Coupled bulk surface free boundary problems arising from a mathematical model of receptor ligand dynamics

MATHEMATICAL MODEL OF RECEPTOR-LIGAND DYNAMICS

CHARLES M. ELLIOTT, THOMAS RANNER, AND CHANDRASEKHAR VENKATARAMAN

ABSTRACT. We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface resident receptors and ligands present in the extra-cellular medium. We prove the existence and uniqueness of solutions. We also consider a number of biologically relevant asymp-totic limits of the model. We prove convergence to limiting problems which take the form of free boundary problems posed on the cell surface. We also report on numerical simulations illustrating convergence to one of the limiting problems as well as the spatio-temporal distributions of the receptors and ligands in a realistic geometry.

1. INTRODUCTION

We start by outlining the mathematical model for receptor-ligand dynamics whose analysis and asymp-totic limits will be the main focus of this work. LetΓbe a smooth, compact closedn-dimensional

hy-persurface contained in the interior of a simply connected domainD ⊂ Rn+1,n = 1,2. The surface

Γseparates the domainDinto an interior domainI and an exterior domainΩ. We will denote by∂0Ω

the outer boundary ofΩ, i.e. the boundary∂D. The vectorsνandνΩdenote the outward pointing unit

normals toΩonΓand∂0Ωrespectively. Fig. 1 shows a cartoon sketch of the setup. We assume that the outer boundary∂0Ωis Lipschitz. We consider the following problem: Findu: ¯Ω_×[0, T) _→ R+ and w: Γ_×[0, T)_→R+such that

δΩ∂tu−∆u= 0 inΩ×(0, T)

(1.1a)

∇u·ν=−_δ1

k

uw onΓ×(0, T)

(1.1b)

u=uDor∇u·νΩ= 0 on∂0Ω×(0, T)

(1.1c)

∂tw−δΓ∆Γw=∇u·ν onΓ×(0, T)

(1.1d)

u(·,0) =u0(·) inΩ

(1.1e)

w(·,0) =w0(·) onΓ,

(1.1f)

whereδΩ, δΓ, δk >0are given model parameters and the initial data are bounded, non-negative func-tions, i.e.,u0

∈L∞_(Ω),w0

∈L∞_(Γ)andu0_{, w}0

≥0. In the above∆Γ denotes the Laplace-Beltrami

operator on the surfaceΓand∆the usual Cartesian Laplacian inRn+1.

We will use either Dirichlet or Neumann boundary conditions on ∂0Ω. For the Dirichlet case, we

assume that the Dirichlet boundary datauDis a positive scalar constant. Our analysis remains valid if we consider bounded positive functions for the Dirichlet boundary data, we restrict the discussion to positive scalar boundary data for the sake of simplicity. The restriction to non-negative solutions is made since we are interested in biological problems whereuandwrepresent chemical concentrations and hence are

non-negative.

Problem (1.1) may be regarded as a basic model for receptor-ligand dynamics in cell biology, model-ling the dynamics of mobile cell surface receptors reacting with a mobile bulk ligand, which is a reduction of the model (2.1) presented in§2. Receptor-ligand interactions and the associated cascades of activation of signalling molecules, so called signalling cascades, are the primary mechanism by which cells sense and respond to their environment. Such processes therefore constitute a fundamental part of many basic

FIGURE1. A sketch of the cell membraneΓand the extra-cellular mediumΩ.

phenomena in cell biology such as proliferation, motility, the maintenance of structure or form, adhesion, cellular signalling, etc. [Bongrand, 1999; Hynes, 1992; Locksley et al., 2001]. Due to the complexity of the biochemistry involved in signalling networks, an integrated approach combining theoretical and com-putational mathematical studies with experimental and modelling efforts appears necessary. Motivated by this need, in this work we focus on understanding a mathematical reduction of theoretical models for receptor-ligand dynamics in cell biology consisting of a coupled system of bulk-surface partial differential equations (PDEs).

A number of recent theoretical and computational studies of receptor-ligand interactions, [e.g., Garc´ıa-Pe˜narrubia et al., 2013; Marciniak-Czochra and Ptashnyk, 2008], employ models which are similar in structure to those considered in this work. Models with similar features arise in the modelling of signalling networks coupling the dynamics of ligands within the cell (e.g., G-proteins) with those on the cell surface [Bao et al., 2014; Jilkine et al., 2007; Levine and Rappel, 2005; Madzvamuse et al., 2015; Morgan and Sharma, 2015; Mori et al., 2008; R¨atz and R¨oger, 2012, 2014]. The ability of cells to create their own chemotactic gradients, i.e., to influence the bulk ligand field, has been conjectured to play a crucial role in collective directed migration for example during neural crest formation [McLennan et al., 2012, 2015a,b] and hence understanding such models is of much biological importance.

