0

Z�

Bð�n 0 ; cÞdx ds ¼ 0: ð4:20Þ

By means of (4.19a)�(4.19b) we can pass to the limit in the last three terms of (4.20).

We can break the ¯rst term in to two terms

�

Z t

0

Z�

�ð�n 0 Þ�n 0;sdx ds ¼ �

Z t

0

Z�

�ð�n 0 Þð�n 0;s � �sÞ dx dsþ �

Z t

0

Z�

�ð�n 0 Þ�s dxds:

Thanks to the continuity of �, we have that

�ð�n 0 Þ! �ð�Þ a:e: in QT

thus, using Lebesgue theorem, ��ð�n 0 Þ! ��ð�Þ strongly inL2ðQT Þwhile �n 0;s ! �sweakly in L2ðQT Þ. Thus,

R t

0

R��ð�n 0 Þð�n 0;s � �sÞ dx ds ! 0 and

�

Z t

0

Z�

�ð�n 0 Þ�n 0;s dx ds ! �

Z t

0

Z�

�ð�Þ�s dx ds:

Hence we have obtained

�

Z t

0

Z�

�ð�Þ�sdx dsþ k

Z t

0

Z�

r�r dx ds

þZ t

0

Z@�

hðÞð�� Þ d ds �Z t

0

Z�

Bð�; cÞdx ds ¼ 0:

As the solution to the parabolic problem (4.17a)�(4.17c) is unique, we have

� ¼ P � a:e: in QT

and, since the limit does not depend on the extracted subsequence, it follows that

P �n ! P �

weakly in H 2;1ðQT Þ and strongly in L2ðQT Þ.Now, let

K :¼ fu 2 L2ðQT Þ : jjujjH 2;1ðQT Þ � Mg:

K is non-empty, convex, closed and relatively compact subset of L2ðQT Þ and F :

K � L2ðQT Þ ! K is a continuous mapping. By Schauder ¯xed point theorem, there


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exists a ¯xed point of the mapping F, i.e. there exists a weak solution � 2 H 2;1ðQT Þ to(3.1a), (3.1e) and (3.1g).

Uniqueness and stability. Let

Jð�Þ :¼Z �

0

��ð�Þd�: ð4:21Þ

Integration of (3.1a) with respect to time leads toZ t

0

Bð�; cÞðx; sÞds ¼ Jð�ðx; tÞÞ � Jð�0ðxÞÞ � k�

Z t

0

�ðx; sÞds: ð4:22Þ

Now, let �1; �2 2 H 2;1ðQT Þ be solutions to (3.1a), (3.1e) and (3.1g) corresponding to

c1; c2 respectively. Inserting these solutions into (4.22), subtracting both equations,

and testing by � :¼ �1 � �2, we ¯ndZ t

0

Z�

Z s

0

Bð�1ðx; �Þ; c1ðx; �ÞÞ � Bð�2ðx; �Þ; c2ðx; �ÞÞd��

�ðx; sÞdx ds

¼Z t

0

Z�

½Jð�1ðx; sÞÞ � Jð�2ðx; sÞÞ��ðx; sÞdx ds

þ k

Z t

0

Z�

rZ s

0

�ðx; �Þd��

r�ðx; sÞdx ds

þZ t

0

Z@�

Z s

0

hðÞ�ð; �Þd��

�ð; sÞd ds: ð4:23Þ

Concerning the last term we can see thatZ t

0

Z@�

Z s

0

hðÞ�ð; �Þd��

�ð; sÞd ds

¼Z t

0

Z@�

hðÞZ s

0

�ð; �Þ d��

�ð; sÞd ds

¼ 1

2

Z t

0

Z@�

hðÞ d

ds

Z s

0

�ð; �Þd�� 2

d ds

¼ 1

2

Z@�

hðÞZ t

0

�ð; sÞds� �2

d:

