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arXiv:2409.14817v1 [math.AP] 23 Sep 2024
A phase field model of Cahn–Hilliard type for tumour
growth with mechanical effects and damage
Giulia Cavalleri
Dipartimento di matematica “F. Casorati”, Universit`a degli Studi di Pavia,
via Ferrata 5, 27100 Pavia, Italy
E-mail: giulia.cavalleri01@universitadipavia.it
Abstract
We introduce a new diffuse interface model for tumour growth in the presence of
a nutrient, in which we take into account mechanical effects and reversible tissue
damage. The highly nonlinear PDEs system mainly consists of a Cahn–Hilliard type
equation that describes the phase separation process between healthy and tumour
tissue coupled to a parabolic reaction-diffusion equation for the nutrient and a hy-
perbolic equation for the balance of forces, including inertial and viscous effects. The
main novelty of this work is the introduction of cellular damage, whose evolution is
ruled by a parabolic differential inclusion. In this paper, we prove a global-in-time
existence result for weak solutions by passing to the limit in a time-discretised and
regularised version of the system.
Key words: Tumour growth, Cahn–Hilliard equation, mechanical effects, viscoelas-
ticity, damage, existence.
AMS (MOS) subject classification: 35A01, 35K35, 35K57, 35K92, 35Q74,
35Q92, 74A45.
1 Introduction
Cancer is one of the leading causes of death worldwide and understanding the primary
mechanisms underlying its development is one of the main challenges scientists face
nowadays. Genetic, biochemical, and mechanical processes come into play simultane-
ously, making it difficult to predict the course of the disease and design specific and
effective treatments. For this reason, it has been understood that mathematics can be
fundamental, offering quantitative tools that can significantly enhance diagnostic and
prognostic applications. Over the last decades, there has been an increasing interest
in mathematical modelling for tumour growth, see e.g. [Byr+06;CL10;AP08] and the
references cited therein. In particular, among the various possible modelling approaches,
we will focus on the so-called phase-field or diffuse interface models. At first glance, it
1
might seem intuitive to model solid tumours as masses separated from healthy tissue
by a sharp interface, employing a free boundary problem (see [BC97;Fri07]). How-
ever, these models present technical limitations in describing situations where there is a
topological change in the tumour, such as coalescence or breaking-up phenomena, which
typically occur both at the early stages of the proliferation (when the tumour is mor-
phologically unstable, see [CLN03]) and at more advanced stages (when it undergoes
metastasis). This difficulty can be overcome by employing a diffuse interface model, in
which the sharp interface is replaced by a thin transition layer with both tumour and
healthy cells. Without attempting to be exhaustive, we refer to [AP08;Byr00;Low+10;
CGH15;FGR15;Gar+16;MRS19] and the references cited therein. This work aims
to introduce and study a new mathematical phase-field model for the evolution of a
young tumour, which implies that the tumour is in the avascular phase and there is no
differentiation between different types of tumour cells (viable, quiescent and necrotic).
As it is common, the tumour growth process is ruled by a Cahn-Hilliard type equation
(see e.g. [Mir19] or [Wu22] for further details on the classical Cahn-Hilliard equation)
coupled with other equations describing the behaviour of relevant quantities. We will
take into account the following aspects.
(i) The presence of the nutrient, a chemical species that feeds tumour cells (such as
oxygen or glucose). In our setting, it is provided by the pre-existing vasculature,
since we assume that the tumour has not developed its own yet.
(ii) The viscoelastic behaviour of biological tissues, which exhibit both elastic (instan-
taneous response to stress) and viscous (time-dependent deformation) properties.
Moreover, it is well known that solid stress can affect tumour growth (see e.g.
[Urc+22]) and, at the same time, tumour growth increases mechanical stress. We
assume infinitesimal displacements, so we will work in the case of linear elasticity.
(iii) The local tissue damage caused by surgery. In many cases, the standard of care
requires surgical resection of the tumour: this causes lesions which, in turn, affect
the proliferation of tumour cells when the growth process eventually restarts. This
may happen for several reasons. First, removing part of the tissue causes damage
to the blood vessels and edema: this must be taken into account in the nutrient
equation. Second, the surgical groove is characterised by different elastic properties
compared to intact tissue (see e.g. [Moe+17]), which must be considered when
choosing a suitable form for the elastic energy.
While the influence of (i) and (ii) on tumour growth is already deeply investigated in
the literature (see e.g. [GLS21;GKT22;GT24]), the role of (iii) is a complete novelty in
this field. However, the impact of the damage and (visco-)elasticity in phase separation
processes has been thoroughly explored in various modelling studies within the field of
materials science (see e.g. [HK11;HK15;Hei+17]).
2
The PDEs system. Explicitly, we derive the following PDEs system
∂tϕ−∆µ=U(ϕ, σ, ε(u), z),(1.1a)
µ=−ǫ∆ϕ+1
ǫΨ′(ϕ) + W,ϕ(ϕ, ε(u), z),(1.1b)
∂tσ−∆σ=S(ϕ, σ, z),(1.1c)
κ∂ttu−div a(z)Vε(∂tu) + W,ε(ϕ, ε(u), z)=0,(1.1d)
∂tz−∆pz+β(z) + π(z) + W,z(ϕ, ε(u), z)∋0,(1.1e)
posed in Q:= Ω ×(0, T ), where Ω is a smooth enough domain in Rdwith d= 2,3
and T > 0 is a fixed time. The Cahn–Hilliard equation given by the combination of
(1.1a)-(1.1b) describes the phase separation process between healthy and tumour tissue,
where ϕdenotes the local difference in volume fraction between tumour and healthy
cells. This means that, at least in principle, the set {ϕ=−1}corresponds the healthy
tissue, {ϕ= 1}to the tumour tissue and {−1< ϕ < 1}is the diffuse interface that sepa-
rates them (see Remark 2.6). The parameter ǫrepresents the thickness of the interfacial
layer. The chemical potential associated to ϕis denoted by µ. The reaction-diffusion
equation (1.1c) rules the diffusion of σ, that is the concentration of the nutrient. The
hyperbolic equation (1.1d) describes the dynamics for u, the small displacement field
of each point with respect to the reference undeformed configuration. Here, ε(u) is the
symmetric gradient of u, i.e., ε(u) = 1
2∇u+ (∇u)t. The fixed and positive parameter
κis supposed to be small and represents the fact that tumour growth occurs at a much
larger timescale than the tissue relaxation into mechanical equilibrium. For simplicity
and without any loss of generality later on we will set ǫ=κ= 1. Finally, the differential
inclusion (1.1e) represents the evolution law for local tissue damage z, which is the main
novelty we introduce. Classically, the damage takes values between 0 and 1: if z(x) is
equal to 1, there is no damage at the point x∈Ω, z(x) equal to 0 means that there
is complete damage and an intermediate value indicates partial damage (see Remark 2.9).
1.1 Derivation of the model
The evolution of our system is driven by classical thermodynamic principles and relies
on a total energy Eand a pseudopotential of dissipation P. The total energy of our
system is
E(ϕ, σ, u, z) = ZΩ
E(ϕ, ∇ϕ, σ, ε(u), ∂tu, z, ∇z) dx
where the energy density Eis the sum of a generalized free energy density and the kinetic
energy density. We postulate it has the following form:
E(ϕ, ∇ϕ, σ, ε(u), ∂tu, z, ∇z)
=1
2|∇ϕ|2+ Ψ(ϕ) + 1
2|σ|2+1
2|∂tu|2+1
p|∇z|p+b
β(z) + bπ(z) + W(ϕ, ε(u), z).
3
The term 1
2|∇ϕ|2+ Ψ(ϕ) is the classical contribution of Ginzburg-–Landau type and
accounts for the interfacial energy of the diffuse interface. The addend 1
2|σ|2results from
the presence of the nutrient, in the sense that higher nutrient concentration corresponds
to higher energy. The system’s kinetic energy is given by 1
2|∂tu|2. Regarding the damage,
1
p|∇z|p+b
β(z) + bπ(z) is an interaction free energy. According to the gradient theories
in damage processes, the gradient term models the influence of damage at a material
point, undamaged in the surrounding. The non-smooth convex b
βallows us to impose
physical constraints on the variable z(such as requiring that z∈[0,1]), while bπis a
smooth concave perturbation. Lastly, as already anticipated, Wis the elastic energy
density. To include dissipation in our model, we define a pseudo-potential of dissipation
P(ε(∂tu), ∂tz) = ZΩ
P(ε(∂tu), ∂tz) dx,
where
P(ε(∂tu), ∂tz) = 1
2a(z)ε(∂tu):Vε(∂tu) + 1
2|∂tz|2.
It depends on the damage time derivative ∂tzand the macroscopic symmetric strain
rate ε(∂tu), which are the dissipative variables of our problem. The fourth-order vis-
cous tensor term Vrepresents the friction between adjacent moving cells with different
velocities. Notice that Pdepends also on the damage z, but we use the notation Pin-
stead of a more precise Pzfor brevity. Following Gurtin’s approach [Gur96], our system
can be derived starting from balance laws for the involved quantities and then imposing
constitutive assumptions so that the system satisfies the second law of thermodynamics,
which, in the case of an isothermal system like ours, is written in the form of an energy
dissipation inequality (see e.g. [Gar+16;Hei+17]).
The Cahn–Hilliard equation of the system (1.1a)-(1.1b) is derived from the mass balance
law
∂tϕ+ div Jϕ=U,
where Jϕis the mass flux and Uis a mass source. As usual, the mass flux is prescribed
by the following constitutive equation
Jϕ:=−∇µ,
where µis the chemical potential associated with ϕand it is defined as the variational
derivative of the energy with respect to ϕ, i.e.,
µ:=δE
δϕ =−∆ϕ+ Ψ′(ϕ) + W,ϕ(ϕ, ε(u), z).
Here we adopt the standard notation according to which W,ϕ is the derivative of Wwith
respect to ϕand the same for the other variables.
The equation (1.1c) for the evolution of the nutrient is also derived from a mass balance
law,
∂tσ+ div Jσ=S,
4
where Sis a source/sink of nutrients and the mass flux is chosen as
Jσ:=−∇[∂σE] = −∇σ.
The equation (1.1d) governing the displacement is a balance law for macroscopic move-
ments in which inertial effects are taken into account and external forces are neglected,
derived from the principle of virtual power
∂ttu−div T= 0.
Here Tis the stress tensor and we postulate it is the sum of a non-dissipative (elastic
stress) and a dissipative part (viscous stress) given by
T=Tnd +Td=∂ε(u)E+∂ε(∂tu)P=W,ε(ϕ, ε(u), z) + a(z)Vε(∂tu).
Finally, the damage differential inclusion (1.1e) is derived from the micro-force balance
law
B−div H= 0,
where we assume that the sum of the external micro-forces acting on the body is equal to
zero. The quantity Brepresents the internal micro-forces and is defined by the following
constitutive assumption
B=Bnd +Bdwith Bnd ∈∂zE=∂b
β(z) + bπ′(z) + W,z(ϕ, ε(u), z),
Bd=∂∂tzP=∂tz.
Without entering into the mathematical details, ∂zEand ∂b
βhave to be interpreted
as subdifferentials in the sense of convex analysis and this justifies the presence of the
belonging symbol instead of equality. In the following, we will employ the notation
β:=∂b
βand π:=bπ′. The term His the internal micro-stress and is defined by
H=Hnd +Hdwith Hnd =∂∇zE=|∇z|p−2∇z,
Hd=∂∇(∂tz)P= 0,
Notice that here we followed Fr´emond’s approach (see [Fr´e02]), assuming that T,Band
Hcan be additively decomposed in a dissipative and a non-dissipative part.
Boundary and initial conditions. We assume the system is isolated from the exterior,
so we prescribe no-flux conditions for ϕ,µ, and z. Regarding σ, we allow a more general
Robin condition that may model also the boundary supply of the nutrient. We assume
that uis zero at the boundary, as in the situation in which the domain is delimited by
a rigid part of the body (e.g. a bone) that prevents displacements. In other terms, we
couple the previous system (1.1) with the following boundary conditions
∇ϕ·ν=∇µ·ν= 0,(1.2a)
∇σ·ν+α(σ−σΓ) = 0,(1.2b)
u=0,(1.2c)
(|∇z|p−2∇z)·ν= 0,(1.2d)
5
on Σ := Γ ×(0, T ), where Γ :=∂Ω is the boundary of the domain and νis the outward
unit normal to Γ. The term σΓis the prescribed concentration of the nutrient at the
boundary and αis a given non-negative constant. Notice that, if α= 0, we gain a no-flux
condition also for the nutrient.
Remark 1.1. Regarding the displacement u, it is possible to consider mixed boundary
conditions, imposing homogeneous Dirichlet and Neumann conditions respectively on
ΓDand ΓNif ΓD∪ΓN= Γ and ΓD∩ΓN=∅. As pointed out later, a key part of
our work is a regularity estimate for the displacement, and it can also be performed
in this more general setting (see [HR15, Remark 3.8, p. 4593]) as well as all the other
arguments we carry on.
The system is supplemented with the initial conditions
ϕ(0) = ϕ0, σ(0) = σ0,u(0) = u0, ∂tu(0) = v0, z(0) = z0,(1.3)
in Ω.