Through proving well-posedness results, this work gives a mathematically sound foundation for the use and simulation of coupled bulk-surface models for receptor-ligand dynamics. Moreover, we justify the consideration of various small parameter asymptotic limits of such models, through non-dimensionalisation using experimentally measured parameter values. We provide a rigorous derivation of the limiting prob-lems and discuss their well-posedness. We also discuss the numerical solution of the original and limiting problems illustrating the asymptotic convergence together with robust and efficient methods for their ap-proximation. This work suggests that models for receptor-ligand dynamics featuring fast reaction kinetics can be derived using classical elements of free boundary methodology as components of the modelling.

et al., 2004; Dancer et al., 1999]. Further details on the cell-biological motivation for studying (1.1), together with the limitsδΩ, δΓ, δk→0, is given in§2.

The main focus of this work is to show the system of partial differential equations (1.1) is well posed and so is meaningful from the mathematical perspective and, furthermore, to obtain reduced models as limits of this system as we send the parameters δΩ, δΓ andδk to zero. Specifically, we establish existence and uniqueness of a solution to (1.1) and show that in the limitsδk → 0,δΩ, δΓ > 0 fixed,

δΓ = δk → 0, δΩ > 0 fixed, δΩ = δΓ = δk → 0, this solution to (1.1) converges to a solution of suitably defined limit problems. Furthermore, in the latter two cases, δΓ = δk → 0 andδΩ =

δΓ = δk → 0, the uniqueness of the solution to the limit problems, respectively constrained parabolic and elliptic problems with dynamic boundary conditions, is also shown. We then show that the limit problems with dynamic boundary conditions may be reformulated as variational inequalities and briefly explore some connections with classical free boundary problems. These reduced models in the form of free boundary problems may be considered as models in their own right and offer simplifications with respect to numerical computation.

That the fast reaction limit (δk → 0) leads to interesting free boundary problems is because of the complementarity nature of the resulting limit

u_≥0, w_≥0, uw= 0 onΓ.

Such limits have been considered for coupled systems of parabolic equations (posed in the same domain) in a number of previous works [e.g., Bothe, 2001; Bothe and Pierre, 2012; Evans, 1980] with the limiting problem corresponding to a Stefan problem [Hilhorst et al., 1996, 2001, 2003]. Here in this paper the main complication in the analysis is that the species reside in different domains and the coupling is on the boundary of the bulk domain which results in added technical complications in passing to the limit.

For the limit problemsδΓ =δk = 0andδΓ =δk =δΩ = 0with dynamic boundary conditions, we obtain Stefan and Hele-Shaw type problems on the hypersurfaceΓwith a differential operator, which may

be interpreted as a non-local fractional differential operator, obtained by using the Dirichlet to Neumann map for the bulk parabolic and elliptic operators. This leads to an interesting variational inequality refor-mulation in the case of the limit bulk elliptic equation consisting of a boundary obstacle problem that is satisfied by the integral in time of the solution. The approach follows that employed for the reformulation of the one-phase Stefan problem and the Hele-Shaw problem for which the transformed variable (integral in time of the solution) satisfies a parabolic [Duvaut, 1973] or elliptic [Elliott, 1980; Elliott and Janovsk`y, 1981] variational inequality respectively.

boundary conditions have been studied in a number of applications. Aitchison et al. [1984] propose a simplified model for an electropaint process that consists of an elliptic equation with nonlinear dynamic boundary conditions involving the normal derivative. The authors formally derive the steady state station-ary problem which consists of a Signorini problem similar to the elliptic variational inequality we derive in_§9. This problem is studied by Caffarelli and Friedman [1985] where the authors prove that the steady state solution (t _{→ ∞}) of an implicit time discretisation solves the Signorini problem proposed as the

formal limit by Aitchison et al. [1984]. A similar problem, which models percolation in gently sloping beaches, that consists of an elliptic equation variational inequality with dynamic boundary conditions involving the normal derivative is proposed and analysed by Aitchison et al. [1983]; Colli and Kenmo-chi [1987]; Elliott and Friedman [1985]. Perthame et al. [2014] derive Hele-Shaw type free boundary problems as limits of models for tumour growth. Finally we mention the work of Nochetto et al. [2015] who consider the numerical approximation of obstacle problems, in particular, they prove optimal con-vergence rates for the thin obstacle (Signorini) problem and prove quasi-optimal concon-vergence rates for the approximation of the obstacle problem for the fractional Laplacian.

Our main results are stated in Theorems 4.2, 5.3, 6.3 and 7.3.

• In Theorem 4.2 we establish the existence of a unique, bounded solution to (1.1).