Thus, from (4.23) we getZ t

0

Z�

Z s

0

Bð�1ðx; �Þ; c1ðx; �ÞÞ �Bð�2ðx; �Þ; c2ðx; �ÞÞðx; �Þd��

�ðx; sÞdx ds

��

Z t

0

Z�

�2ðx; sÞdx dsþ k

2

Z�

rZ t

0

�ðx; sÞds��

��2

dx

þ 1

2

Z@�

hðÞZ t

0

�ð; sÞds� �2

d ��

Z t

0

Z�

�2ðx; sÞdx ds:


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Using H€older's and Young's inequalities and Lemma 4.1 it follows thatZ t

0

Z�

Z s

0

Bð�1ðx; �Þ; c1ðx; �ÞÞ � Bð�2ðx; �Þ; c2ðx; �ÞÞd��

�ðx; sÞdx ds��

�� 1

4

Z t

0

Z�

Z s

0

Bð�1ðx; �Þ; c1ðx; �ÞÞ � Bð�2ðx; �Þ; c2ðx; �ÞÞd�� 2

dx ds

þ

Z t

0

jj�ð�; sÞjj2L 2ð�Þ ds

� T

4

Z t

0

Z�

Z s

0

K1j�1ðx; �Þ � �2ðx; �Þj2 þ K2jc1ðx; �Þ � c2ðx; �Þj2� �

d�dx ds

þ

Z t

0

jj�ð�; sÞjj2L 2ð�Þ ds

� CT

4

Z t

0

jj�jj2L 2ð0;s;L 2ð�ÞÞdsþCT

4

Z t

0

jjcjj 2L2ð0;s;L2ð�ÞÞ ds

þ

Z t

0

jj�ð�; sÞjj2L 2ð�Þ ds:

Thus, we have

��

Z t

0

Z�

�2ðx; sÞdx ds � CT

4

Z t

0

jj�jj 2L2ð0;s;L 2ð�ÞÞds

þ CT

4

Z t

0

jjcjj 2L2ð0;s;L 2ð�ÞÞ dsþ

Z t

0

jj�ð�; sÞjj2L 2ð�Þ ds:

Choosing > 0 such that �� > 0 we have:

jj�jj2L 2ð0;t;L 2ð�ÞÞ � �

Z t

0

jj�jj2L 2ð0;s;L 2ð�ÞÞds þ �

Z t

0

jjcjj2L 2ð0;s;L 2ð�ÞÞds

with constants �; � > 0.

Hence, applying Gronwall lemma, we ¯nd a constant C1 such that

jj�1 � �2 jj2L 2ð0;t;L 2ð�ÞÞ � C1

Z t

0

jjc1ðsÞ � c2ðsÞjj2L 2ð0;s;L 2ð�ÞÞds: ð4:24Þ

Inequality (4.16) follows immediately from Lemma 4.1(b) and estimate (4.15).

The proof of Lemma 4.3 is thus complete.

Now, we are in a position to prove Theorem 3.1.

Let us denote

�ð�; cÞ :¼ ð1� p�mÞDð�; cÞ:

We note that, in view of (A6) and Lemma 4.1(b), � is Lipschitz-continuous with

respect to � and c.


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We de¯ne an operator

T : L2ðQT Þ ! L2ðQT Þ;

T c ¼ c;ð4:25Þ

by demanding c to be the solution of the parabolic problem

@c

@t� divð�crcÞ ¼ 0 in QT ; ð4:26aÞ

��c@c

@�¼ �ðc� cpÞ on @�� ð0;T Þ; ð4:26bÞ

cðx; 0Þ ¼ c0 in �; ð4:26cÞ

where �c ¼ ð1� pc �mcÞDð�c; cÞ; ð�c; pc;mcÞ being the solution to (3.1a), (3.1c) and

(3.1d) with respect to given c. Denoting

aðc; ; tÞ :¼Z�

�crcr dxþZ@�

� c d;

hfðtÞ; i :¼Z@�

�cp d; 2 H 1ð�Þ;

we have that problem (4.26a)�(4.26c) is equivalent to the following. We seek a

function c such that, for all 2 H 1ð�Þ and a.e. in t 2 ð0;T Þd

dtcðtÞ;

� �þ aðcðtÞ; ; tÞ ¼ hfðtÞ; i; ð4:27aÞ

cð0Þ ¼ c0; ð4:27bÞc 2 W ð0;T Þ; ð4:27cÞ

where h ; i denotes the duality between H 1ð�Þ and ðH 1ð�ÞÞ�.In view of (A3), (A6), (A7), admits a unique solution c (cf. Ref. 27, Prop. 30.10).