Choice of the sources. The nonlinear source Uin equation (1.1a) accounts for bio-
logical mechanisms related to tumour cells proliferation and death. Explicitly, we make
the following choice
U(ϕ, σ, ε(u), z):=λpσ
1 + |W,ε(ϕ, ε(u), z)|−λa+fg(ϕ, z),(1.4)
referring to [GL17;GLS21]. As it is common, we assume the mechanisms controlling
cell division to be suppressed in tumour cells, so proliferation is limited only by the
availability of nutrients. We model it with the term λpσ, where λpis a fixed prolifera-
tion coefficient. We also suppose that tumour cells only die because of apoptosis, and
we denote with λathe constant apoptosis rate. Furthermore, we consider the presence
of mechanical stress caused by surrounding tissues as a factor that can reduce tumour
growth. This is expressed by the fact that, if the mechanical stress W,ε grows in modulus,
the proliferation term λpσreduces. We also allow the presence of a medical treatment,
modeled by the prescribed function f, that affects proliferation. The function gguar-
antees that proliferation and apoptosis occur only in the tumour tissue, as well as the
effectiveness of the medical care f. A good modeling choice is a non-negative function
that vanishes in {ϕ=−1}, is equal to 1 where {ϕ= 1}and is increasing in the variable
ϕ. We also allow the dependence of gon the damage z.
For the choice of the nutrient source Sin equation (1.1c), we refer to the aforementioned
literature, assuming
S(ϕ, σ, z):=−λcσg(ϕ, z) + Λs(z)(σs−σ).(1.5)
The term −λcσg(ϕ, z) models the fact that the nutrient consumption is higher where
the tumour cells density is higher. Here λcis a fixed consumption rate. The term
Λs(z)(σs−σ) is a supply term that takes into account the nutrients provided by nearby
6
pre-existing vasculature. Note that Λsis a supply rate that may depend on the local
damage zsince the damage, in the sense of a lesion caused by a surgical procedure,
affects the blood vessels.
Choice of the elastic energy density. Accordingly to the classical theory of linear
elasticity (see e.g. [Sla02]) and to the previous literature (see e.g. [GLS21;GKT22]), we
assume that the elastic energy density has the following expression
W(ϕ, ε(u), z):=W(x, ϕ, ε(u), z) = 1
2h(z)C(x)(ε(u)− Rϕ):(ε(u)− Rϕ).(1.6)
Here Cis a fourth-order elasticity tensor, whose mathematical requirements will be
specified later on. We include the multiplicative, non-negative, and possibly degenerate
function hto add dependence on the damage. Notice that, from the modeling point of
view, Cshould also depend on the phase ϕbecause tumour tissue and healthy tissue could
have a different elastic response to solicitations. However, we weren’t able to handle such
dependence from the mathematical point of view (see Section 1.1 below). Finally, the
term Rϕis the stress-free strain (also called eigenstrain) which is the strain the material
would attain if the tissue were uniform and unstressed at a phase configuration ϕ. In
other terms, it is the strain due to growth. As it is common, we assume that it satisfies
Vegard’s law, i.e., it is given by a linear function of ϕ, where R ∈ Rd×dis a fixed matrix.
With such a choice, the partial derivatives of Wthat appear in the equations of the
PDEs system are:
W,ϕ(ϕ, ε(u), z) = −h(z)C(ε(u)− Rϕ):R,(1.7)
W,ε(ϕ, ε(u), z) = h(z)C(ε(u)− Rϕ),(1.8)
W,z(ϕ, ε(u), z) = 1
2h′(z)C(x)(ε(u)−Rϕ):(ε(u)− Rϕ).(1.9)
1.2 Aim of the paper
The purpose of this work is to prove the existence of weak solutions to the problem
(1.1)-(1.2)-(1.3). To do so, we will introduce an appropriate time-discretised and regu-
larized version of our system. Then, we will show that the discrete problem is well-posed
and that its solution satisfies some a priori estimates. Finally, employing compactness
results, we will pass to the limit as the time-step tends to 0 and prove that the limit we
find solves the original PDEs system.
Mathematical difficulties. The main mathematical challenges that we faced are the
following.
•The presence of the mass source in the Cahn–Hilliard equation (1.1a)-(1.1b), which
implies that there is no mass conservation, i.e., the mean value of ϕis not constant.
This is expected from the modeling point of view, however, it requires being able
to handle the term ZΩ
U(ϕ, σ, ε(u), z)µdx
7
in the energy estimate (see the proof of Proposition 3.8).
•The non-linear coupling between the single equations. In particular, in the damage
equation (1.1e), the term
W,z(ϕ, ε(u), z) = 1
2h′(z)C(ε(u)− Rϕ):(ε(u)− Rϕ)
is quadratic in ε(u). In order to pass to the limit in this term from the discrete
to the continuous problem, we have to perform a suitable regularity estimate for
the displacement uto obtain strong convergence for ε(u). This estimate, in turn,
requires a L∞(0, T ;W1,p) uniform bound for the damage z, with p > d: although
the p−Laplacian in equation (1.1e) is a non-linear operator which complicates the
analysis, it has a fundamental regularising role. For the same reason, since we do
not have uniform estimates for ϕin equally strong spaces, we cannot allow a de-
pendence of the elasticity tensor on the phase. In the literature (see e.g. [Hei+17])
this issue has been addressed by putting a ∆pϕinstead of ∆ϕin equation (1.1b).
However, we will not follow this strategy here.
•The damage equation is highly non-linear, due to the presence of −∆pzand the
subdifferential β=∂b
β(z). In particular, the p−Laplcacian operator seems to affect
the possibility of gaining uniqueness due to its degenerate character. As already
pointed out in [RR14] for a similar equation, this difficulty may be overcome by
replacing the degenerate p−Laplacian operator −div(|∇z|p−2∇z) with the non-
degenerate one −div((1 + |∇z|2)p−2
2∇z) or with the fractional s-Laplacian. How-
ever, we do not include such analysis in this paper and uniqueness remains an open
problem.
Plan of the paper. The paper is organized as follows. In Section 2, after introducing
some notation and preliminary results, we list the hypotheses under which we work. Then
we state the weak formulation of our problem and our main result, i.e., Theorem 2.13.
Section 3is completely devoted to the proof of the existence result.
2 Main result
2.1 Notation and preliminaries
Notation. In what follows, for any real Banach space Xwith dual space X′, we
indicate its norm as k·kXand the dual pairing between Xand X′as h·,·iX. We de-
note the Lebesgue and Sobolev spaces over Ω as Lp:=Lp(Ω), Wk,p :=Wk,p(Ω) and
Hk:=Wk,2(Ω), while for the Lebesgue spaces over Γ we use Lp
Γ:=Lp(Γ). We use H1
0to
denote the functions of H1that have zero trace at the boundary. Moreover, to keep the
notation as simple as possible, we will often not distinguish between scalar, vector, and
matrix-valued spaces (for example, we will use Lpinstead of Lp(Ω) but also Lp(Ω; Rd)).
However, we will use bold font for vectors and calligraphic font for tensors. For the
8
sake of brevity, the norm of the Bochner space Wk,p(0, T ;X) is indicated as k·kWk,p (X),
omitting the time interval (0, T ). Sometimes, for p∈[1,+∞), we will identify Lp(Q)
with Lp(0, T ;Lp). With the notation C0([0, T ]; X) we mean the space of continuous
X-valued functions, while with C0
w([0, T ]; X) we mean the space of weakly continuous
X-valued functions. Regarding the constants, as it is common, we will use the notation
Cto indicate a constant that depends only on the assigned data of the problem and
whose value might change from line to line. If we want to highlight the dependency on
a certain parameter, we put it as a subscript (e.g. Cτindicates a constant that depends
on τ,C0a constant that depends on the initial data, and so on).
Useful inequalities. We will make use of classical inequalities such as H¨older, Young,
Poincar´e, and Poincar´e–Wirtinger. For convenience, we recall a special case of the
Gagliardo–Nirenberg interpolation inequality (see e.g. [Nir59]).
Theorem 2.1 (Gagliardo–Nirenberg inequality).Let Ω⊆Rdbe a Lipschitz bounded
domain. Given
r > q ≥1, s > d 1
2−1
r,1
r=α
q+ (1 −α)1
2−s
d,
there exists a constant Csuch as for every v∈Hs, the following inequality holds true:
kvkLr≤Ckvkα
Lqkvk1−α
Hs.
Another inequality we will employ is the following Ehrling’s Lemma, also known as
Aubin–Lions inequality (see [LM12, Theorem 16.4, p. 102]).
Theorem 2.2 (Ehrling’s lemma).Let (X, k·kX),(Y, k·kY)and (Z, k·kZ)be Banach
spaces with Xcompactly embedded in Yand Ycontinuously embedded in Z. Then, for
every ε > 0there exists a C(ε)>0such that
kxkY≤εkxkX+C(ε)kxkZ
for every x∈X.
Finally, a key part of our main result’s proof is based on an estimate we obtain through
a discrete version of the well-known Gronwall inequality. For the sake of completeness,
we include its statement.
Lemma 2.3 (Discrete Gronwall inequality).Let {xn}n∈Nbe a real numbers sequence
satisfying
xn≤θ+
n−1
X
k=0
γkxk∀n∈N
for a constant θand a non-negative sequence {γn}n∈N. Then, it holds
xn≤θexp n−1
X
k=0
γk
for every n∈N.
9
A proof can be found in [Cla87], making the additional and trivial observation that
1 + γ≤exp(γ) for every γ∈R.
Preliminary on mathematical visco-elasticity. Let C= (chijk ) be a fourth-order
tensor such that:
(i) Cis symmetric, i.e.,
chijk (x) = cihjk (x) = cjk hi(x) (2.1)
for a.e. x∈Ω and for every indices i, j, k, h = 1,...,d.
(ii) Csatisfies the strong ellipticity condition, i.e., there exists a positive constant C
such that for all e∈Rd×d
sym and for a.e. x∈Ω
C(x)e:e≥C|e|2,(2.2)
where :denotes the standard Frobenius inner product between matrices.
As it will be specified in Section 2.2, both the elasticity tensor Cand the viscosity tensor
Vin equation (1.1d) satisfy (2.1) and (2.2). Moreover, the following regularity result
holds true.
Lemma 2.4. Let Ωbe a C2domain in Rdand C= (chijk )∈W1,∞(Ω; Rd×d×d×d)be a
symmetric and strongly elliptic fourth-order tensor. Then, there exist C1, C2>0such
that for every uin H2with u=0on ∂Ω
C1kukH2≤ kdiv[Cε(u)]kL2≤C2kukH2.
For more details, c.f. [MH94, Proposition 1.5, p. 318] and [Neˇc12, Lemma 3.2., p. 263].
Subdifferentials of convex functions. Here we will introduce some notation and
we will recall some facts. For more details, the interested reader may refer to [Br´e73,
Proposition 2.16, p. 47]. Let (S, Σ, ω) a positive finite measure space and Hbe a Hilbert
space. Then, a proper, convex, and lower semicontinuous function φ:H→(−∞,+∞]
induces a proper, convex, and lower semicontinuous function Φ over L2(S;H) in a natural
way, defining
Φ(v):=(RSφ(v) dωif φ(v)∈L1(S),
+∞otherwise.
It is well known that the subdifferentials of φand Φ are maximal monotone operators.
Moreover, it also holds that ξ∈∂Φ(v) if and only if ξ(s)∈∂φ(v(s)) for a.e. s∈
S. Furthermore, the Moreau–Yosida approximations of parameter εof the previous
functions are linked by the following relation:
Φε(v) = ZS
φε(v) dω.
10
In light of all these properties, with a slight abuse of notation, we will write φinstead
of Φ and ∂φ instead of ∂Φ. Moreover, a function φ:R→(−∞,+∞] induces both a
function defined over L2(Ω) and over L2(0, T ;L2(Ω)) but, for the sake of simplicity, we
will not make any distinction even in this case.
The p-Laplace operator with homogeneous Neumann conditions. Let p≥2
and define
Φp:L2(Ω) →[0,+∞],Φp(v):=
1
pRΩ|∇v|pdxif v∈W1,p(Ω),
+∞otherwise.
Then Φphas domain D(Φp) = W1,p(Ω) and it is proper, convex and lower semicontinuous
on L2(Ω). Hence its subdifferential −∆p:=∂Φpis a maximal monotone operator.
Moreover, every vin the natural domain
D(−∆p) = {v∈W1,p(Ω) :−∆pv∈L2(Ω),|∇v|p−2∇v·ν= 0 on Γ},
satisfies ZΩ−∆pv ω dx=ZΩ|∇v|p−2∇v· ∇ωdx
for every ω∈ D(Φp). In particular,
−∆pv=−div(|∇v|p−2∇v)
in the sense of distributions. Finally, the following regularity result holds true (the
interested reader can refer to [Sav98, Theorem 2, Remark 3.5]).
Lemma 2.5. For all 0< δ < 1
p, the inclusion D(−∆p)⊆W1+δ,p(Ω) holds. Moreover,
it exists Cδ>0such that, for all v∈W1+δ,p(Ω),
kvkW1+δ,p ≤Cδ(k−∆pvkL2+kvkL2).
2.2 Hypotheses
Let d= 2,3 denote the space dimension and Ω a bounded C2-domain in Rd.
(H1) Regarding the nonlinear sources Uand Sdefined in (1.4) and (1.5), we consider
λp, λa, λcnon-negative constants,(2.3)
g∈C0(R2),non-negative and bounded,(2.4)
f∈L∞(0, T ;L2),(2.5)
Λs∈C0(R),non-negative and bounded,(2.6)
σs∈L∞(Q), non-negative.(2.7)
11
(H2) Regarding the smooth potential Ψ ∈C1(R), we suppose that the following growth
conditions hold
Ψ(r)≥C1|r|2−C2,(2.8)
|Ψ′(r)| ≤ C3Ψ(r) + C4(2.9)
for some fixed positive constants C1,C2,C3,C4and for every r∈R.
Moreover, we assume that there exists a convex-concave splitting Ψ = `
Ψ+a
Ψsuch
that
`
Ψ,a
Ψ∈C1(R) (2.10)
`
Ψis convex and its derivative satisfies `
Ψ′(0) = 0,(2.11)
a
Ψis concave and its derivative a
Ψ′is Lipschitz continuous.(2.12)
Remark 2.6. Note that the Hypothesis (H2) is compatible with the classical and phys-
ically relevant choice Ψ(r) = 1
4(1 −r2)2. However, it does not allow us to consider
singular potential, such as logarithm type. This means that we can not guarantee that
ϕhas values in the physically relevant interval [−1,1].