• In Theorem 5.3 we present a rigorous derivation that in the limitδk →0,δΩ, δΓ >0fixed, the solution to (1.1) converges to a solution of a system of constrained coupled bulk-surface parabolic equations (c.f., (5.1)).

• In Theorem 6.3 we present a rigorous derivation that in the limitδΓ = δk → 0, with δΩ >0 fixed, the solution to (1.1) converges to the unique solution of constrained parabolic problem with dynamic boundary condition (c.f., (6.1)).

• In Theorem 7.3 we present a rigorous derivation that in the limitδΩ=δΓ=δk→0, the solution to (1.1) converges to the unique solution of constrained elliptic problem with dynamic boundary condition (c.f., (7.1)).

We conclude the paper by providing some numerical experiments employing a coupled bulk-surface finite element method where we support numerically the theoretical convergence results to a limiting problem and investigate the resulting free boundary problem on a surface.

2. BIOLOGICAL MOTIVATION

We now present a model for receptor-ligand dynamics and justify, through non-dimensionalisation of the model using parameter values previously measured in experimental studies, the simplifications and limiting problems considered in this work.

We start with the following model, that corresponds to one of the models presented by Garc´ıa-Pe˜narrubia et al. [2013] if one neglects the terms involving internalisation of receptors and complexes. The reaction under consideration is between mobile receptors that reside on the cell surface with ligands present in the extra-cellular medium (the bulk region surrounding the cell). We assume a single species of mobile surface (cell membrane) resident receptor whose concentration (surface density) is denoted bycrand a single species of bulk resident diffusible ligand whose concentration (bulk concentration) is denoted by

with outer boundary∂0Ω(c.f., Fig. 1), we have in mind models of the following form,

∂tcL−DL∆cL= 0 inΩ×(0, T)

(2.1a)

DL∇cL·ν=−koncLcr+koffcrl onΓ×(0, T) (2.1b)

cL =cDorDL∇cL·νΩ= 0 on∂0Ω×(0, T)

(2.1c)

∂tcr−Dr∆Γcr=−koncLcr+koffcrl onΓ×(0, T) (2.1d)

∂tcrl−Drl∆Γcrl=koncLcr−koffcrl onΓ×(0, T). (2.1e)

The model is closed by suitable (bounded, non-negative) initial conditions. For the outer boundary condi-tion we take either a Dirichlet or a Neumann boundary condicondi-tion. The Dirichlet boundary condicondi-tion, with

cD>0a positive constant, arises under the modelling assumption that the background concentration of ligands sufficiently far away from the cell is uniform. Alternatively, the Neumann boundary condition arises from assuming zero flux across∂0Ω.

2.1. Non-dimensionalisation and limit problems. We are interested in different limit problems arising from model (2.1) for ligand-receptor binding. To simplify notation, we write the unknowns as u =

cL, w=crandχ=crland the parametersDΩ=DL,DΓ=Dr=Drl,k=kon,k−1=koff. The first problem we consider is to findu: Ω_×[0, T)→Randw, χ: Γ×[0, T)→Rsuch that

∂tu−DΩ∆u= 0 inΩ×(0, T)

(2.2a)

DΩ_∇u_·ν=₋kuw+k−1χ onΓ×(0, T)

(2.2b)

u=uDorDΩ∇u·νΩ= 0 on∂0Ω×(0, T)

(2.2c)

∂tw−DΓ∆Γw=DΩ∇u·ν onΓ×(0, T)

(2.2d)

∂tχ−DΓ∆Γχ=−DΩ∇u·ν onΓ×(0, T)

(2.2e)

u(_·,0) =u0(_·) inΩ

(2.2f)

w(_·,0) =w0(_·) onΓ

(2.2g)

χ(_·,0) =χ0(_·) onΓ.

(2.2h)

In order to determine the sizes of each coefficient, we take the following rescaling. We set

˜

x=x/L, ˜t=t/S, u˜=u/U, w˜=w/W, χ˜=χ/X,

whereL is a length scale, S is a time scale, U, W andX are typical concentrations for u, w andχ

respectively.

Applying the chain rule, this leads to a non-dimensional form of (2.2):

δΩ∂˜tu˜−∆˜u= 0 inΩ˜×(0,T˜)

(2.3a)

∇u˜_·ν=₋1

δk

˜

uw˜+δχχ˜ onΓ˜×(0,T˜)

(2.3b)

˜

u= ˜uDor∇u˜·νΩ˜ = 0 on∂0Ω˜×(0,T˜)

(2.3c)

∂˜tw˜−δΓ∆Γ˜w˜=µ∇u˜·ν onΓ˜×(0,T˜),

(2.3d)

∂˜tχ˜−δΓ∆Γ˜χ˜=−µ

0

∇u˜_·ν onΓ˜_×(0,T˜)

(2.3e)

˜

u(_·,0) = ˜u0(_·) :=u0(_·)/U inΩ

(2.3f)

˜

w(_·,0) = ˜w0(_·) :=w0(_·)/W onΓ

(2.3g)

˜

χ(_·,0) = ˜χ0(_·) :=χ0(_·)/X onΓ.