Moreover, there exists a constant M independent of c, such that

jjcjjWð0;T Þ � M : ð4:28Þ

To derive the continuity of the operator T , let fcng � L2ð0;T ;L2ð�ÞÞ, with cn ! c

strongly in L2ðQT Þ. De¯ning cn ¼ T cn, thanks to (4.28), we have jjcn jjWð0;T Þ � M .

Thus, there exists a subsequence fcn 0 g such that

cn 0 ! c weakly in W ð0;T Þ: ð4:29Þ

We test (4.26a) by

�ðx; tÞ ¼ ðtÞðxÞ with 2 C 1½0;T �; ðT Þ ¼ 0; 2 H 1ð�Þ: ð4:30Þ

Denoting T cn 0 :¼ cn 0 , we haveZ T

0

Z�

cn 0;s�dx dsþZ T

0

Z�

�cn 0 rcn 0r�dx dsþZ T

0

Z@�

�ðcn 0 � cpÞ�d ds ¼ 0: ð4:31Þ


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Concerning the ¯rst term in (4.31) we haveZ T

0

Z�

cn 0;s � dx ds ¼ �Z�

cn 0 ðx; 0Þ�ðx; 0Þdx�Z T

0

Z�

cn 0 �s dx ds:

Now, Z�

cn 0 ðx; 0Þ�ðx; 0Þdx ¼Z�

c0 �ðx; 0Þdx;

and, by virtue of (4.29),Z T

0

Z�

cn 0 �s dx ds !Z T

0

Z�

c�s dx ds a:e: in QT :

The second term can be rearranged asZ T

0

Z�

�cn 0 rcn 0r�dx ds

¼Z T

0

Z�

�cn 0 ðrcn 0 � rcÞr� dx dsþZ T

0

Z�

�cn 0 rcr�dx ds:

Since � is continuous and bounded as a function of c, possibly extracting a sub-

sequence, we obtain: �cn 0 ðx; tÞ ! �cðx; tÞ a.e. in QT , therefore �cn 0 ðx; tÞr� !�cðx; tÞr� pointwise, moreover �cn 0 ðx; tÞr� is bounded in L2 thus, using Lebesgue

theorem, we have

�cn 0 r� ! �cr� strongly in L2ðQT Þ:

Moreover, ðrcn 0 � rcÞ ! 0 weakly in L2ðQT Þ because of (4.29), thus we obtainZ T

0

Z�

�cn 0 rcn 0r� dx ds !Z T

0

Z�

�crcr� dx ds:

Applying the trace theorem, the last term in (4.31) converges too.

Thus, we can pass to the limit in (4.31) obtaining

� ð0ÞZ�

c0 ðxÞdx�Z T

0

Z�

c s dx ds

þZ T

0

Z�

�crcr dx dsþZ T

0

Z@�

�ðc� cpÞ d ds ¼ 0: ð4:32Þ

ConsequentlyZ T

0

Z�

cs dxþZ�

�crcr dxþZ@�

�ðc� cpÞd� �

ds ¼ 0:

The above is true for ; satisfying (4.32). Therefore (4.32) gives, a.e. in t 2 ð0;T Þ

d

dtcðtÞ;

� �þ aðt; cðtÞ; Þ ¼ hF ðtÞ; i 8 2 H 1ð�Þ:


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Since the solution of (4.26a)�(4.26c) is unique, we can conclude

T c ¼ c;

and, since the limit does not depend on the extracted subsequence, it follows that

T cn ! T c ð4:33Þ

weakly in W ð0;T Þ and strongly in L2ðQT Þ.Now, let

K :¼ fv 2 L2ðQT Þ : jjvjjWð0;T Þ � Mg:

K is convex and compact in L2ðQT Þ and F : K � L2ðQT Þ ! K is a continuous

mapping. By the Schauder ¯xed point theorem the proof is concluded.