(H3) We assume that the fourth-order elasticity tensor Cin (1.6) belongs to the space
C1(Ω; Rd×d×d×d) and is
Lipschitz continuous and bounded, (2.13)
symmetric (i.e. it satisfies (2.1)),(2.14)
strongly elliptic (i.e. it satisfies (2.2)).(2.15)
Regarding the fourth-order viscous tensor V, we suppose that it is of the form
V=ωC(2.16)
for a positive constant ω.
Remark 2.7. It is worth pointing out that the viscosity tensor is usually assumed to
be only symmetric and positively defined. The stronger assumption (2.16) is made in
order to prove the desired regularity for the displacement u. Without it, our argument
does not apply anymore (see the proof of Proposition 3.5 below).
(H4) We require that the scalar function hin (1.6) is C2(R) and that
hand h′are Lipschitz continuous,(2.17)
his bounded with 0 ≤h≤h∗.(2.18)
12
We postulate that the viscosity coefficient ais C1(R) and that it satisfies
ais Lipschitz continuous, (2.19)
ais bounded with 0 < a∗≤a≤a∗.(2.20)
(H5) We assume that the constant pthat occurs in the p-Laplacian −∆pin the damage
equation (1.1e) satisfies
p > d (2.21)
where dis the space dimension.
(H6) We consider a function bπ∈C1(R) with derivative π:=bπ′that satisfies
bπis concave,(2.22)
πis Lipschitz continuous.(2.23)
(H7) Let b
β:R→[0,+∞] be a function
proper, convex and lower semicontinuous (2.24)
with int(D(b
β)) 6=∅(2.25)
and denote by β:=∂b
β:R⇒Rits subdifferential.
Remark 2.8. We remind to the reader that βis a maximal monotone operator.
Remark 2.9. Note that Hypothesis (H7) is quite general, and is compatible with a
large class of potentials. A simple and classical example to keep in mind is the following
b
β(r) = I[0,1](r) = (0 if r∈[0,1],
+∞otherwise.
In particular, it would ensure that the damage zhas values in the physically significant
range [0,1].
(H8) Regarding the boundary conditions (1.2b) for the nutrient, we assume that
σΓ∈L∞(Σ) and σΓ≥0,(2.26)
α≥0.(2.27)
(H9) Regarding the initial conditions (1.3), we assume that
ϕ0∈H1,Ψ(ϕ0)∈L1,(2.28)
σ0∈L2,0≤σ0≤M:= max{kσskL∞(Q),kσΓkL∞(Σ)},(2.29)
u0∈H2∩H1
0,v0∈H1
0,(2.30)
z0∈ D(−∆p),b
β(z0)∈L1.(2.31)
13
2.3 Weak formulation and existence result
Definition 2.10. We say that a quintuplet (ϕ, µ, σ, u, z)is a weak solution to the
PDEs system (1.1)–(1.3)if it has the regularity
ϕ∈L2(0, T ;H2)∩L∞(0, T ;H1)∩H1(0, T ; (H1)′), µ ∈L2(0, T ;H1),
σ∈L2(0, T ;H1)∩H1(0, T ; (H1)′),
u∈H1(0, T ;H2∩H1
0)∩W1,∞(0, T ;H1
0)∩H2(0, T ;L2),
z∈L∞(0, T ;W1,p)∩H1(0, T ;L2),−∆pz∈L2(0, T ;L2)
and there exists a subgradient
ξ∈L2(0, T ;L2)with ξ∈β(z)a.e. in Q,
such that the following equations are satisfied a.e. in (0, T )
h∂tϕ, ζiH1+ZΩ∇µ· ∇ζdx=ZΩ
U(ϕ, σ, ε(u), z)ζdx, (2.32a)
ZΩ
µζ dx=ZΩ∇ϕ· ∇ζdx+ZΩ
Ψ′(ϕ)ζdx+ZΩ
W,ϕ(ϕ, ε(u), z)ζdx, (2.32b)
h∂tσ, ζiH1+ZΩ∇σ· ∇ζdx+αZΓ
(σ−σΓ)ζdA=ZΩ
S(ϕ, σ, z)ζdx, (2.32c)
ZΩ
∂ttu·ωdx+ZΩa(z)Vε(∂tu) + W,ε(ϕ, ε(u), z):ε(ω) dx= 0,(2.32d)
ZΩ
∂tzρ dx+ZΩ|∇z|p−2∇z· ∇ρdx+ZΩ
ξρ dx+ZΩ
π(z)ρdx
+ZΩ
W,z(ϕ, ε(u), z)ρdx= 0
(2.32e)
for all ζ∈H1,ω∈H1
0and ρ∈W1,p. Moreover, we require that the quintuplet complies
with the initial conditions, i.e.,
ϕ(0) = ϕ0, σ(0) = σ0,u(0) = u0, ∂tu(0) = v0, z(0) = z0a.e. in Ω.
Remark 2.11. Notice that, with the regularity that we demand, requiring (2.32b)
is equivalent to ask that equation (1.1b) is satisfied in L2(0, T ;L2) and the boundary
condition ∇ϕ·ν= 0 in (1.2) is satisfied in the sense of the traces. The same holds also
for the damage equation. Similarly, equation (2.32d) is equivalent to ask that
∂ttu−a′(z)Vε(∂tu)∇z−a(z) div Vε(∂tu)
−h′(z)Cε(∂tu)− Rϕ∇z−h(z) div Cε(∂tu)−CRϕ=0
is satisfied in L2(0, T ;L2) and that the boundary condition u=0in (1.2) holds in the
sense of the traces.
14
Remark 2.12. Note that, by standard embedding results (see [Str66] and [LM12]),
ϕ∈L∞(0, T ;H1)∩C0([0, T ]; (H1)′)֒→C0
w([0, T ]; H1),
σ∈L2(0, T ;H1)∩H1(0, T ; (H1)′)֒→C0([0, T ]; L2),
u∈H1(0, T ;H2∩H1
0)֒→C0([0, T ]; H2∩H1
0),
∂tu∈L∞(0, T ;H1
0)∩C0([0, T ]; L2)֒→C0
w([0, T ]; H1
0),
z∈L∞(0, T ;W1,p)∩C0(0, T ;L2)֒→C0
w([0, T ]; W1,p),
so ϕ(0) makes sense in H1,σ(0) in L2,u(0) in H2∩H1
0,∂tu(0) in H1
0and z(0) in W1,p.
This justifies the initial data regularities that we prescribed.
Theorem 2.13. Let Hypotheses (H1)–(H9) be satisfied. Then, there exists a weak solu-
tion to the system (1.1)–(1.3)in the sense of Definition 2.10 with the additional property
that
0≤σ≤Ma.e. in Q.
3 Proof of the existence theorem
To prove the existence theorem, we will introduce a semi-implicit Euler scheme that is
a time-discrete and regularised version of our system.
3.1 Time discretisation
We consider a uniform partition of [0, T ] with time-step τsuch that T=Kτ·τand
nodes tk
τ:=kτ for k= 0,...,Kτ. We also introduce the notation:
Ik
τ:=([0, τ ] if k= 1,
(tk−1
τ, tk
τ] if k= 2 . . . Kτ.
We approximate f,σsand σΓwith their local means, i.e., we define
fk
τ:=1
τZtk
τ
tk−1
τ
fds, σk
s,τ :=1
τZtk
τ
tk−1
τ
σsds, σk
Γ,τ :=1
τZtk
τ
tk−1
τ
σΓds,
for every k= 1,...,Kτ.
Remark 3.1. It is obvious that, since f∈L∞(0, T ;L2), σs∈L∞(Q), and σΓ∈L∞(Σ),
then fk
τ∈L2,σk
s,τ ∈L∞, and σk
Γ,τ ∈L∞
Γwith
kfk
τkL2≤ kfkL∞(L2),kσk
s,τ kL∞≤ kσskL∞(Q),kσk
Γ,τ kL∞
Γ≤ kσΓkL∞(Σ),(3.1)
for every k= 1,...,Kτ. In addition, 0 ≤σk
s,τ , σk
Γ,τ ≤M.
15
We employ a convex-concave splitting for h. More explicitly, we define
`
h(z):=h(z) + 1
2 sup
x∈R|h′′(x)|!z2,a
h(z):=−1
2 sup
x∈R|h′′(x)|!z2,
and we note that `
his convex, a
his concave and h=`
h+a
h. Furthermore, we observe that,
since h′is Lipschitz by Hypothesis (H4), the same holds for `
h′and a
h′and, since h > 0,
also `
h>0. However, `
hand a
hare not bounded. Moreover, we extend the convex-concave
decomposition to W, with respect to its third variable:
`
W3(ϕ, ε(u), z):=1
2
`
h(z)C(ε(u)− Rϕ):(ε(u)− Rϕ),
a
W3(ϕ, ε(u), z):=1
2
a
h(z)C(ε(u)− Rϕ):(ε(u)− Rϕ).
We do not need to do the same for its first and second variables because Wis already
convex with respect to ϕand e. These splittings will have a key role in carrying out the
discrete a priori estimates in Proposition 3.8, where we will employ the following trivial
result, the proof of which is just a simple application of convex and concave inequalities.
Lemma 3.2. Let s:R→Rbe a differentiable function that admits a convex-concave
decomposition s=`
s+a
swith differentiable `
sand a
s. Then,
(`
s′(x) + a
s′(y))(x−y)≥s(x)−s(y)
for every x, y ∈R.
We replace b
βwith its Moreau–Yosida approximation b
βτdefined by
b
βτ(z):= min
y∈R1
2τ|y−z|2+b
β(z)∀z∈R,
and consequentially the maximal monotone operator βwith βτ:= (b
βτ)′. Note that we
set the regularisation parameter equal to the time step τso that we will pass to the limit
simultaneously in the Yosida regularisation and in the time discretisation as τ→0.
Remark 3.3. We recall that b
βτ∈C1(R) is still convex and that βτis non-decreasing and
Lipschitz continuous with Lipschitz constant bounded by τ−1(see [Br´e73, Proposition
2.6, p. 28 and Proposition 2.11 p. 39]). Moreover, since b
βis non-negative, b
βτis non-
negative. Finally, it is obvious by the definition of Moreau–Yosida approximation that
b
βτ(z)≤b
β(z) for every z∈R.
For every sequence of scalar or vector-valued functions {wk}Kτ
k=0 defined over Ω, we
introduce the notation:
Dτ,k w=wk−wk−1
τ, D2
τ,k w=wk−2wk−1+wk−2
τ2,
16
for every k= 1 . . . Kτ. Moreover, we will sometimes make use of the special notation
vk
τ:=Dτ,k u
to denote the time-discrete velocity at the time-step k. We introduce here the time-
discrete approximation of our problem, which is posed in Ω:
Dτ,k ϕ−∆µk
τ=Uk−τDτ,k µ, (3.2a)
µk
τ=−∆ϕk
τ+`
Ψ′(ϕk
τ) + a
Ψ′(ϕk−1
τ) + W,ϕ(ϕk
τ, ε(uk−1
τ), zk−1
τ) + τDτ,k ϕ, (3.2b)
Dτ,k σ−∆σk
τ=Sk,(3.2c)
D2
τ,k u−div ha(zk
τ)Vε(Dτ,k u) + W,ε(ϕk
τ, ε(uk
τ), zk
τ)i=0,(3.2d)
Dτ,k z−∆pzk
τ+βτ(zk
τ) + π(zk−1
τ) + `
W3,z(ϕk
τ, ε(uk−1
τ), zk
τ)
+a
W3,z(ϕk
τ, ε(uk−1
τ), zk−1
τ) = 0.(3.2e)
Here, for brevity, we introduced the following notation for the source terms:
Uk:=λpσk
τ
1 + |W,ε(ϕk−1
τ, ε(uk−1
τ), zk−1
τ)|−λa+fk
τg(ϕk−1
τ, zk−1
τ),
Sk:=−λcσk
τg(ϕk−1
τ, zk−1
τ) + Λs(zk−1
τ)(σk
s,τ −σk
τ).
The system (3.2) is coupled with the boundary conditions:
∇ϕk
τ·ν=∇µk
τ·ν= 0,(3.3a)
∇σk
τ·ν+α(σk
τ−σk
Γ,τ ) = 0,(3.3b)
uk
τ=0,(3.3c)
(|∇zk
τ|p−2∇zk
τ)·ν= 0.(3.3d)
For every τ > 0 we employ a recursive procedure that, starting from the initial values
ϕ0
τ:=ϕ0, σ0
τ:=σ0,u0
τ:=u0, z0
τ:=z0,(3.4)
gives (ϕk
τ, µk
τ, σk
τ,uk
τ, zk
τ) for every k= 1,...,Kτthat satisfies the previous system
(3.2)-(3.3) in the following sense.
Definition 3.4. We say that a quintuplet (ϕk
τ, µk
τ, σk
τ,uk
τ, zk
τ)is a weak solution to the
system (3.2)–(3.3)if it has the regularity
(ϕk
τ, µk
τ, σk
τ,uk
τ, zk
τ)∈H2×H2×H1×H2× D(−∆p),
it satisfies the boundary conditions (3.3)in the sense of traces, equations (3.2a)-(3.2b)-
(3.2d)-(3.2e)hold a.e. in Ω, and equation (3.2c)plus boundary condition (3.3b)hold in
the weak sense
ZΩ
Dτ,k σζ dx+ZΩ∇σk
τ· ∇ζdx+αZΓ
(σk
τ−σk
Γ,τ )ζdA=ZΩ
Skζdx
for all ζ∈H1.