Parameter Value Source

L 7.5_·10−6_{m Garc´ıa-Pe˜narrubia et al. [2013]} U 1.0_·10−3_{mol m}−3 _{Garc´ıa-Pe˜narrubia et al. [2013]} W 2.3_·10−8_{mol m}−2 _{Garc´ıa-Pe˜narrubia et al. [2013]} X 2.3·10−8_{mol m}−2 _{limited by total receptor concentration} DΩ 1.0·10−11_m2_s−1 _{Linderman and Lauffenburger [1986]} DΓ 1.0·10−15_m2_s−1 _{Linderman and Lauffenburger [1986]} kon 1.0·103m3mol−1s−1 Garc´ıa-Pe˜narrubia et al. [2013]

koff 5.0·10−3s−1 Garc´ıa-Pe˜narrubia et al. [2013] TABLE1. Parameters used for rescaling equations. The values forU andW are

ex-treme values taken from within a physical range from Garc´ıa-Pe˜narrubia et al. [2013].

Here we have six non-dimensional coefficients:

δΩ= L

2

DΩS, δk =

DΩ

kLW, δχ=

k−1LX

DLU

, δΓ =DΓS

L2 ,

µ= DΩSU

LW , µ

0₌ DΩSU

LX .

Taking values from Table 1, we infer that

δΩ= (5.6 s)_·S−1, δk = 5.7·10−2, δχ= 8.7·10−2,

δΓ = (1.8_·10−4s−1)_·S, µ= (5.7_·10−2s−1)_·S, µ0 = (5.7_·10−2s−1)_·S.

First, we note thatδχ 1. Considering the limitδχ →0by dropping the termsδχχdecouples the equations foru,˜ w˜from the equation forχ˜. This results in the problem, which we have written in terms

of the original variables:

∂tu−δ_Ω−1∆u= 0 inΩ

(2.4a)

∇u·ν=−_δ1

k

uw onΓ×(0, T)

(2.4b)

u=uDor∇u·νΩ= 0 on∂0Ω

(2.4c)

∂tw−δΓ∆Γw=µ∇u·ν onΓ

(2.4d)

u(_·,0) =u0(_·) inΩ

(2.4e)

w(_·,0) =w0(_·) onΓ,

(2.4f)

This is the first problem we consider in Section 4. Similar methods to those shown in the remaining sections can be used to show well posedness of the system (2.3) and rigorously take the limitδχ→0for

δk, δΩ, δΓ, µ >0fixed. The existence and uniqueness theory of (2.3) and the limit to obtain (2.4) in the more general case of time dependent domains are considered by Alphonse et al. [2016].

We see thatδk 1. Again using the original variables, we consider the limit problem: Findu: Ω×

[0, T)_→Randw: Γ×[0, T)→Rsuch that

∂tu−δ_Ω−1∆u= 0 inΩ

(2.5a)

uw= 0 onΓ

(2.5b)

u=uDor∇u·νΩ= 0 on∂0Ω

(2.5c)

∂tw−δΓ∆Γw=µ∇u·ν onΓ

(2.5d)

u(_·,0) =u0(_·) inΩ

(2.5e)

w(_·,0) =w0(_·) onΓ,

whereδΩ, δΓandµare positive parameters. We consider this problem as a large ligand–receptor binding rate limit of (2.2). We consider the well posedness of the problem and the justification of the limit in Section 5.

We can consider different problems by choosing different time scalesS. We can achieve two different

problems by resolving the timescale of the volumetric diffusion (δΩ_≈1) or the timescale of the surface

adsorption flux (µ_≈1).

ForS =L2_{/DΩ= 5.6 s, we have}

δΩ= 1, δΓ= 1.0_·10−31, µ= 3.2_·10−1_≈1.

This leads to a parabolic limit problem with dynamic boundary condition: Findu: Ω_×[0, T)_→Rand w: Γ_×[0, T)_→Rsuch that

∂tu−δ−_Ω1∆u= 0 inΩ

(2.6a)

uw= 0 onΓ

(2.6b)

u=uD on∂0Ω

(2.6c)

∂tw=∇u·ν onΓ

(2.6d)

u(_·,0) =u0(_·) inΩ

(2.6e)

w(_·,0) =w0(_·) onΓ.