5. Proof of Theorem 3.2

We commence with the following regularity result:

Lemma 5.1. Under the assumptions of Theorem 3.2, the solutions �, c to the

initial�boundary value problems related to Eqs. (3.1a)�(3.1b) are in W 2;15 ðQT Þ.

Proof. Since we proved the existence of at least one solution for the initial-boundary

value problems related to Eqs. (3.1a), (3.1b), we can now follow the approach

developed by Griepentrog, in Refs. 8 and 9 about linear parabolic equations with

nonsmooth bounded coe±cients, in order to improve the regularity of the solutions

under consideration.

The coe±cients and the right-hand sides of the equations are indeed functions in

L1ðQT Þ and the coe±cients in the boundary conditions too.

Moreover, the initial conditions are Lipschitz-continuous functions and we can

apply Th. 3.4 and Th. 6.8 of Ref. 8 and Th. 6.1 of Ref. 9, whence we obtain that � and

c are in Cð�QT Þ.It follows that the right-hand sides of the ODEs (3.1c)�(3.1d) are continuous

functions, therefore the corresponding solutions are continuously di®erentiable.

Thus, the PDEs (3.1a)�(3.1b) have continuous coe±cients and we can apply

a classical result of Ladyzenskaja (Ref. 19, Th. 9.1, p. 341) which yields: �, c 2W 2;1

5 ðQT Þ.

Lemma 5.2. Assuming that � is constant, we have that, for every c1; c2 2 L2ðQT Þ,there exists a constant M > 0 such that, for the corresponding �1; �2, it holds:

jj�1 � �2 jj2H 2;1ðQT Þ � M jjc1 � c2 jj 2L2ðQT Þ: ð5:1Þ

Proof. We consider the heat equation of our system:

��t ¼ k��þ �Lppt þ �Lmmt: ð5:2Þ


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We write (5.2) for �1; c1; p1;m1 and �2; c2; p2;m2. Subtracting and denoting as usual

� ¼ �1 � �2, we see that the di®erence satis¯es the following system:

��t � k�� ¼ �ðLpð�1Þp1;t � Lpð�2Þp2;tÞ þ �ðLmð�1Þm1;t � Lmð�2Þm2;tÞ ¼: f;

�k@�

@�¼ h�;

�ðx; 0Þ ¼ 0:

Applying again standard parabolic theory (cf. Ref. 21 Th. 6.2, Chap. 6), we know

that there exists a positive constant K such that we can estimate the norm of the

solution as follows:

jj�jjH 2;1ðQT Þ � KjjfjjL2ðQT Þ:

Now we can estimate the term jjfjjL 2ðQT Þ, by means of Lemma 4.1(b) and assumptions

(A2), as:

jj�ðLpð�1Þp1;t � Lpð�2Þp2;tÞ þ �ðLmð�1Þm1;t � Lmð�2Þm2;tÞjjL 2ðQT Þ

� K1jj�1 � �2 jjL 2ðQT Þ þK2jjc1 � c2 jjL2ðQT Þ;

with K1;K2 positive constants. Now, using Lemma 4.3, we can estimate the term

jj�1 � �2 jjL 2ðQT Þ and therewith ¯nish the proof.

Lemma 5.3. Let u 2 L1ð0;T ;L2ð�ÞÞ \ L2ð0;T ;H 1ð�ÞÞ, then there holds

Z T

0

jjuðtÞjj 10=3L10=3ð�Þ

dt �Z T

0

jjuðtÞjj2L 6ð�Þ

dt

� �jjujj4=3

L1ð0;T ;L 2ð�ÞÞ:

Proof. Owing to Riesz' convexity theorem (cf. Ref. 27, A113), we have

jjujjLrð�Þ � jjujj1��Lq1 ð�Þ

jjujj�Lq2 ð�Þ

;

for all u 2 Lq1ð�Þ \ Lq2ð�Þ with 1 � q1; q2 <1; 0 < � < 1, and 1r ¼ 1��

q1þ �

q2.