17
Notice that, due to the regularising term τ(Dτ,kµ) = µk
τ−µk−1
τin the discrete Cahn–
Hilliard equation (3.2a), at the step k= 1 the term µ0
τappears. So, we define
µ0
τ:= 0.
Similarly, to give a meaning to the term D2
τ,k uin the displacement equation (3.2d) at
the step k= 1, we introduce
u−1
τ:=u0−τv0.
Proposition 3.5. Let Hypotheses (H1)–(H9) be satisfied. Then, for every k= 1 . . . Kτ,
there exists a unique weak solution to the system (3.2)–(3.3)in the sense of Defini-
tion 3.4.
Proof. Nutrient equation. First of all, we can rewrite the system
(Dτ,k σ−∆σk
τ=−λcσk
τg(ϕk−1
τ, zk−1
τ) + Λs(zk−1
τ)(σk
s,τ −σk
τ) in Ω
∇σk
τ·ν+α(σk
τ−σk
Γ,τ ) = 0 on Γ (3.5)
in the more convenient form
(−∆σk
τ+ckσk
τ=dkin Ω
∇σk
τ·ν+α(σk
τ−σk
Γ,τ ) = 0 on Γ,(3.6)
where
ck:=1
τ+λcg(ϕk−1
τ, zk−1
τ) + Λs(zk−1
τ), dk=σk−1
τ
τ+ Λs(zk−1
τ)σk
s,τ (3.7)
are known terms in L∞and L2respectively, with ck≥0 a.e. in Ω. The variational
formulation of the problem is the following:
Find a σk∈H1such that ∀ζ∈H1
ZΩ∇σk
τ· ∇ζdx+αZΓ
σk
τζdA+ZΩ
ckσk
τζdx=ZΩ
dkζdx+ZΓ
σk
Γ,τ ζdA. (3.8)
Using Lax–Milgram theorem, one can show that there exists a unique weak solution
σk
τ∈H1.
Cahn–Hilliard equation. We consider the problem:
Dτ,k ϕ−∆µk
τ=Uk−τDτ,k µin Ω
µk
τ=−∆ϕk
τ+`
Ψ′(ϕk
τ) + a
Ψ′(ϕk−1
τ) + W,ϕ(ϕk
τ, ε(uk−1
τ), zk−1
τ) + τDτ,k ϕin Ω
∇ϕk
τ·ν=∇µk
τ·ν= 0 on Γ.
(3.9)
The first equation in (3.9) can be reformulated in the equivalent form
1
τ(I−∆)−1ϕk
τ+µk
τ= (I−∆)−1Uk+µk−1
τ+1
τϕk−1
τ,(3.10)
18
observing that I−∆:D(−∆) ⊆L2→L2(with Neumann homogeneous boundary
condition) is a bijective operator, so γ:= (I−∆)−1:L2→L2is injective. Moreover,
−∆:D(−∆) ⊆L2→L2is a linear single-valued maximal monotone operator and, as a
consequence, γis a linear, single-valued, monotone and contractive operator defined on
all L2. Substituting µk
τin the second equation of (3.9) and recalling the expression of
W,ϕ from (1.7), we obtain:
1
τγ(ϕk
τ)−∆ϕk
τ+`
Ψ′(ϕk
τ) + h(zk−1
τ)CR :R+ 1ϕk
τ
=γUk+µk−1
τ+1
τϕk−1
τ−a
Ψ′(ϕk−1
τ) + ϕk−1
τ+h(zk−1
τ)Cε(uk−1
τ):R.
For brevity, we introduce the known functions
jk:=γUk+µk−1
τ+1
τϕk−1
τ−a
Ψ′(ϕk−1
τ) + ϕk−1
τ+h(zk−1
τ)Cε(uk−1
τ):R,
lk:=h(zk−1
τ)CR :R+ 1
and we notice that jk∈L2, that lkis bounded from above (since hand Care bounded by
Hypotheses (H4) and (H3) respectively) and satisfies lk≥1 (because his non-negative
and Cis strongly elliptic again by Hypotheses (H4) and (H3)). To find a solution for
1
τγ(ϕk
τ)−∆ϕk
τ+`
Ψ′(ϕk
τ) + lkϕk
τ=jk,(3.11)
we introduce `
Ψδ, the Moreau–Yosida approximation of `
Ψwith regularisation parameter
δ > 0. We define the operator
Bδ
τ,k :=1
τγ+`
Ψ′
δ+lkI:L2→L2.
We can reformulate the regularised system in the abstract form:
(Bδ
τ,k −∆)(ϕδ) = jk.(3.12)
The operator −∆ is maximal monotone. Bδ
τ,k is monotone (because it is the sum of
monotone operators) and hemicontinuous (because it is continuous). Finally, it is easy
to show that Bδ
τ,k −∆ is coercive. So, we can apply [Bar76, Corollary 1.3, p. 48], and
conclude that Bδ
τ,k −∆ is maximal monotone and that R(Bδ
τ,k −∆) = L2. This leads to
the fact that it exists a
ϕδ∈ D(Bδ
τ,k −∆) = L2∩ D(−∆) = {v∈H2|∇v·ν= 0 on Γ}
that satisfies (3.12). Note that, obviously, ϕδalso depends on kand τ, but at this level,
they are fixed, so we omit this dependence to not overload the notation. Now it only
remains to pass to the limit for δ→0 and show that the limit satisfies (3.11). We need
some a priori estimates.
19
First a priori estimate. We test (3.12) with ϕδ:
ZΩ
1
τγ(ϕδ)ϕδ+|∇ϕδ|2+`
Ψ′
δ(ϕδ)ϕδ+lkϕ2
δdx=ZΩ
jkϕδdx.
Using the fact that γis monotone with γ(0) = 0, that `
Ψ′
δis monotone with
`
Ψ′
δ(0) = 0 and lk≥1, we have
kϕδk2
H1≤CkϕδkL2kjkkL2,
from which we get kϕδkH1≤CkjkkL2=Cτ, where Cτdoes not depend on δ.
Second a priori estimate. We test (3.12) with −∆ϕδ+`
Ψ′
δ(ϕδ) and, since lkis
uniformly bounded from above and γis a contraction, by the first a priori estimate
we get:
k−∆ϕδ+`
Ψ′
δ(ϕδ)kL2≤ k−1
τγ(ϕδ)−lkϕδ+jkkL2≤Cτ.
On the other hand, we have
k−∆ϕδ+`
Ψ′
δ(ϕδ)k2
L2=ZΩ| − ∆ϕδ|2dx+ZΩ|`
Ψ′
δ(ϕδ)|2dx+ 2 ZΩ−∆ϕδ
`
Ψ′
δ(ϕδ) dx
=ZΩ| − ∆ϕδ|2dx+ZΩ|`
Ψ′
δ(ϕδ)|2dx+ 2 ZΩ
`
Ψ′′
δ(ϕδ)|∇ϕδ|2dx
≥ k−∆ϕδk2
L2+k`
Ψ′
δ(ϕδ)k2
L2
because `
Ψ′′
δ≥0 a.e. (recall that `
Ψ′
δis Lipschitz and non-decreasing).
From the first and the second a priori estimates, we get that kϕδkH2+k`
Ψ′
δ(ϕδ)kL2≤Cτ,
so there exist a ϕk
τ∈H2and a ρk
τ∈L2such that, along a non-relabelled sub-sequence,
ϕδ⇀ ϕk
τin H2,ϕδ→ϕk
τin L2and `
Ψ′
δ(ϕδ)⇀ ρk
τin L2. Furthermore, because of these
convergences,
lim
δ→0ZΩ
`
Ψ′
δ(ϕδ)ϕδdx=ZΩ
ρk
τϕk
τdx.
So, thanks to [Bar76, Proposition 1.1, p. 42], we have that ρk
τ=`
Ψ′(ϕk
τ). Pointing out
that γ(ϕδ)→γ(ϕk
τ) in L2(because γis a contraction), we can pass to the weak limit in
(3.11) and deduce that
γ(ϕk
τ)−∆ϕk
τ+`
Ψ′(ϕk
τ) + lkϕk
τ=jk
in L2. Additionally, we remark that ∇ϕk
τ·ν= 0 on Γ in the sense of traces because
∇ϕδ·ν= 0 for every δand the normal trace operator is linear and continuous over H2.
Finally, we define µk
τas in the second equation of the system (3.9) and claim that it
belongs to
R((I−∆)−1) = D(I−∆) = {v∈H2|∇v·ν= 0 on Γ}
20
by comparison in (3.10). It remains to prove that the solution (ϕk
τ, µk
τ) is unique. We
take two solutions and the components ϕ1and ϕ2solving (3.11). They satisfy
(γ(ϕ1)−γ(ϕ2)) −∆(ϕ1−ϕ2) + `
Ψ′(ϕ1)−`
Ψ′(ϕ2) + lk(ϕ1−ϕ2) = 0
in L2. Testing this equation with ϕ1−ϕ2, we have
ZΩ
(γ(ϕ1)−γ(ϕ2))(ϕ1−ϕ2) dx+ZΩ|∇(ϕ1−ϕ2)|2dx
+ZΩ
(`
Ψ′(ϕ1)−`
Ψ′(ϕ2))(ϕ1−ϕ2) dx+ZΩ
lk(ϕ1−ϕ2)2dx= 0.
Recalling that γand `
Ψ′are monotone (so the first and third addend are non-negative)
and that lk≥1, we get
ZΩ
(ϕ1−ϕ2)2dx+ZΩ|∇(ϕ1−ϕ2)|2dx≤0
from which ϕ1=ϕ2follows. Consequentially, also the components µ1and µ2must
coincide from (3.10).
Damage differential equation. We want to find a weak solution of
Dτ,k z−∆pzk
τ+βτ(zk
τ) + π(zk−1
τ)
Dτ,k z+
`
h′(zk
τ) + a
h′(zk−1
τ)
2C[ε(uk−1
τ)− Rϕk
τ]:[ε(uk−1
τ)− Rϕk
τ] = 0 in Ω
(|∇zk
τ|p−2∇zk
τ)·ν= 0 on Γ
(3.13)
using a minimizing procedure. So we introduce the functional Fτ,k :W1,p →Rdefined
as follows:
Fτ,k (z):=1
2τZΩ|z|2dx−1
τZΩ
zk−1
τzdx+1
pZΩ|∇z|pdx+ZΩb
βτ(z) dx
+ZΩ
π(zk−1
τ)zdx+ZΩ
`
h(z)
2C[ε(uk−1
τ)− Rϕk
τ]:[ε(uk−1
τ)− Rϕk
τ] dx
+ZΩ
a
h′(zk−1
τ)
2C[ε(uk−1
τ)− Rϕk
τ]:[ε(uk−1
τ)− Rϕk
τ]zdx
and we use the direct method of the Calculus of Variations. We consider a minimizing
sequence {zj}jand prove that it admits a subsequence that converges to a minimizer
for Fτ,k . We will need coercivity and weakly lower semi-continuity of Fτ,k.
Coercivity. Reminding that b
βτand `
hare nonnegative, πand a
h′are Lipschitz and
zk−1
τ∈W1,p ֒→L∞(since pis strictly bigger then d), Cis bounded and strongly
elliptic, we obtain:
Fτ,k (z)≥1
2τZΩ|z|2dx−CZΩ|z|dx+1
pZΩ|∇z|pdx−CZΩ|ε(uk−1
τ)−Rϕk
τ|2|z|dx.
21
Using Young inequality and ε(uk−1
τ)− Rϕk
τ∈H1֒→L4, the previous inequality
becomes:
Fτ,k (z)≥CZΩ|z|2dx+CZΩ|∇z|pdx−C.
Weakly lower semi-continuity. All terms are convex and continuous in the strong
topology and therefore weakly lower semi-continuous (see [Br´e11, Corollary 3.9, p.
61]).
We note that it exists C∈Rsuch that infW1,p Fτ ,k < C, so we can suppose without loss
of generality that Fτ,k (zj)< C for every j. Thanks to coercivity, it trivially follows that
{zj}jis bounded in W1,p; so there exists a sub-sequence (that we do not relabel) and a
zk
τ∈W1,p such that zj⇀ zk
τin W1,p. From weakly lower semi-continuity we get that:
Fτ,k (zk
τ)≤lim inf
j→+∞Fτ,k (zj) = inf
W1,p Fτ,k ,
so zk
τis a minimizer for Fτ,k . To conclude, we observe that Fτ,k is Fr´echet differentiable,
so the minimum zk
τsatisfies the following associated Euler-Lagrange equation
0 = 1
τZΩ
zk
τwdx−1
τZΩ
zk−1
τwdx
+ZΩ|∇zk
τ|p−2∇zk
τ· ∇wdx+ZΩ
βτ(zk
τ)wdx+ZΩ
π(zk−1
τ)wdx
+ZΩ
`
h′(zk
τ) + a
h′(zk−1
τ)
2C[ε(uk−1
τ)− Rϕk
τ]:[ε(uk−1
τ)−Rϕk
τ]wdx
(3.14)
for every w∈W1,p. By comparison in (3.14), we can also claim that −∆pzk
τ=
−div(|∇zk
τ|p−2∇zk
τ) is in L2because
zk
τ−zk−1
τ
τ+βτ(z) + π(zk−1
τ) +
`
h′(zk
τ) + a
h′(zk−1
τ)
2C[ε(uk−1
τ)−Rϕk
τ]:[ε(uk−1
τ)− Rϕk
τ]
belongs to L2. This means that zk
τ∈ D(−∆p). Now we prove that the solution is unique.