(2.6f)

In this case, we have resolved the timescale of the diffusion of ligand, but the effect of the diffusion of surface bound receptor is lost. We consider the well posedness of this problem and the justification of the limit in Section 6.

Alternatively, takingS= 102_{s, we have} δΩ= 5.7_·10−2

1, δΓ= 1.8_·10−2

1, µ= 5.7_≈1.

This leads to an elliptic problem with dynamic boundary condition: Find u: Ω_×[0, T) _→ R and w: Γ_×[0, T)_→Rsuch that

−∆u= 0 inΩ

(2.7a)

uw= 0 onΓ

(2.7b)

u=uD on∂0Ω

(2.7c)

∂tw=∇u·ν onΓ

(2.7d)

w(_·,0) =w0(_·) onΓ.

(2.7e)

In this regime, we have chosen a time scale so that the diffusion of ligand has no memory of its previous value, except via the boundary condition. This means this problem no longer requires an initial condition foruto be a closed system. We do not consider the exterior Neumann boundary condition in this case,

since we arrive at a trivial problem where the solution isu= 0andw=w0. The well posedness of this

problem and a rigorous justification of limit is given in Section 7. We also show in Section 9 that we can reformulate problems (2.6) and (2.7) by integrating forwards in time to derive variational inequalities. 2.2.Remark. In the large ligand-receptor binding rate limit, the nonlinear constraint(2.5b)(uw = 0)

implies that the domain Γ is separated into two regions, for positive times, whereu = 0and where

u >0. In the regionu >0, we have a Neumann boundary condition∇u·ν= 0. This can be interpreted

3. PRELIMINARIES

In this section, we define some of our notation and collect some technical results that will be used in the subsequent sections. We also prove some compact embedding results in Lemmas 3.7, 3.8 and 3.9 that are used to deduce strong convergence from weak convergence in suitable spaces.

Given a Hilbert spaceY we denote the dual space of a linear functionals onY by(Y)0. As we consider

functions defined on surfaces, along with the surface function spacesL2_(Γ)andH1_{(Γ), we will also use}

the spaceH1/2_{(Γ)and its dualH}1/2_(Γ)0_{. For a Hilbert spaceY, we consistently use the notation} h·,_·iY to denote the duality pairing between the spaceY and its dual(Y)0.

3.1.Definition. The spaceH1/2_{(Γ)is defined by}

(3.1) H1/2_{(Γ) :=}n_ξ

∈L2(Γ) :kξkH1/2_(Γ)<+∞

o ,

where

(3.2) kξ_kH1/2_(Γ):=

Z

Γ

ξ2 dσ+

Z

Γ Z

Γ

|ξ(x)₋ξ(y)_|2

|x₋y_|n dσ(x) dσ(y)

!12

.

The space can be characterised via the following result.

3.2.Proposition(Trace Theorem). The trace operator fromH1(Ω)toH1/2_{(Γ)is bounded and}

surject-ive.

Proof. The result can be found in [Grisvard, 2011, Thm 1.5.1.3].

We recall the following interpolated trace inequality.

3.3.Proposition(Interpolated trace theorem). For allφ∈H1_{(Ω)and for anyδ >0}

(3.3) _kφ_k2_L2_(Γ)≤δk∇φk

2

L2_(Ω)+cδkφk2_L2_(Ω).

Proof. See e.g. [Grisvard, 2011, Thm 1.5.1.10].

Note that forξ _∈ L2_(Γ)andρ

∈ H1/2_{(Γ)the following duality pairing is equal toL}2_(Γ)

inner-product:

(3.4) hξ, ρ_iH1/2_(Γ)=

Z

Γ ξρ dσ

3.4. Compact embeddings. Since we are dealing with nonlinear problems, we will need to use some compact embeddings of Bochner spaces.

We recall that if_{fk}is a sequence of bounded functions inLp(0, T;B), withBa Banach space, for

1_≤p <_∞, then there exists a subsequence_{fkj} ⊂ {fk}andf ∈L

p_{(0, T;B)such that}

(3.5) fkj * f inL

p_{(0, T;B).}

Here we are interested to show under what conditions we may assert the existence of a strongly convergent subsequence. The basic results we require are summarised by Simon [1986].

3.5.Lemma(Aubin-Lions-Simons compactness theory [Simon, 1986]). Let_{fk}be a bounded sequence

of functions inLp_{(0, T;B)whereBis a Banach space and1}

≤p≤ ∞. If

(1) the sequence of functions{fk} is bounded inLp(0, T;X)whereX is compactly embedded in

B; (2) either

(b) for eachk, the time translates of{fk}are such that

(3.6)

Z T−τ

0

fk(t+τ)−fk(t)

p

B dt→0 as τ→0.