Invoking the continuous embedding H 1ð�Þ � L6ð�Þ, the assertion follows by

de¯ning q1 ¼ 6; q2 ¼ 2; � ¼ 25, and r ¼ 10

3 .

We are now in a position to prove Theorem 3.2. We write Eq. (3.1b) for c1 and c2,

subtract, integrate over QT and test by c1 � c2. In the sequel we will use the following

notations: c ¼ c1 � c2; � ¼ �1 � �2; p ¼ p1 � p2; m ¼ m1 �m2. We have

1

2

Z�

c2ðtÞdxþZ t

0

Z�

ðð1� p1 �m1ÞDð�1Þrc1 � ð1� p2 �m2ÞDð�2Þrc2Þrc dx ds

þZ t

0

Z@�

�c2d ds ¼ 0:


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Now,

Z t

0

Z�

ðð1� p1 �m1ÞDð�1Þrc1 � ð1� p2 �m2ÞDð�2Þrc2Þrc dx ds

¼Z t

0

Z�

ð1� p1 �m1ÞDð�1Þjrcj2dx ds�Z t

0

Z�

ðpþmÞDð�1Þrc2rc dx ds

þZ t

0

Z�

ð1� p2 �m2ÞðDð�1Þ �Dð�2ÞÞrc2rc dx ds: ð5:3Þ

Denoting, in the sequel, by Ki generic positive constants independent of � and c, we

obtain

1

2

Z�

c2ðtÞdxþK1

Z t

0

jjrcjj 2L2ð�Þdx ds

�Z t

0

Z�

jpþmjjDð�1Þjjrc2jjrcj dx ds

þZ t

0

Z�

j1� p2 �m2jjDð�1Þ �Dð�2Þjjrc2jjrcj dx ds: ð5:4Þ

By means of Lemma 5.1, we know that c2 2 W 2;15 ðQT Þ. According to Amann (cf.

Ref. 2, Th. 1.1), we have the embedding W 2;15 ðQT Þ ,! Cð½0;T �;W 1

5 ð�ÞÞ.Thus, we can estimate the ¯rst term in the right-hand side of (5.4) as:

Z t

0

Z�

jpþmjjDð�1Þjjrc2jjrcj dx ds

�Z t

0

jjpþmjjL 10=3ð�Þjjrc2 jjL 5ð�ÞjjDð�1ÞjjL1ð�ÞjjrcjjL 2ð�Þds

� K1

4

Z t

0

jjrcjj2L 2ð�Þdsþ4K2

K1

Z t

0

jjpþmjj2L 10=3ð�Þds: ð5:5Þ

Moreover, thanks to Lemma 4.1(b), we get

Z t

0

jjpþmjj2L 10=3ð�Þds ¼Z t

0

Z�

jpþmj10=3dx� 3=5

ds

� K3

Z t

0

Z s

0

Z�

�10=3

dx d�

� 3=5ds

þK4

Z t

0

Z s

0

Z�

c10=3

dx d�

� 3=5ds: ð5:6Þ


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Now, we apply Lemma 5.3, Young's inequality and the embedding H 1ð�Þ ,! L6ð�Þto the right-hand side of (5.6), obtainingZ t

0

Z s

0

Z�

�10=3

dx d�

� 3=5ds �

Z t

0

Z s

0

K5jj�jj2H 1ð�Þd�

� 3=5jj�jj 4=5L1ð0;s;L 2ð�ÞÞds

� 3K5

5

Z t

0

Z s

0

jj�jj2H 1ð�Þd� ds þ 2

5

Z t

0

jj�jj 2L1ð0;s;L 2ð�ÞÞds:

ð5:7Þ

Analogously, it holdsZ t

0

Z s

0

Z�

c10=3

dx d�

� 3=5ds � 3K5

5

Z t

0

Z s

0

jjcjj 2H 1ð�Þd� dsþ2

5

Z t

0

jjcjj2L1ð0;s;L 2ð�ÞÞds:

ð5:8Þ

Regarding the second term on the right-hand side of (5.4), using Lemma 4.1,

assumption (A6) and Young's inequality again, we haveZ t

0

Z�

j1� p2 �m2jjDð�1Þ �Dð�2Þjjrc2jjrcj dx ds

� jjrc2 jjL1ð0;t;L3ð�ÞÞ

Z t

0

jj�jjL 6ð�ÞjjrcjjL 2ð�Þds

� K6

Z t

0

jj�jj2H 1ð�Þds þ K1

4

Z t

0

jjrcjj2L 2ð�Þds: ð5:9Þ

Using (5.4) to (5.9), we ¯nd that

min1

2;K1

2

�Z�

c2ðtÞdxþZ t

0

jjrcjj2L 2ð�Þds

� K7

Z t

0

jj�jj 2H 1ð�ÞdsþZ t

0

jj�jj2L1ð0;s;L 2ð�ÞÞds

� �þ 3K5

5

Z t

0

Z s

0

jj�jj2H 1ð�Þd� ds

þ K8

Z t

0

Z s

0

jjcjj2H 1ð�Þd�dsþZ t

0

jjcjj 2L1ð0;s;L2ð�ÞÞds

� �: ð5:10Þ

Thanks to the embedding W 2;12 ðQT Þ ,! Cð½0;T �;W 1

2 ð�ÞÞ (see Ref. 2, Th. 1.1) and

Lemma 5.2, we obtain for the ¯rst term in the right-hand side of (5.10):Z t

0

jj�ðsÞjj 2H 1ð�Þds �Z t

0

jj�jj 2L1ð0;s;H 1ð�ÞÞds

� K9

Z t

0

jj�jjH 2;1Qs

ds

�Z t

0

K10

Z s

0

jjcjj 2L2ð�Þd� ds: ð5:11Þ


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Analogous estimates hold for the other terms involving �. Thus we end up with

jjcðtÞjj 2L2ð�Þ þZ t

0

jjrcjj 2L2ð�Þds

� K11

Z t

0

jjcjj2L1ð0;s;L 2ð�ÞÞdsþK12

Z t

0

Z s

0

jjrcjj2L2ð�Þd� ds

þK13

Z t

0

Z s

0

jjcjj2L2ð�Þd� ds 8 t 2 ½0;T �: ð5:12Þ

Taking the essential supremum over ½0; t� for some t 2 ½0;T � in the previous

inequality, we obtain

jjcðtÞjj 2L1ð0;tL 2ð�ÞÞ þZ t

0

jjrcjj2L 2ð�Þ

� K11

Z t

0

jjcjj2L1ð0;s;L 2ð�ÞÞdsþK12

Z t

0

Z s

0

jjrcjj2L 2ð�Þd� ds

þK13

Z t

0

Z s

0

jjcjj2L 2ð�Þd� ds: ð5:13Þ

SinceR t

0

R s

0jjcjj2L 2ð�Þd� ds � T

R t

0jjcjj 2L1ð0;s;L 2ð�ÞÞds, we can apply the Gronwall lemma

and conclude the proof of Theorem 3.2.

The following table contains the parameters involved in the complete process.

6. Numerical Results

In this section we present some numerical simulations to demonstrate the e®ect of gas

carburizing on a sample workpiece. The simulations are based on our model (3.1a)�(3.1j). As a sample con¯guration, we consider the cross section of a cylinder of radius

50mm. Note that our initial temperature is chosen above the austenitization tem-

perature such that we may assume it to be homogeneously austenitic. Material

parameters are taken from the data tables for the low-carbon steel AISI 4130. The

interval time ð0;T Þ of the whole process is divided as ð0;Tc� [ ½Tc;T Þ, where Tc

denotes the ending time of carburization.