If we suppose to have two solutions to (3.13)z1and z2, they both are minimizers of Fτ,k
and satisfy (3.14). If we consider the difference between the two equations and we take
w=z1−z2as test function, we obtain:
0 = 1
τZΩ
(z1−z2)2dx+ZΩ
(|∇z1|p−2∇z1− |∇z2|p−2∇z2)· ∇(z1−z2) dx
+ZΩ
`
h′(z1)−`
h′(z2)
2(z1−z2)C[ε(uk−1
τ)− Rϕk
τ]:[ε(uk−1
τ)− Rϕk
τ] dx
+ZΩβτ(z1)−βτ(z2)(z1−z2) dx≥1
τkz1−z2k2
L2,
where the last inequality follows from the fact that −∆p,βτ, and `
h′are monotone op-
erators, so the related terms are non-negative. Thus, it turns out that z1=z2.
22
Displacement equation. First of all, we rewrite the system
D2
τ,k u−div ha(zk
τ)Vε(Dτ,k u) + h(zk
τ)C(ε(uk
τ)− Rϕk
τ)i=0in Ω
uk
τ=0on Γ (3.15)
as
−div hTkε(uk
τ)i+uk
τ=tkin Ω
uk
τ=0on Γ,(3.16)
where we have introduced the following known terms:
Tk:=τa(zk
τ)V+τ2h(zk
τ)C= (τωa(zk
τ) + τ2h(zk
τ))C=θ(zk
τ)C,(3.17)
tk:= 2uk−1
τ−uk−2
τ−div hτ2h(zk
τ)ϕk
τCR +τ a(zk
τ)Vε(uk−1
τ)i.(3.18)
Since Tkis bounded and coercive and tkis in L2, it is easy to prove using Lax–Milgram
theorem that system (3.16) has a (unique) weak solution uk
τ∈H1
0. It remains to be
proved that uk
τ∈H2and it can be done exactly as in [HR15, Lemma 4.1, p. 4596],
using a bootstrap argument. Notice that here is where we need to require V=ωC.
Given a sequence of scalar or vector-valued functions {wk
τ}Kτ
k=0 defined over Ω, we intro-
duce the piecewise constant interpolations wτ, wτand the piecewise linear interpolation
wτover the time interval [0, T ] as
wτ(t):=wk
τ, wτ(t):=wk−1
τ, wτ(t):=t−tk−1
τ
τwk
τ+tk
τ−t
τwk−1
τ(3.19)
for every t∈Ik
τ. With this new notation, the time-discretised and regularized system
(3.2) can be written as
∂tϕτ−∆µτ=λpστ
1 + |W,ε(ϕτ, ε(uτ), zτ)|−λa+fτg(ϕτ, zτ)−(µτ−µτ),(3.20a)
µτ=−∆ϕτ+`
Ψ′(ϕτ) + a
Ψ′(ϕτ)−h(zτ)(ε(uτ)− Rϕτ):CR + (ϕτ−ϕτ),(3.20b)
∂tστ−∆στ=−λcστg(ϕτ, zτ) + Λs(zτ)(σs,τ −στ),(3.20c)
∂tvτ−div a(zτ)Vε(vτ) + h(zτ)C(ε(uτ)−Rϕτ)=0,(3.20d)
∂tzτ−∆pzτ+βτ(zτ) + π(zτ)
+
`
h′(zτ) + a
h′(zτ)
2(ε(uτ)− Rϕτ):C(ε(uτ)−Rϕτ) = 0.
(3.20e)
3.2 A priori estimates for the time-discrete system
In the following, we will need the boundedness of the nutrient variable σk
τ, so we prove
a comparison principle.
23
Lemma 3.6. The function σk
τsatisfies 0≤σk
τ≤Mfor every k= 0,...,Kτ.
Proof. Knowing that σ0
τ=σ0satisfies this property by Hypothesis (H9), we proceed by
induction on k, so we suppose that 0 ≤σk−1
τ≤Mand we prove that the same stands
for σk
τ. We remind that 0 ≤σk
s,τ , σk
Γ,τ ≤Mand that, using the notation introduced
in (3.7), ck≥1/τ and dk≥0. We also recall that, given a function s, its positive and
negative parts are defined as
s+(x):= max{s(x),0}, s−(x):= max{−s(x),0},
and that, if s∈H1, the following relations hold
ZΩ
s s+dx=ks+k2
L2,ZΩ
s s−dx=−ks−k2
L2,
ZΩ∇s· ∇s+dx=k∇s+k2
L2,ZΩ∇s· ∇s−dx=−k∇s−k2
L2.
(3.21)
Testing (3.2c) with −(σk
τ)−, we obtain
−ZΩ∇σk
τ·∇[(σk
τ)−] dx−ZΓ
α(σk
τ−σk
Γ,τ )(σk
τ)−dA−ZΩ
ckσk
τ(σk
τ)−dx=−ZΩ
dk(σk
τ)−dx.
Using (3.21), it holds
1
τk(σk
τ)−k2
L2≤ k∇[(σk
τ)−]k2
L2+k√ck(σk
τ)−k2
L2+k√α(σk
τ)−k2
L2
Γ
=−ZΩ
dk(σk
τ)−dx−ZΓ
α σk
Γ,τ (σk
τ)−dA≤0,
so k(σk
τ)−k2
L2= 0 (or, equivalently, σk
τ≥0 a.e. in Ω).
In the same way, we test (3.2c) with (σk
τ−M)+, obtaining
ZΩ∇σk
τ· ∇[(σk
τ−M)+] dx+ZΓ
α(σk
τ−σk
Γ,τ )(σk
τ−M)+dA
+ZΩ
ckσk
τ(σk
τ−M)+dx=ZΩ
dk(σk
τ−M)+dx,
that can be rewritten as
ZΩ|∇[(σk
τ−M)+]|2dx+ZΓ
α[(σk
τ−M)+]2dA−ZΓ
α(σk
Γ,τ −M)(σk
τ−M)+dA
+ZΩ
ck[(σk
τ−M)+]2dx+ZΩ
(ckM−dk)(σk
τ−M)+dx= 0.
Noticing that
ckM−dk=1
τ(M−σk−1
τ) + Λs(zk−1
τ)(M−σk
s,τ ) + λcMg(ϕk−1
τ, zk−1
τ)≥0,
and recalling that ck≥1/τ , from the previous inequality it follows that k(σk
τ−M)+k2
L2
is equal to 0, so σk
τ≤Ma.e. in Ω.
24
Remark 3.7. From Lemma 3.6 and (3.1) it follows that:
•Uk=λpσk
τ
1 + |W,ε(ϕk−1
τ, ε(uk−1
τ), zk−1
τ)|−λa+fk
τg(ϕk−1
τ, zk−1
τ) belongs to L2with
kUkkL2≤Cfor a positive Cindependent of τand k.
•Sk=−λcσk
τg(ϕk−1
τ, zk−1
τ) + Λs(zk−1
τ)(σk
s,τ −σk
τ) is in L∞with kSkkL∞≤Cfor a
positive Cindependent of τand k.
Proposition 3.8. The time-discrete solution to the problem (3.20)constructed from
Proposition 3.5 satisfies the following a priori estimates uniformly in τ:
kϕτkL∞(H1)∩L2(H2)+kϕτkL∞(H1)≤C, (3.22)
τ−1/2kϕτ−ϕτkL2(L2)≤C, (3.23)
k∂tϕτkL2((H1)′)≤C, (3.24)
k`
Ψ′(ϕτ)kL2(L2)+k`
Ψ′(ϕτ)kL2(L2)≤C, (3.25)
kµτkL2(H1)+kµτkL2(H1)≤C, (3.26)
kστkL∞(L2)∩L2(H1)+kστkL∞(L2)∩L2(H1)≤C, (3.27)
k∂tστkL2((H1)′)≤C, (3.28)
kuτkL∞(H2)+kuτkL∞(H2)≤C, (3.29)
kuτkW1,∞(H1)∩H1(H2)≤C, (3.30)
kvτkL∞(H1)∩L2(H2)+kvτkL∞(H1)≤C, (3.31)
kvτkL∞(H1)∩H1(L2)≤C, (3.32)
kzτkL∞(W1,p)∩L2(W1+δ,p)+kzτkL∞(W1,p )∩L2(W1+δ,p)≤C, (3.33)
kzτkL∞(W1,p)∩L2(W1+δ,p)∩H1(L2)≤C, (3.34)
k−∆pzτkL2(L2)+k−∆pzτkL2(L2)≤C, (3.35)
kβτ(zτ)kL2(L2)≤C, (3.36)
where δ∈(0,1/p).
Notice that in equations (3.22) and (3.31) the estimates for the retarded piecewise con-
stant interpolants hold in weaker spaces because they are equal to the initial data in
[0, τ ] and the initial data are less regular than the corresponding discrete solutions at
the step k= 1,...,Kτ.
Proof. Energy estimate. Testing (3.2a) with τµk
τ, we obtain:
τZΩ
Dτ,k ϕ µk
τdx+τZΩ|∇µk
τ|2dx=τZΩ
Ukµk
τ−τZΩ
(µk
τ−µk−1
τ)µk
τdx.
Using Young inequality to handle the last term, we have:
τZΩ
Dτ,k ϕ µk
τdx+τZΩ|∇µk
τ|2dx+τ
2ZΩ|µk
τ|2dx−τ
2ZΩ|µk−1
τ|2dx≤τZΩ
Ukµk
τ.(3.37)
25
Testing (3.2b) with −(ϕk
τ−ϕk−1
τ),
−τZΩ
µk
τDτ,k ϕdx+ZΩ∇ϕk
τ· ∇(ϕk
τ−ϕk−1
τ) dx+ZΩh`
Ψ′(ϕk
τ) + a
Ψ′(ϕk−1
τ)i(ϕk
τ−ϕk−1
τ) dx
+ZΩ
W,ϕ(ϕk
τ, ε(uk−1
τ), zk−1
τ)(ϕk
τ−ϕk−1
τ) dx+ZΩ|ϕk
τ−ϕk−1
τ|2dx= 0.
Employing Young inequality for the second term and Lemma 3.2 for Ψ = `
Ψ+a
Ψ, we get
−τZΩ
µk
τDτ,k ϕdx+1
2ZΩ|∇ϕk
τ|2dx−1
2ZΩ|∇ϕk−1
τ|2dx
+ZΩ
Ψ(ϕk
τ) dx−ZΩ
Ψ(ϕk−1
τ) dx
+ZΩ
W,ϕ(ϕk
τ, ε(uk−1
τ), zk−1
τ)(ϕk
τ−ϕk−1
τ) dx+ZΩ|ϕk
τ−ϕk−1
τ|2dx≤0.
(3.38)
Testing (3.2c) with τσk
τand applying Young inequality for the first term, we obtain:
1
2ZΩ|σk
τ|2dx−1
2ZΩ|σk−1
τ|2dx+τZΩ|∇σk
τ|2dx+τZΓ
α|σk
τ|2dA
≤τZΩ
Skσk
τdx+τZΓ
ασk
Γ,τ σk
τdA.
(3.39)
Testing (3.2d) with uk
τ−uk−1
τ=τvk
τ, we get:
ZΩ
(vk
τ−vk−1
τ)·vk
τdx+τZΩ
a(zk
τ)Vε(vk
τ):ε(vk
τ) dx
+ZΩ
W,ε(ϕk
τ, ε(uk
τ), zk
τ):(ε(uk
τ)−ε(uk−1
τ)) dx= 0.
Exploiting Young inequality for the first term, the fact that a∗≤aand that Vis
uniformly elliptic for the second term, we have:
1
2ZΩ|vk
τ|2dx−1
2ZΩ|vk−1
τ|2dx+Cτ ZΩ|ε(vk
τ)|2dx
+ZΩ
W,ε(ϕk
τ, ε(uk
τ), zk
τ):(ε(uk
τ)−ε(uk−1
τ)) dx≤0.
(3.40)
Finally, we test (3.2e) with zk
τ−zk−1
τ, obtaining:
τZΩ|Dτ,k z|2dx+ZΩ|∇zk
τ|p−2∇zk
τ· ∇(zk
τ−zk−1
τ) dx
+ZΩ
βτ(zk
τ)(zk
τ−zk−1
τ) dx+ZΩ
π(zk−1
τ)(zk
τ−zk−1
τ) dx
+ZΩh`
W3,z(ϕk
τ, ε(uk−1
τ), zk
τ) + a
W3,z(ϕk
τ, ε(uk−1
τ), zk−1
τ)i(zk
τ−zk−1
τ) dx= 0.
26
Employing Young for the second term, the convexity of b
βτfor the third and moving the
term with πto the right-hand side, we get:
τZΩ|Dτ,k z|2dx+1
pZΩ|∇zk
τ|pdx−1
pZΩ|∇zk−1
τ|pdx
+ZΩb
βτ(zk
τ) dx−ZΩb
βτ(zk−1
τ) dx
+ZΩh`
W3,z(ϕk
τ, ε(uk−1
τ), zk
τ) + a
W3,z(ϕk
τ, ε(uk−1
τ), zk−1
τ)i(zk
τ−zk−1
τ) dx
≤ − ZΩ
π(zk−1
τ)(zk
τ−zk−1
τ) dx=−τZΩ
π(zk−1
τ)Dτ,k zdx.
(3.41)
Now we notice that, since Wis convex with respect to its first variable ϕand its second
variable e, and since we can apply Lemma 3.2 to W=`
W3+a
W3, we have:
W,ϕ(ϕk
τ, ε(uk−1
τ), zk−1
τ)(ϕk
τ−ϕk−1
τ)≥W(ϕk
τ, ε(uk−1
τ), zk−1
τ)−W(ϕk−1
τ, ε(uk−1
τ), zk−1
τ),
W,ε(ϕk
τ, ε(uk
τ), zk−1
τ):(ε(uk
τ)−ε(uk−1
τ)) ≥W(ϕk
τ, ε(uk
τ), zk
τ)−W(ϕk
τ, ε(uk−1
τ), zk
τ),
h`
W3,z(ϕk
τ, ε(uk−1
τ), zk
τ) + a
W3,z(ϕk
τ, ε(uk−1
τ), zk−1
τ)i(zk
τ−zk−1
τ)
≥W(ϕk
τ, ε(uk−1
τ), zk
τ)−W(ϕk
τ, ε(uk−1
τ), zk−1
τ).