Then there exists a subsequence{fkj} ⊂ {fk}andf ∈L

p_{(0, T;X)such that}

(3.7) fkj * f inL

p_{(0, T;X)}

fkj →f inL

p

(0, T;B).

3.6.Remark. Using the criterion 2(a), we see that if_{ηk} ⊂L2(0, T;L2(Ω))with a constantC > 0

such that

kηkk_L2_(0,T;H1_(Ω))+k∂tηkk_L2_0,T;(H1_(Ω))0≤C for allk,

then, there exists a subsequence, for which will use the same subscript{ηk}, andη ∈L2(0, T;H1(Ω))

such that

(3.8) ηk* η inL

2_{(0, T;H}1_(Ω)) ηk→η inL2(0, T;L2(Ω)).

This follows from the compact embedding ofH1_(Ω)inL2_(Ω).

However, we wish to recover strong convergence of a subsequence with less control over the time derivatives. The generality of criterion 2(b) allows a more general weak in time notion of solution to be used.

We will apply this result for sequences to derive strongly convergent subsequences inL2

0, T;H1/2_(Γ)0

.

3.7.Lemma. Let_{ξk} be a bounded sequence inH1/2(Γ). Then there exists a subsequence{ξkj} ⊂ {ξk}andξ∈H1/2(Γ)such that

(3.9) ξkj * ξ inH

1/2_(Γ) ξkj →ξ inL

2_(Γ).

Proof. For anyρ∈H1/2_{(Γ), we define an extension toΩ, writtenEρ∈H}1_{(Ω), as the unique solution}

of:

−∆(Eρ) = 0 inΩ

Eρ= 0 on∂0Ω

Eρ=ρ onΓ.

We note that for a constant independent ofρ, we have

kEρ_kH1_(Ω)≤ckρk_H1/2_(Γ).

This implies we have a sequence{Eξk}which is uniformly bounded inH1(Ω): There existsC0>0 such that

kEξkk_H1_(Ω)≤C0.

From the compact embedding ofH1_(Ω)intoL2_{(Ω), we know that there exists a subsequence{ξ}

kj} ⊂ {ξk}, andη∈H1(Ω)such that

Eξkj →η inL

2_(Ω).

Denote byξ=η_|Γ. Fixε >0and chooseδ≤ε/(4C0). From the strong convergence of{Eξkj}, we

know there existsKsuch that forj_≥K,

Eξkj −η _L2_(Ω)≤

ε

2cδ

wherecδ is from Proposition 3.3. It follows that forj ≥ K, we can infer by applying the interpolated trace inequality (Proposition 3.3), withδas above, that

ξkj −ξ

_L2_(Γ)≤δ

Eξkj −η

_H1_(Ω)+cδ

Eξkj −Eξ _L2_(Ω)

≤2δC0+ε 2 ≤ε.

Thus, we have shown the strong convergence ofξktoξinL2(Γ). 3.8.Lemma. Let_{ξk}be a bounded sequence inL2(Γ). Then there exists a subsequence{ξkj} ⊂ {ξk}

andξ_∈L2_{(Γ)such that}

(3.10) ξkj * ξ inL

2_(Γ) ξkj →ξ in

H1/2_(Γ)0_.

Proof. Since {ξk} is uniformly bounded inL2(Γ), we know that is has a subsequence {ξkj} which

weakly converges to someξ_∈L2_{(Γ). We suppose, for contradiction, that there exists no subsequence of} {ξkj}that strongly converges toξin

H1/2_(Γ)0_{. This implies that there existsδ >0such that}

ξkj −ξ

_(H1/2_(Γ))0 ≥δ.

Using the definition ofH1/2_(Γ)0 _{as the dual space toH}1/2_{(Γ), this implies there exists a sequence} {ρj} ⊂H1/2(Γ), with

ρj

_H1/2_(Γ)= 1, such that for allj

hξkj −ξ, ρjiH1/2_(Γ)=

Z

Γ

(ξkj−ξ)ρj dσ≥ δ

2.

From Lemma 3.7, we know that a subsequence{ρjl} ⊂ {ρj}converges strongly to ρ ∈ H

1/2_(Γ)in L2_{(Γ). Hence, we can infer}

Z

Γ

(ξkj −ξ)ρ dσ≥ δ

2.

However, this contradicts the supposition thatξkj converges weakly toξinL 2_(Γ).

We conclude this section with a result which is similar in nature to the previous results.

3.9.Lemma. Let_{ηk}be a bounded sequence inL2(0, T;H1(Ω))andη∈L2(0, T;H1(Ω))such that (3.11) ηk →η inL2(0, T;L2(Ω)).

Then the trace sequence converges to the trace of the limit:

ηk|Γ →η|Γ inL2(0, T;L2(Γ)).