For the process parameters we refer to Table 1. The expression for Dð�; cÞ is takenfrom Ref. 24, the value of h is taken from Ref. 15. For the function g1 we took the data

of Ref. 6, cf. Fig. 4. g2 has been taken constant as in Ref. 13, which has been found

su±cient to describe the kinetics of the phase transition. The main coupling e®ect is

through the carbon dependent start and end temperature of the martensite for-

mation, MsðcÞ and MfðcÞ respectively, which have been identi¯ed from Fig. 3. The

simulations were performed with Femlab, a software based on the ¯nite element

method.

Figure 5 is a view of a sector of the sample con¯guration that we considered, after

carburizing for about eight hours.


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0 100 200 300 400 500 600 700 800 9000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Temperature, oC

g 1(θ)

Fig. 4. Plot of the transformation function g1, depending on the temperature �.

Table 1. Process parameters.

Value Unit Value Unit

� 7800 kg=m3 �0 1150 K

� 385 J=kg K c0 0.25 weight %

Lp 77,000 J=kg cp 1.2 weight %

Lm 82,000 J=kg �� 300 K

� ðif t � T1Þ 6e-5 m=s � ðif T1 < t � T Þ 0 m=sh ðif t � T1Þ 0 W=m2K h ðif T1 < t � T Þ 10,000 W=m2KD 0:47 expð�1:6c� ð37000� 6600cÞ=ð1:987T ÞÞ1e� 4 m2=s

k 35 W=mK

Fig. 5. Snapshot of the simulation at time t ¼ 30;120 s (after 30,000 seconds carburizing and 120 seconds

quenching) showing the carburizing e®ect. In the right column carbon percentage is indicated.


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As already mentioned in the Introduction the process consists of at least two

stages: ¯rst, the workpiece is immersed in a carbon-rich atmosphere furnace (the

so-called carburizing); secondly, quenching is performed, through which austenite is

transformed into the hard phase martensite m, where the temperature gradient is

high and into pearlite p where the temperature gradient is lower. In other words, the

hardening occurs close to the boundary, while in the core the softer phase pearlite is

formed. The e®ect of time and temperature on total case depth (which is usually

speci¯ed as the layer at carbon content 0.4%) is shown in Fig. 6.

In Fig. 7 we can observe the distribution of phase fractions at the end of a cycle of

carburizing and quenching.

In the same ¯gure we can see how the formation of martensite depends on the

carbon concentration, in accordance with the graphic of Fig. 3 of the ¯rst section.

Indeed, as we can see in Fig. 3, the martensite terminal temperature is well below

zero, because of the residual austenite at room temperature which cannot be trans-

formed into martensite, thus 100% of martensite is not achieved; in Fig. 7, derived

from our simulations, the maximum of the martensite phase fraction is about 65%.

The maximum of the martensite fraction is not achieved on the surface, but at the

total case depth, i.e. where the carbon concentration corresponds to 0.4%.

Fig. 6. Plot of total case depth vs. carburizing time at four selected temperatures. Graph based on data in

table.


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7. Concluding Remarks

In this paper we have studied a mathematical model of case hardening, including the

coupling between carbon di®usion equation, temperature evolution and phase tran-

sitions. From mathematical point of view, we have proved existence and uniqueness

of a solution. First numerical results con¯rm qualitative agreement with experiments.

A more detailed comparison requires more precise data. To this end a cooperation

with some engineering institutes has been started. The results will be published in a

forthcoming paper.

From a practical point of view, a reduction of energy consumption and of process

time as well as increasing the process stability are of great interest. Therefore the

development of an optimal control strategy is under study.

References

1. Atlas of Isothermal Transformation and Cooling Transformation Diagram (Amer. Soc. forMetals, 1977).

Fig. 7. Phase fractions of martensite, pearlite and carbon percentage curve, plotted against the radius of

the circle, for di®erent carburizing times Tc and end times T , after a quenching time of 100 s.


Mat

h. M

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2. H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat.35 (2000) 161�177.