So, by summing the three above inequalities, we obtain that the left-hand side is greater
or equal than
W(ϕk
τ, ε(uk
τ), zk
τ)−W(ϕk−1
τ, ε(uk−1
τ), zk−1
τ).
Adding (3.37), (3.38), (3.39), (3.40), (3.41) and employing the previous inequality re-
garding W, we infer that:
τ
2ZΩ|µk
τ|2dx−τ
2ZΩ|µk−1
τ|2dx+1
2ZΩ|∇ϕk
τ|2dx−1
2ZΩ|∇ϕk−1
τ|2dx
+ZΩ
Ψ(ϕk
τ) dx−ZΩ
Ψ(ϕk−1
τ) dx+1
2ZΩ|σk
τ|2dx−1
2ZΩ|σk−1
τ|2dx
+1
2ZΩ|vk
τ|2dx−1
2ZΩ|vk−1
τ|2dx+1
pZΩ|∇zk
τ|pdx−1
pZΩ|∇zk−1
τ|pdx
+ZΩb
βτ(zk
τ) dx−ZΩb
βτ(zk−1
τ) dx+ZΩ
W(ϕk
τ, ε(uk
τ), zk
τ) dx
−ZΩ
W(ϕk−1
τ, ε(uk−1
τ), zk−1
τ) dx+τZΩ|∇µk
τ|2dx+ZΩ
τ−1|ϕk
τ−ϕk−1
τ|2dx
+ZΩ|∇σk
τ|2dx+ZΓ
α|σk
τ|2dA+CZΩ|ε(vk
τ)|2dx+ZΩ|Dτ,k z|2dx
≤τZΩ
Ukµk
τdx+ZΩ
Skσk
τdx+ZΓ
ασk
Γ,τ σk
τdA−ZΩ
π(zk−1
τ)Dτ,k zdx
≤τZΩ
Ukµk
τdx+C+ZΩ|π(zk−1
τ)||Dτ,k z|dx,
(3.42)
27
where the latter inequality follows from the fact that Sk,σk
τand σk
Γ,τ are bounded in
L∞uniformly with respect to kand τ. Then, by the H¨older, Poincar´e–Wirtinger, and
Young inequalities, recalling that kUkkL2≤C, we have
ZΩ
Ukµk
τdx≤ kUkkL2kµk
τkL2≤Ckµk
τ− hµk
τikL2+|Ω|1
2|hµk
τi|
≤Ck∇µk
τkL2+|Ω|1
2|hµk
τi|≤ηZΩ|∇µk
τ|2dx+Cη+C|hµk
τi|,
(3.43)
where hµk
τidenotes the mean value of µk
τand ηis a fixed and sufficiently small positive
constant. Testing (3.2b) with 1 and dividing by |Ω|, we obtain
hµk
τi=1
|Ω|ZΩ
µk
τdx=1
|Ω|ZΩ
`
Ψ′(ϕk
τ) + a
Ψ′(ϕk−1
τ) + W,ϕ(ϕk
τ, ε(uk−1
τ), zk−1
τ) + τDτ,k ϕdx.
Adding and subtracting a
Ψ′(ϕk
τ) and Rϕk−1
τ, employing growth assumption (2.9) of Ψ,
the Lipschitz continuity of a
Ψ′, and the boundedness of h, we have
|hµk
τi| ≤ 1
|Ω|ZΩ`
Ψ′(ϕk
τ) + a
Ψ′(ϕk−1
τ)+Ch(zk−1
τ)|ε(uk−1
τ)− Rϕk
τ|+τ|Dτ,k ϕ|dx
≤CZΩΨ′(ϕk
τ)+a
Ψ′(ϕk
τ)−a
Ψ′(ϕk−1
τ)+h(zk−1
τ)|ε(uk−1
τ)− Rϕk−1
τ|
+h∗|Rϕk−1
τ− Rϕk
τ|+τ|Dτ,k ϕ|dx
≤CZΩ
Ψ(ϕk
τ) + h(zk−1
τ)|ε(uk−1
τ)− Rϕk−1
τ|+τ|Dτ,k ϕ|dx.
(3.44)
Using Young inequality twice and Cstrong ellipticity from Hypothesis (H3), from the
above inequality we obtain
|hµk
τi| ≤ CZΩ
Ψ(ϕk
τ) + h(zk−1
τ)|ε(uk−1
τ)− Rϕk−1
τ|2dx+τZη|Dτ,k ϕ|2+Cηdx
≤CZΩ
Ψ(ϕk
τ) + W(ϕk−1
τ, ε(uk−1
τ), zk−1
τ) dx+ηZΩ
τ−1|ϕk
τ−ϕk−1
τ|2dx+Cη.
So, substituting in (3.43), we deduce that
ZΩ
Ukµk
τdx≤CZΩ
Ψ(ϕk
τ) + W(ϕk−1
τ, ε(uk−1
τ), zk−1
τ) dx
+ηZΩ|∇µk
τ|2dx+ηZΩ
τ−1|ϕk
τ−ϕk−1
τ|2dx+Cη.
(3.45)
Moreover, recalling that, by Hypothesis (H6), πis Lipschitz continuous, using H¨older
inequality and Young inequality with a small η, we get
ZΩ|π(zk−1
τ)||Dτ,k z|dx≤CZΩ
(|zk−1
τ|+ 1)|Dτ,k z|dx
≤C(kzk−1
τkL2+ 1)kDτ,k zkL2≤ηkDτ,k zk2
L2+Cη(kzk−1
τk2
L2+ 1)
≤ηkDτ,k zk2
L2+Cηk−1
X
i=1
τkDτ,i zk2
L2+ 1,
(3.46)
28
where we have also used the fact that zk−1
τ=z0+Pk−1
i=1 τDτ,i zif k≥2 and zk−1
τ=z0
if k= 1. Finally, using inequalities (3.45) and (3.46) in (3.42), moving to the left-hand
side the terms with η(fixing ηsmall enough) and summing from k= 1 to j, we obtain
τ
2ZΩ|µj
τ|2dx+1
2ZΩ|∇ϕj
τ|2dx+ZΩ
Ψ(ϕj
τ) dx+1
2ZΩ|σj
τ|2dx+1
2ZΩ|vj
τ|2dx
+1
pZΩ|∇zj
τ|pdx+ZΩb
βτ(zj
τ) dx+ZΩ
W(ϕj
τ, ε(uj
τ), zj
τ) dx
+
j
X
k=1 τ1
2ZΩ|∇µk
τ|2dx+1
2ZΩ
τ−1|ϕk
τ−ϕk−1
τ|2dx+ZΩ|∇σk
τ|2dx(3.47)
+ZΓ
α|σk
τ|2dA+CZΩ|ε(vk
τ)|2dx+1
2ZΩ|Dτ,k z|2dx
≤C0+C
j
X
k=1 τZΩ
Ψ(ϕk
τ) dx+ZΩ
W(ϕk−1
τ, ε(uk−1
τ), zk−1
τ) dx
+
k−1
X
i=1
τZΩ|Dτ,iz|2dx,
where Cdoes not depend on the initial data while
C0=1
2ZΩ|∇ϕ0|2dx+ZΩ
Ψ(ϕ0) dx+1
2ZΩ|σ0|2dx+1
2ZΩ|v0|2dx+1
pZΩ|∇z0|pdx
+ZΩb
βτ(z0) dx+ZΩ
W(ϕ0, ε(u0), z0) dx
≤Ckϕ0k2
H1+kσ0k2
L2+kv0k2
L2+ku0k2
H1+kz0kp
W1,p +ZΩ
Ψ(ϕ0) dx+ZΩb
β(z0) dx.
Here we used the fact that b
βτ(z0)≤b
β(z0) a.e. that comes directly from the definition
of b
βτ(see [Br´e73, Proposition 2.11, p. 39]) and the following inequality regarding the
elastic energy
ZΩ
W(ϕ0, ε(u0), z0) dx=ZΩ
h(z0)
2C(ε(u0)− Rϕ0):(ε(u0)− Rϕ0) dx
≤Ch∗ZΩ|ε(u0)− Rϕ0|2dx≤C(ku0k2
H1+kϕ0k2).
Applying the discrete Gronwall inequality stated in Lemma 2.3 to (3.47) leads to the
boundedness of the left-hand side, from which we have
kϕj
τkH1+kΨ(ϕj
τ)kL1+kσj
τkL2+kvj
τkL2+k∇zj
τkLp+kb
β(zj
τ)kL2+kε(uj
τ)kL2
+
Kτ
X
k=1 Ztk
τ
tk−1
τk∇µk
τk2
L2+kτ−1/2(ϕk
τ−ϕk−1
τ)k2
L2+k∇σk
τk2
L2+kε(vk
τ)k2
L2
+kDτ,k zk2
L2ds≤C
(3.48)
29
and, as a consequence, (3.23) and (3.27).
Energy estimate consequences. From the equality
zj
τ=z0+
j
X
k=1 Ztk
τ
tk−1
τ
Dτ,k zds,
and (3.48), we have kzj
τkL2≤C. By Poincar´e–Wirtinger inequality,
kzj
τkW1,p ≤Ck∇zj
τkLp+|hzj
τi|≤C. (3.49)
Here we used k∇zj
τkLp≤Cby (3.48) and we controlled the mean value of zj
τwith its
bounded L2norm. We can also gain a mean value estimate for µk
τ. Combining the
first line from (3.44) with (3.48), we immediately obtain |hµk
τi| ≤ C. As a consequence,
exploiting Poincar´e–Wirtinger inequality, it follows that
kµj
τkL2≤ kµj
τ− hµj
τikL2+C|hµj
τi| ≤ Ck∇µj
τkL2+ 1
and, thanks to (3.48), we get (3.26). Notice that here we also employ µτ= 0 in [0, τ ]
to obtain the estimate for µτ. Finally, by comparison in (3.20a) and (3.20c), we have
(3.24) and (3.28).
Higher order estimate for the displacement. We test the equation (3.2d) with
−τdiv hVε(vk
τ)i, obtaining
−ZΩ
(vk
τ−vk−1
τ)·div hVε(vk
τ)idx+τZΩ
div hh(zk
τ)Cε(uk
τ)i·div hVε(vk
τ)idx
+τZΩ
div ha(zk
τ)Vε(vk
τ)i·div hVε(vk
τ)idx=τZΩ
div hh(zk
τ)CRϕk
τi·div hVε(vk
τ)idx.
Developing the obvious calculations in the second and third terms on the left-hand side
and moving some terms to the right-hand side, we have
−ZΩ
(vk
τ−vk−1
τ)·div hVε(vk
τ)idx+τZΩ
a(zk
τ) div hVε(vk
τ)i·div hVε(vk
τ)idx
=−τZΩh′(zk
τ)Cε(uk
τ)∇zk
τ·div hVε(vk
τ)idx−τZΩ
h(zk
τ) div hCε(uk
τ)i·div hVε(vk
τ)idx
−τZΩa′(zk
τ)Vε(vk
τ)∇zk
τ·div hVε(vk
τ)idx+τZΩ
div hh(zk
τ)CRϕk
τi·div hVε(vk
τ)idx.
Concerning the first term on the left-hand side, recall that for every kit holds uk
τ= 0
on Γ and, consequently, vk
τ= 0 on Γ. Thus, it can be estimated as follows:
−ZΩ
(vk
τ−vk−1
τ)·div hVε(vk
τ)idx=ZΩhε(vk
τ)−ε(vk−1
τ)i:hVε(vk
τ)idx
≥1
2ZΩ
ε(vk
τ):Vε(vk
τ) dx−1
2ZΩ
ε(vk−1
τ):Vε(vk−1
τ) dx,
(3.50)
30
where the inequality holds because Vis symmetric and positive-defined, so the associated
quadratic form is convex. For the second left-hand term, since ais bounded from below
by a strictly positive constant by Hypothesis (H4), using Lemma 2.4 (and the fact that
vk
τ= 0 on Γ), we obtain
τZΩa(zk
τ) div hVε(vk
τ)i·div hVε(vk
τ)idx
≥a∗τZΩdiv hVε(vk
τ)i2dx≥C∗τkvk
τk2
H2.
(3.51)
The first term on the right-hand side can be estimated as follows:
−τZΩh′(zk
τ)Cε(uk
τ)∇zk
τ·div hVε(vk
τ)idx≤Cτkε(uk
τ)kLqk∇zk
τkLpkdiv[Vε(vk
τ)]kL2,
thanks to H¨older inequality. Here qis chosen to satisfy 1
q+1
p+1
2= 1 and, since p > d
and d= 2 or d= 3, it is easy to prove that q∈[2,6). So, because of the embedding
H1֒→Lq, Lemma 2.4, the energy estimate (3.48) and Young inequality, it follows:
−τZΩh′(zk
τ)Cε(uk
τ)∇zk
τ·div hVε(vk
τ)idx≤Cητkuk
τk2
H2+ητ kvk
τk2
H2,(3.52)
where η > 0 is small and yet to be chosen.
Regarding the second term on the right-hand side, since his bounded, we deduce that
−τZΩh(zk
τ) div hCε(uk
τ)i·div hVε(vk
τ)idx
≤Cτkdiv[Cε(uk
τ)]kL2kdiv[Vε(vk
τ)]kL2≤Cτ kuk
τkH2kvk
τkH2
≤τCηkuk
τk2
H2+ητ kvk
τk2
H2.