(3.12)

Proof. Denote byC0>0the upper bound of{ηk}andηinL2(0, T;H1(Ω)):

kηkkL2_(0,T;H1_(Ω))+kηk_L2_(0,T;H1_(Ω))≤C0.

Fixε >0and chooseδ≤ε/(2C0). Then from the convergence of{ηk}inL2(0, T;L2(Ω)), there exists

Ksuch that fork≥K,

kηk−ηkL2_(0,T;L2_(Ω))≤

ε

2cδ

,

wherecδ is from Proposition 3.3. It follows that fork ≥K, we can infer by applying the interpolated trace inequality (Proposition 3.3), withδas above, that

kηk−ηkL2_(0,T;L2_(Γ))≤δkηk−ηkL2_(0,T;H1_(Ω))+cδkηk−ηkL2_(0,T;L2_(Ω))

Thus, we have shown the strong convergence ofηktoηinL2(0, T;L2(Γ)). 4. LIGAND-RECEPTOR MODEL

In this section, we establish an existence and uniqueness theory for (1.1). As described in§2.1, (1.1) arises from (2.1) if one neglects the receptor-ligand complexes, non-dimensionalises as in (2.3) and (for simplicity) sets the surface interchange fluxµ= 1.

In order to introduce the concept of a weak solution to (1.1), forγ _∈ R, we introduce the Sobolev

space

He1γ(Ω) :={v∈H 1_(Ω)

|v=γon∂0Ω},

where the boundary values are understood in the sense of traces and we adopt the notation, of using the same symbol for a function and its trace. We now introduce our concept of a weak solution to (1.1). 4.1.Definition(Weak solution of (1.1)). For the Dirichlet boundary data case, we say that a pair(u, w)∈ L2_{(0, T;H}1

e_uD(Ω))×L2(0, T;H1(Γ))withu, w ≥0and with(∂tu, ∂tw) ∈L2

0, T; H1

e0(Ω)

0 × L2_{0, T; H}1_(Γ)0

is a weak solution of (1.1) if for all(η, ρ)_∈H1

e0(Ω)×H

1_{(Γ)and for a.e.t}

∈(0, T)

δΩ

H1

e₀(Ω)

0h∂_tu, ηi

H1

e₀(Ω)+ Z

Ω∇

u_{· ∇}η dx=₋1

δk

Z

Γ

uwη dσ

(4.1a)

(H1_(Γ))0h∂tw, ρiH1_(Γ)+δΓ

Z

Γ∇

Γw· ∇Γρ dσ=−_δ1

k

Z

Γ

uwρ dσ.

(4.1b)

In the case of Neumann boundary data, we say that a pair(u, w)_∈L2_{(0, T;H}1_(Ω))

×L2_{(0, T;H}1_(Γ))

withu, w≥0and with(∂tu, ∂tw)∈L2

0, T , H1(Ω)0

×L20, T; H1(Γ)0

is a weak solution of (1.1) if for all(η, ρ)_∈H1_(Ω)

×H1_{(Γ)and for a.e.t}

∈(0, T)

δΩ_(H1_(Ω))0h∂tu, ηiH1_(Ω)+

Z

Ω∇

u· ∇η dx=−_δ1

k

Z

Γ

uwη dσ

(4.2a)

(H1_(Γ))0h∂tw, ρiH1_(Γ)+δΓ

Z

Γ∇

Γw· ∇Γρ dσ=−_δ1

k

Z

Γ

uwρ dσ,

(4.2b)

We note that ifu∈L2_{(0, T;H}1_{(Ω))then by the trace theoremu∈L}2_{(0, T;H}1/2_{(Γ)). We now show}

the well posedness of problem (1.1) in the sense of the following Theorem.

4.2. Theorem (Existence and uniqueness of a bounded solution pair to (1.1)). Given bounded,

non-negative initial datau0andw0, there exists a unique solution pair(u, w)to the systems(4.1)and(4.2). Furthermore, we have that in the case of Dirichlet data

(4.3) 0≤u(x, t)≤max(ku0kL∞(Ω), uD) for a.e.(x, t)∈Ω×(0, T)

0≤w(x, t)≤ kw0kL∞_(Γ) for a.e.(x, t)∈Γ×(0, T),

or in the case of Neumann data

(4.4) 0≤u(x, t)≤ ku0kL∞(Ω) for a.e.(x, t)∈Ω×(0, T)

0≤w(x, t)≤ kw0kL∞_(Γ) for a.e.(x, t)∈Γ×(0, T).