3. S. H. Avner, Introduction to Physical Metallurgy, 2nd edn. (McGraw-Hill, 1974).4. M. Avrami, Kinetics of phase change, J. Chem. Phys. 8 (1940) 812�819.5. R. Chatterjee-Fischer, Überblick über die M€oglichkeiten zur Verkürzung der Aufkoh-

lungsdauer, HTM H€arterei-Techn. Mitt. 40 (1985) 7�11.6. K. Chelminski, D. H€omberg and D. Kern, On a thermomechanical model of phase tran-

sitions in steel, Adv. Math. Sci. Appl. 18 (2008) 119�140.7. J. Fuhrmann and D. H€omberg, Numerical simulation of the surface hardening of steel, Int.

J. Numer. Meth. Heat Fluid Flow 9 (1999) 705�724.8. J. A. Griepentrog, Sobolev�Morrey spaces associated with evolution equations, Adv. Di®.

Eqns. 12 (2007) 781�840.9. J. A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-

Morrey spaces, Adv. Di®. Eqns. 12 (2007) 1031�1078.10. D. H€omberg, A mathematical model for the phase transitions in euctectoid carbon steel,

IMA J. Appl. Math. 54 (1995) 31�57.11. D. H€omberg, A mathematical model for induction hardening including mechanical e®ects,

Nonlinear Anal. Real World Appl. 5 (2004) 55�90.12. D.H€omberg, Irreversible phase transitions in steel,Math.Meth.Appl. Sci.20 (1997) 59�77.13. D. H€omberg and W. Wol®, PID-control of laser surface hardening of steel, IEEE Trans.

Control Syst. Tech. 14 (2006) 896�904.14. H. P. Hougardy, Die Darstellung des Umwandlungsverhaltens von Stählen in den ZTU-

Schaubildern, HTM H€arterei-Techn. Mitt. 33 (1978) 63�70.15. M. Hunkel, T. Lübben, F. Ho®mann and P. Mayr, Using the jominy end-quench test for

validation of thermo-metallurgical model parameters, J. Phys. IV France 120 (2004)571�579.

16. M. Hunkel, T. Lübben, O. Belkessam, U. Fritsching, F. Ho®mann and P. Mayr, Modelingand simulation of coupled gas and material behavior during gas quenching, Proc. ASM2001, Heat Treatment (2001) 5�11.

17. H. W. Knobloch and F. Kappel, Gew€ohnliche Di®erentialgleichungen (Teubner, 1974).18. T. Inoue, T. Yamaguchi and Z. Wang, Stresses and phase transformations occurring in

quenching of carburized steel gear wheel, Mat. Sci. Tech. 1 (1985) 872�876.19. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear

Equations of Parabolic Type, Amer. Math. Soc. Transl., Vol. 23 (Amer. Math. Soc., 1968).20. J. L. Lions, Quelques methodes de resolution des probl�emes aux limites non lineaires

(Gauthier-Villars, 1969).21. J. L. Lions and E. Magenes, Probl�emes aux limites non homog�enes (Dunod, 1968).22. M. Motoyama, R. E. Ricklefs and J. A. Larson, The e®ect of carburizing variables on

residual stresses in hardened chromium steel, SAE Technical Paper Series 750050 (Soc. ofAutomotive Engineers, 1975).

23. A. Ochsner, J. Gegner and G. Mishuris, E®ect of di®usivity as a function of the method ofcomputation of carbon concentration pro¯les in steel,Met. Sci. Heat. Treat. 46 (2004) 3�4.

24. C. A. Stickels, Gas carburizing, in ASM Handbook, Heat Treatment (Amer. Soc. forMetals, 1997), pp. 312�324.

25. G. G. Tibbetts, Di®usivity of carbon in iron and steels at high temperatures, J. Appl.Phys. 51 (1980) 4813�4816.

26. A. Visintin, Mathematical models of solid-solid phase transitions in steel, IMA J. Appl.Math. 30 (1987) 143�157.

27. E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II (Springer-Verlag,1990).


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A MATHEMATICAL MODEL FOR CASE HARDENING OF STEEL

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