(3.53)
We handle the third term on the right-hand side using the fact that ais Lipschitz
continuous by Hypothesis (H4), H¨older inequality, previous estimates, Young inequality,
the embedding H2֒→Lq, and Ehrling’s Lemma (Theorem 2.2):
−τZΩha′(zk
τ)Vε(vk
τ)∇zk
τi·div hVε(vk
τ)idx
≤Cτkε(vk
τ)kLqk∇zk
τkLpkdiv[Vε(vk
τ)]kL2
≤Cτkε(vk
τ)kLqkvk
τkH2≤Cητkε(vk
τ)k2
Lq+ηkvk
τk2
H2
≤ητ kvk
τk2
H2+θτ kvk
τk2
H2+Cη,θτkvk
τk2
L2≤(η+θ)τkvk
τk2
H2+Cη,θτ
(3.54)
where η, θ > 0 are small and yet to be chosen.
Finally, for the last term on the right-hand side, we use H¨older and Young inequalities
31
and the previous estimates, obtaining
τZΩ
div hh(zk
τ)CRϕk
τi·div hVε(vk
τ)idx
=τZΩCRh′(zk
τ)ϕk
τ∇zk
τ+h(zk
τ)∇ϕk
τ+h(zk
τ)ϕk
τdiv(CR)·div hVε(vk
τ)idx
≤Cτ kϕk
τkLqk∇zk
τkLp+k∇ϕk
τkL2+kϕk
τkL2kdiv[Vε(vk
τ)]kL2
≤Cτkvk
τkH2≤ητ kvk
τk2
H2+Cητ.
(3.55)
Putting together (3.50)–(3.55) and fixing ηand θsmall enough lead to
1
2ZΩ
ε(vk
τ):Vε(vk
τ) dx−1
2ZΩ
ε(vk−1
τ):Vε(vk−1
τ) dx+C∗
2τkvk
τk2
H2≤Cτ(1 + kuk
τk2
H2).
So, summing for k= 1 to jand recalling that Vis coercive by Hypothesis (H3), we get
kε(vk
τ)k2
L2+
j
X
k=1
τkvk
τk2
H2≤C0+C
j
X
k=1
τkuk
τk2
H2
≤C0+C
j
X
k=1
τ"k
X
s=1
τkvs
τk2
H2+ku0k2
H2#=C0+C
j
X
k=1
τ"k
X
s=1
τkvs
τk2
H2#,
where the last equality holds changing the constant C0. So, applying the discrete Gron-
wall inequality stated in Lemma 2.3 leads to
kε(vk
τ)k2
L2+
j
X
k=1
τkvk
τk2
H2≤C.
Since we already know that kvk
τkL2≤Cthanks to (3.47), (3.31) follows. Moreover,
recalling the trivial identity
uk
τ=u0+
k
X
s=1
τvs
τ,
also (3.29) holds true. Finally, by comparison in (3.20d), we deduce that ∂tvτis uni-
formly bounded in L2(0, T ;L2). Indeed,
∂tvτ= div h(zτ)Cε(uτ)−div h(zτ)CRϕτ+ div a(zτ)Vε(vτ),
where each term on the right-hand side is uniformly bounded in L2(0, T ;L2). Indeed, h,
a,Cand Vare bounded and Lipschitz continuous. The term k∇zτkL∞(Lp)is uniformly
bounded thanks to (3.48). Moreover,
kε(uτ)kL∞(H1)+kϕτkL∞(H1)+kε(vτ)kL2(H1)≤C
32
thanks to the estimates (3.29)-(3.48)-(3.31), and H1֒→Lqwere qis the H¨older conju-
gate of 2p
p+2 . So, the estimate (3.32) follows. Since ∂tuτ=vτ, (3.29) and (3.31) imply
(3.30).
Higher order estimate for the order parameter. Equation (3.20b) can be rewritten
as `
Ψ′(ϕτ)−∆ϕτ=µτ−a
Ψ′(ϕτ) + h(zτ)C(ε(uτ)− Rϕτ):R − (ϕτ−ϕτ).
The right-hand side belongs to L2(0, T ;L2) because a
Ψ′is Lipschitz continuous by Hy-
pothesis (H2), hand Care bounded by Hypotheses (H4) and (H3). More specifically,
the following estimate holds:
k`
Ψ′(ϕτ)−∆ϕτkL2(L2)≤kµτkL2(L2)+CkϕτkL2(L2)+ 1
+Ckε(uτ)kL2(L2)+kϕτkL2(L2)+kϕτ−ϕτkL2(L2)≤C.
On the other hand, we have that
k`
Ψ′(ϕτ)−∆ϕτk2
L2(L2)
=k`
Ψ′(ϕτ)k2
L2(L2)+k−∆ϕτk2
L2(L2)+ 2 ZT
0ZΩ−∆ϕτ
`
Ψ′(ϕτ) dxds
≥ k`
Ψ′(ϕτ)k2
L2(L2)+k−∆ϕτk2
L2(L2),
(3.56)
from which estimate (3.25) follows. Observe that the inequality in (3.56) stands because
`
Ψ′is an increasing and continuous function (then, a maximal monotone graph). More
explicitly, if we consider its Yosida approximation `
Ψ′
δ, we have
ZT
0ZΩ−∆ϕτ
`
Ψ′
δ(ϕτ) dxds=ZT
0ZΩ
`
Ψ′′
δ(ϕτ)|∇ϕτ|2dxds≥0,
because `
Ψ′
δis monotone and Lipschitz continuous, so `
Ψ′′
δexists a.e. and it is non-
negative. Moreover, `
Ψ′
δ(ϕτ)→`
Ψ′(ϕτ) strongly in L2(0, T ;L2) as δ→0+(see [Br´e73,
Proposition 2.6, p. 28]). So, passing to the limit in the previous expression, we deduce
what we claimed. Taking into account (3.48), we deduce that (3.22) holds. Notice that
the asymmetry between ϕτand ϕτin the estimate (3.22) is a consequences of the fact
that ϕτ=ϕ0in [0, τ ] and ϕ0belongs to H1, not to H2.
More estimates for the damage. From (3.20e), we have
−∆pzτ+βτ(zτ) = −∂tzτ−π(zτ)−
`
h′(zτ) + a
h′(zτ)
2C[ε(uτ)−Rϕτ]:[ε(uτ)− Rϕτ]
in L2(0, T ;L2). More specifically, we know that
k−∆pzτ+βτ(zτ)kL2(L2)
≤ k∂tzτkL2(L2)+CkzτkL2(L2)+ 1 + kε(uτ)k2
L4(L4)+kϕτk2
L4(L4)
≤Ck∂tzτkL2(L2)+kzτkL2(L2)+kuτk2
L∞(H2)+kϕτk2
L∞(H1)+ 1≤C,
33
making use of the previous estimates, the Hypothesis (H6) according to which πis
Lipschitz continuous, the fact that `
hand a
hare continuous, the uniform boundedness of
kzτkL∞(Q)+kzτkL∞(Q)from (3.49), and the embedding W1,p ֒→C0(Ω). On the other
hand,
k−∆pzτ+βτ(zτ)k2
L2(L2)
=k−∆pzτk2
L2(L2)+kβτ(zτ)k2
L2(L2)+ 2 ZT
0ZΩ−∆pzτβτ(zτ) dxds
=k−∆pzτk2
L2(L2)+kβτ(zτ)k2
L2(L2)+ 2 ZT
0ZΩ
β′
τ(zτ)|∇zτ|pdxds
≥ k−∆pzτk2
L2(L2)+kβτ(zτ)k2
L2(L2),
where the inequality stands because β′
τis monotone and Lipschitz continuous (so it is
a.e. differentiable with positive derivative). Thus, we have proved (3.35) and (3.36),
employing the fact that z0∈ D(−∆p) by Hypothesis (H9). Finally, to conclude the
estimate (3.33), we make use of the inequality stated in Lemma 2.5, for which
kzτkW1+δ,p +kzτkW1+δ,p ≤Cδk−∆pzτkL2+k−∆pzτkL2+kzτkL2+kzτkL2,
for any δ∈(0,1/p). Thanks to (3.49) that we have already proved, we get (3.33).
Combining (3.33) with the energy estimate (3.48), we obtain (3.34).
3.3 Compactness assertions
Lemma 3.9. There exist a quintuplet (ϕ, µ, σ, u, z)that satisfy the regularity of The-
orem 2.13 such that, for a non-relabelled subsequence, we have
ϕτ→ϕweakly-∗in L∞(0, T ;H1)∩H1(0, T ; (H1)′),(3.57)
strongly in C0([0, T ]; Lr)∀r∈[1,6),(3.58)
ϕτ→ϕweakly-∗in L∞(0, T ;H1)∩L2(0, T ;H2),(3.59)
ϕτ, ϕτ→ϕstrongly in Lr(0, T ;Lr)∀r∈[1,6) and a.e. in Q, (3.60)
`
Ψ′(ϕτ)→`
Ψ′(ϕ)weakly in L2(0, T ;L2),(3.61)
a
Ψ′(ϕτ)→a
Ψ′(ϕ)strongly in L2(0, T ;L2),(3.62)
µτ, µτ→µweakly in L2(0, T ;H1),(3.63)
στ→σweakly-∗in L∞(0, T ;L2)∩L2(0, T ;H1)∩H1(0, T ; (H1)′),(3.64)
strongly in L2(0, T ;Lr)∀r∈[1,6) and a.e. in Q, (3.65)
στ→σweakly-∗in L∞(0, T ;L2)∩L2(0, T ;H1),(3.66)
uτ→uweakly-∗in W1,∞(0, T ;H1)∩H1(0, T ;H2),(3.67)
strongly in C0([0, T ]; X)for all Xs.t. H2֒→֒→X ֒→L2,(3.68)
34
uτ,uτ→uweakly-∗in L∞(0, T ;H2),(3.69)
strongly in L∞(0, T ;X)for all Xs.t. H2֒→֒→X ֒→L2,(3.70)
vτ→∂tuweakly-∗in L∞(0, T ;H1)∩H1(0, T ;L2),(3.71)
vτ→∂tuweakly-∗in L∞(0, T ;H1)∩L2(0, T ;H2),(3.72)
zτ→zweakly-∗in L∞(0, T ;W1,p)∩L2(0, T ;W1+δ,p)∩H1(0, T ;L2),
(3.73)
strongly in Ls(0, T ;W1,p)∀s∈[1,+∞)and a.e. in Q, (3.74)
zτ, zτ→zweakly-∗in L∞(0, T ;W1,p)∩L2(0, T ;W1+δ,p)∀δ∈0,1/p,(3.75)
strongly in Ls(0, T ;W1,p)∀s∈[1,+∞)and a.e. in Q, (3.76)
−∆pzτ→ −∆pzweakly in L2(0, T ;L2),(3.77)
βτ(zτ)→ξweakly in L2(0, T ;L2)with ξ∈β(z).(3.78)
Proof. Most of the convergences are obvious from Proposition 3.8 and standard com-
pactness results (Banach—Alaoglu theorem and Aubin–Lions theorem); this way, we
immediately obtain (3.59), (3.57)–(3.58), (3.63)–(3.69), (3.72)–(3.75), (3.73). Notice
that it is easy to identify the limit of a piecewise constant interpolant and its retarded
function. For example, let’s prove that µτ, µτconverges to the same limit. From (3.26),
we know that µτ→µand µτ→νweakly in L2(0, T ;H1)֒→L2(0, T ;L2). Moreover, we
recall that, by definition,
µτ(t) = µτ(t+τ) for a.e. t∈(0, T −τ).(3.79)
Take a test function ρ∈C∞
c(Ω ×(0, T )). Since it has compact support, there exists a
ǫ > 0 such that supp(ρ)⊆Ω×(ǫ, T −ǫ) and definitively 2τ < ǫ. By a simple change of
variables, taking (3.79) into account, we have
ZT
0ZΩ
µτρdxdt=ZT−ǫ
ǫZΩ
µτ(x, t +τ)ρ(x, t) dxdt=ZT+τ−ǫ
ǫ+τZΩ
µτ(x, s)ρ(x, s −τ) dxds
=ZT
0ZΩ
µτ(x, s)ρ(x, s −τ) dxds→ZT
0ZΩ
µρ dxds.
Here we have used the fact that ρ(·,·−τ) is still a test function with compact support in
(ǫ+τ, T +τ−ǫ)⊆(0, T −τ) and then we pass to the limit because we have the product
of a weak convergent sequence and a strong convergent one in L2(0, T ;L2). But we also
know that ZT
0ZΩ
µτρdxdt→ZT
0ZΩ
νρ dxdt,
so, by uniqueness of the limit and the Fundamental Lemma of the Calculus of Variations,
we conclude that µ=ν. In the following, we will discuss the less immediate limits of the
statement. To prove (3.60), we initially show that ϕτ, ϕτ→ϕstrongly in L2(0, T ;L2).
Rewriting the piecewise linear interpolant ϕτas
ϕτ(t) = ϕτ(t) + t−tk−1
τhϕτ(t)−ϕτ(t)i
35
for every t∈Ik
τ, then
kϕτ−ϕτk2
L2(L2)=
Kτ
X
k=1 Ztk
τ
tk−1
τZΩt−tk−1
τ2
(ϕτ(t)−ϕτ(t))2dxdt
≤ kϕτ−ϕτk2
L2(L2)≤τC →0,
where the last inequality is due to (3.23). On the other hand, by (3.58), ϕτgoes to ϕ
strongly in L2(0, T ;L2), so also ϕτ→ϕin L2(0, T ;L2) and, using again (3.23), the same
stands for ϕτ. We can also deduce that, along a non-relabelled subsequence, ϕτ, ϕτ→ϕ
a.e. in Q. Since kϕτkL6(L6),kϕτkL6(L6)≤C(thanks to (3.22) and the embedding
L∞(0, T ;H1)֒→L6(0, T ;L6)) and since ϕτ, ϕτ→ϕpointwise a.e., then ϕτ, ϕτ→ϕ
in Lr(0, T ;Lr) for every r∈[1,6), so (3.60) stands. From (3.25), k`
Ψ′(ϕτ)kL2(L2)≤C;
moreover, `
Ψis continuous and ϕτ→ϕa.e., so `
Ψ′(ϕτ)→`
Ψ′(ϕ) a.e. in Q. Hence, we
have also (3.61). By (3.60) and the Lipschitz continuity of a
Ψ′, we get (3.62). In order
to prove (3.70), we start by noticing that for every t∈Ik
τ
uτ(t) = uτ(t) + t−tk−1
τuτ(t)−uτ(t)=uτ(t) + t−tk−1
τZIk
τ
vτdt,
from which, using (3.31) and H2֒→X, it follows that
kuτ−uτkX≤ kvτkL2(X)τ1/2≤Cτ 1/2→0.
Since we already know that uτ→ustrongly in L∞(0, T ;X) by (3.68), this inequality
leads to (3.70). Finally, we prove the convergences regarding the damage. From Aubin–
Lions compactness result, L2(0, T ;W1+δ,p)∩H1(0, T ;L2)) ֒→֒→L2(0, T ;W1,p) so, using
(3.34) and (3.33), along a subsequence zτ→zstrongly in L2(0, T ;W1,p). Since zτis
bounded in L∞(0, T ;W1,p), we obtain (3.74). As we have already observed before, for
every t∈Ik
τit holds
zτ(t) = zτ(t)−tk−t
τZIk
τ
∂tzτdt
and, as a consequence,
kzτ−zτkL∞(L2)≤ k∂tzτkL2(L2)τ1/2≤Cτ 1/2→0.
Hence, we deduce that, along a subsequence, zτ−zτ→0 a.e. in Q. Since we know that
kzτ−zτkL∞(Q)≤CkzτkL∞(W1,p)+kzτkL∞(W1,p)≤C,
we obtain that zτ−zτ→0 strongly in Ls(0, T ;Ls) for every s∈[0,+∞). It trivially
follows that zτ−zτ→0 strongly in Ls(0, T ;Lt) for every s, t ∈[0,+∞). Now we want
to prove that a subsequence converges strongly in L2(0, T ;W1,p). To reach our purpose,
we employ the following inequality of Gagliardo–Nirenberg type for fractional Sobolev
spaces (see [BM18, Theorem 1, p. 1356] for further details)
kzτ−zτkW1,p ≤Ckzτ−zτkθ
Lpkzτ−zτk1−θ
W1+δ,p
36
with θ=δ/(1+δ). Taking the square of this inequality, integrating over the time interval
(0, T ) and using H¨older inequality leads to
ZT
0kzτ−zτk2
W1,p dt≤CZT
0kzτ−zτk2θ
Lpkzτ−zτk2(1−θ)
W1+δ,p dt
≤C"ZT
0kzτ−zτk2θq
Lpdt#1/q "ZT
0kzτ−zτk2(1−θ)q′
W1+δ,p dt#1/q′
where q= 1/θ = (1 + δ)/δ and q′=q/(1 −q) = 1/(1 −θ) (so that 2θq′= 2(1 −θ)q= 2).
Hence, we have
kzτ−zτkL2(W1,p)≤Ckzτ−zτk2/q
L2(Lp)kzτ−zτk2/q′
L2(W1+δ,p)≤Ckzτ−zτk2/q
L2(Lp)→0.
This strong convergence, combined with the boundedness of zτ−zτin L∞(0, T ;W1,p)
(that we have from (3.33)), gives us kzτ−zτkLs(W1,p)→0 for every s∈[0,+∞). Since we
already know (3.74), we have (3.76). Because of (3.35), it exists a w∈L2(0, T ;L2) such
that, along a non-relabeled subsequence, −∆pzτ⇀ w in L2(0, T ;L2). Then, recalling
that zτ→zstrongly in L2(0, T ;L2) and that the operator −∆p:L2→L2is maximal
monotone so it is strong-weak closed (see [Br´e73, Proposition 2.5, p. 27]), we may
identify w=−∆pz, which proves (3.77). Finally, from (3.36), we deduce that it exits a
ξ∈L2(0, T ;L2) such that βτ(zτ)⇀ ξ in L2(0, T ;L2). Since βis maximal monotone, βτis
its Yosida approximation and zτ→zstrongly in L2(0, T ;L2), using [Bar76, Proposition
1.1, p. 42], we deduce that ξ∈β(z) so (3.78) stands.
3.4 Passage to the limit in the discrete system
Now we have all the instruments necessary to prove our main result, Theorem 2.13. We
want to exploit the compactness result Lemma 3.9 proving that the limit we found is a
weak solution to our problem in the sense of Definition 2.10.
Cahn–Hilliard equation. In (3.20a), we can easily pass to the weak limit in the terms
on the left-hand side and in the second term on the right-hand side using convergence
(3.57) and (3.63). Given a ζ∈L2(0, T ;H1), we want to prove that
ZT
0ZΩλpστ
1 + |W,ε(ϕτ, ε(uτ), z τ)|−λa+fτg(ϕτ, zτ)ζdtdx
→ZT
0ZΩλpσ
1 + |W,ε(ϕ, ε(u), z)|−λa+fg(ϕ, z)ζdtdx.
First, we note that ϕτ(resp. zτ,ε(uτ)) converges to ϕ(resp. z,ε(u)) a.e. in Q
because of (3.60) (resp. (3.76), (3.70)). Since gand W,ε are continuous, g(ϕτ, zτ)→
g(ϕ, z) and W,ε (ϕτ, ε(uτ), zτ)→W,ε(ϕ, ε(u), z) a.e. in Q. Moreover, gis bounded and
1+|W,ε| ≥ 1. Finally, we recall that στ⇀ σ weakly in L2(0, T ;L2) (thanks to (3.66)) and
37
fτ→fstrongly in L2(0, T ;L2), so the above convergence holds. In (3.20b), exploiting
convergences (3.63), (3.59), (3.61) and (3.62) we can immediately pass to the weak limit
in all the terms except in W,ϕ(ϕτ, ε(uτ), z τ). But, for every ρ∈L2(0, T ;L2), we have
−ZT
0ZΩ
h(zτ)(ε(uτ)− Rϕτ):CRρdxdt→ −ZT
0ZΩ
h(z)(ε(u)− Rϕ):CRρdxdt,
because ε(uτ)− Rϕτ⇀ ε(u)− Rϕweakly in L2(0, T ;L2) (from (3.69) and (3.59)) and
CRh(zτ)ρ→ CRh(z)ρstrongly in L2(0, T ;L2). This last convergence holds true since C
is bounded, his continuous and bounded, zτ→za.e. in Qfrom (3.76), so we can apply
Dominated Convergence Theorem.
Nutrient equation. Rewriting explicitly (3.20c), it holds
ZT
0h∂tστ, ζiH1dt+ZT
0ZΩ∇στ· ∇ζdxdt+αZT
0ZΓ
(στ−σΓ,τ )ζdxdt
=ZT
0ZΩh−λcστg(ϕτ, zτ) + Λs(zτ)(σs,τ −στ)iζdxdt
for every ζ∈L2(0, T ;H1). As we have already pointed out, g(ϕτ, zτ)ζ→g(ϕ, z)ζ
strongly in L2(0, T ;L2) and in the same way one can prove that Λs(zτ)ζ→Λs(z)ζ
strongly in L2(0, T ;L2). So,
ZT
0ZΩh−λcστg(ϕτ, zτ) + Λs(zτ)(σs,τ −στ)iζdxdt
→ZT
0ZΩ−λcσg(ϕ, z ) + Λs(z)(σs−σ)ζdxdt
because we also know that στ⇀ σ weakly in L2(0, T ;L2) thanks to (3.66). Regarding
the term with the boundary integral, we recall that the trace operator H1→L2
Γis linear
and continuous so, the weak convergence στ−σΓ,τ ⇀ σ −σΓin L2(0, T ;H1) (that we
have from (3.66) and by construction of σΓ,τ ) leads to the weak convergences of the
traces in L2(0, T ;L2
Γ). All the other terms converge using (3.64) and (3.66). Finally,
0≤σ≤Mbecause στsatisfies this property and, thanks to (3.65), we have pointwise
convergence a.e. in Q.
Displacement equation. Equation (3.20d) can be rewritten as
∂tvτ−h′(zτ)Cε(uτ)− Rϕτ∇zτ−h(zτ) div Cε(uτ)− CRϕτ
−a′(zτ)Vε(vτ)∇zτ−a(zτ) div Vε(vτ)=0,
and it is satisfied in L2(0, T ;L2). Thanks to (3.71), we can pass to the weak limit in the
first term. For the other more complicated addends, we proceed explicitly and consider
a function ρ∈L2(0, T ;L2). Regarding the second term, we want to prove that
ZT
0ZΩ
h′(zτ)Cε(uτ)− Rϕτ∇zτ·ρdxdt→ZT
0ZΩ
h′(z)Cε(u)− Rϕ∇z·ρdxdt.
38
As we have already exploited, zτ→za.e. in Qand h′is continuous, so h′(zτ)→h′(z)
a.e. in Q. Moreover, from (3.33) we know that kzτkL∞(Q)≤Cso, since h′is continuous,
kh′(zτ)kL∞(Q)≤C. From (3.76), choosing s=p, we get that ∇zτ→ ∇zin Lp(0, T ;Lp).
Hence, h′(zτ)∇zτ·ρ→h′(z)∇z·ρstrongly in Lq(0, T ;Lq) with q=2p
p+2 . Let q′be the
H¨older conjugate of q, then it is easy to verify that q′∈[2,6) if d= 3 and q′∈[2,+∞) if
d= 2. From the boundedness of C, (3.59) and (3.69), we have that Cε(uτ)− Rϕτ⇀
Cε(u)−Rϕweakly in Lq′(0, T ;Lq′), so the desired convergence follows. To prove that
ZT
0ZΩ
h(zτ) div Cε(uτ)− CRϕτ·ρdxdt→ZT
0ZΩ
h(z) div Cε(u)− CRϕ·ρdxdt,
we observe that his continuous and bounded and zτ→za.e. so, thanks to Dominated
Convergence Theorem, h(zτ)ρ→h(z)ρin L2(0, T ;L2). Moreover, by (3.69), (3.59), and
since Cis bounded and Lipschitz, div Cε(uτ)− CRϕτ⇀div Cε(u)− CRϕweakly in
L2(0, T ;L2). Now we take into consideration the fourth term and we are going to show
that ZT
0ZΩ
a′(zτ)Vε(vτ)∇zτ·ρdxdt→ZT
0ZΩ
a′(z)Vε(v)∇z·ρdxdt.
Since a′is continuous and zτ→za.e. by (3.76), a′(zτ)→a′(z) a.e. in Q. Exploiting a′
boundedness (that follows from the fact that ais Lipschitz), by Dominated Convergence
Theorem a′(zτ)ρ→a′(z)ρstrongly in L2(0, T ;L2). Moreover, by (3.72) and (3.59),
ε(vτ)∗
⇀ ε(∂tu) weakly-∗in L∞(0, T ;L2)∩L2(0, T ;H1)֒→L2p/d(0, T ;L2p/(p−2)), where
the embedding holds true because of Gagliardo–Nirenberg’s inequality. More precisely,
we apply Theorem 2.1 with
r∈2,2d
d−2, q = 2, s = 1, α =d1
r−d−2
2d.
Finally, from (3.76) with s= 2p/(p−d), we get ∇zτ→ ∇zstrongly in L2p/(p−d)(0, T ;Lp).
So, we have concluded, because
1
2+1
2p/d +1
2p/(p−d)= 1,1
2+1
2p/(p−2) +1
p= 1.
Lastly, we aim to show that
ZT
0ZΩ
a(zτ) div Vε(vτ)·ρdxdt→ZT
0ZΩ
a(z) div Vε(v)·ρdxdt.
Continuity and boundedness of a, convergences a.e. of zτfrom (3.76) and Dominated
Convergence Theorem lead to a(zτ)ρ→a(z)ρstrongly in L2(0, T ;L2). From (3.72) we
have that vτ⇀ ∂tuweakly in L2(0, T ;H2) and from Hypothesis (H4)Vis bounded and
Lipschitz. Thus, the last term of the displacement equation passes to the limit.
39
Damage equation. We discuss only the less immediate term. Consider a test function
ρ∈L2(0, T ;L2). We will prove that
ZT
0ZΩ
`
h′(zτ) + a
h′(zτ)
2(ε(uτ)− Rϕτ):C(ε(uτ)− Rϕτ)ρdxdt
→ZT
0ZΩ
h′(z)
2(ε(u)− Rϕ):C(ε(u)− Rϕ)ρdxdt.
Since zτ,zτ→za.e. in Ω, `
h′,a
h′are continuous, and `
h′+a
h′=h′, we have
`
h′(zτ)+a
h′(zτ)
2→
h′(z)
2a.e. in Ω. Moreover, kzτkL∞(Q),kzτkL∞(Q)are uniformly bounded by (3.33). It
follows that k`
h′(zτ)kL∞(Q),k`
h′(zτ)kL∞(Q)≤C. Using the Dominated Convergence
Theorem, we deduce that
`
h′(zτ)+a
h′(zτ)
2ρ→h′(z)
2ρstrongly in L2(0, T ;L2). From (3.70),
choosing X=W1,4, and from (3.60), choosing r= 4, we get that ε(uτ)− Rϕτ→
ε(u)−Rϕstrongly in L4(0, T ;L4). Since Cis bounded, we have the desired convergence.
Acknowledgments
The author wishes to express her gratitude to Professor Elisabetta Rocca and Professor
Pierluigi Colli for introducing her to this research area and for the numerous insight-
ful comments and suggestions, without which this work would not have been possible.
The author is a member of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la
Probabilit`a e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).
This research activity has been supported by the MIUR-PRIN Grant 2020F3NCPX
“Mathematics for industry 4.0 (Math4I4)”.
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