We start by replacingwbyM(w)in the nonlinear coupling terms, whereM: R→R+is the cut off

function

(4.5) M(r) =

      

0 r <0

r 0_≤r_≤M

M r > M,

withM _{≥ k}w0_kL∞_(Γ). This leads us to consider the following problem. Find(u, w), in the same spaces

as Definition 4.1, that satisfy for all(η, ρ)_∈H1

e0(Ω)×H

1_{(Γ)and for a.e.t}

∈(0, T)

δΩ

H1

e₀(Ω)

0h∂_tu, ηi

H1 0(Ω)+

Z

Ω∇

u_{· ∇}η dx=₋1

δk

Z

Γ

uM(w)η dσ

(4.6)

(H1_(Γ))0h∂tw, ρiH1_(Γ)+δΓ

Z

Γ∇

Γw· ∇Γρ dσ=−_δ1

k

Z

Γ

uM(w)ρ dσ.

(4.7)

AsM(w)is bounded, existence for this problem with the cutoff nonlinearity can be shown via a Galerkin

method and standard energy arguments. We now show positivity of the solutions to (4.6), (4.7):u, w_≥0

almost everywhere in their domains and that the trace ofu_≥0onΓ. Testing (4.6) withu− = min(u,0) and using the fact thatM(w)_≥0, we have

δΩ 2 d dt Z Ω

(u−)2 dx+

Z

Ω|∇

(u−)|2 dx=−

1

δk

Z

Γ

(u−)2M(w) dσ≤0.

Sinceu0_≥0, we haveu_≥0almost everywhere inΩ_×(0, T). Moreover, by the trace inequality, applied tou−, we have that the trace ofuis non-negative. We next test (4.7) withw−= min(w,0)to get

1 2 d dt Z Γ

(w−)2 dσ+δΓ

Z

Γ|∇

Γ(w−)|2 dσ=−

1

δk

Z

Γ

uM(w)w− dσ= 0,

asM(w)w−= 0from the definition ofM()(4.5). Sincew0≥0, we see thatw≥0almost everywhere inΓ ×(0, T). We now show pointwise bounds. Let (u, w) be solutions of (4.6) and (4.7) and set

θw_{= (w}

− kw0kL∞_(Γ)). The variableθwsatisfies

(H1_(Γ))0h∂tθ

w_{, ρ}

iH1_(Γ)+δΓ

Z

Γ∇ Γθw

· ∇Γρ dσ=₋1

δk

Z

Γ

uwρ dσ.

We test withρ= (θw₎

+≥0and recall thatu, w≥0then

1 2 d dt Z Γ

(θw

+)2 dσ+δΓ Z

Γ|∇ θw

+|2 dσ=−

1

δk

Z

Γ

uwθw

+ dσ≤0.

This implies that θw

+ = 0 and hencew ≤ kw0kL∞_(Γ). The same argument for u withθu = (u−

max(uD,kukL∞_(Ω))so thatθ₊u ∈ He10, givesu ≤max(uD,kukL∞(Ω)). AsM was chosen such that

M _{≥ k}w0_kL∞_(Γ)andw ≥0, we have thatM(w) =w, hence we have constructed a solution to (4.1)

which satisfies

(4.8) 0≤u≤max(ku0k∞, uD) and0≤w≤ kw0k∞.

It remains to show that the solution is unique. To do this, we argue as follows. Let(u1, w1)and (u2, w2)be two (weak) solutions of (4.1). Definingeu _{:= u1}

−u2 andew _{:= w1}

−w2 we have that eu_{, e}w_{satisfy for all(η, ρ)}

∈H1

e0(Ω)×H

1_{(Γ)and for a.e.t∈(0, T)}

δΩ

H1

e₀(Ω)

0h∂_teu, ηi

H1

e₀(Ω)+ Z

Ω∇ eu

· ∇η dx=₋1

δk

Z

Γ

(u1w1₋u2w2)η dσ

(4.9a)

(H1_(Γ))0h∂tew, ρiH1_(Γ) dσ+

Z

Γ

δΓ∇Γew· ∇Γρ dσ=−1 δk

Z

Γ

(u1w1−u2w2)ρ dσ

Letψ : R→ Rbe a smooth convex function satisfyingψ(0) =ψ0(0) = 0. Settingη =ψ0(eu)and ρ=ψ0_(ew_{)in (4.9) and combining the equations gives}

d dt

Z

Ω

δΩψ(eu_{) dx+}

Z

Γ

ψ(ew_{) dσ}

+

Z

Ω ψ00(eu₎

|∇eu

|2 dx+

Z

Γ

δΓψ00(ew₎

|∇Γew

|2 dσ

(4.10)

=₋1

δk

Z

Γ

(u1w1₋u2w2) ψ0(eu_{) +ψ}0_(ew₎

dσ.

Hence asψis convex we have

d dt

Z

Ω

δΩψ(eu_{) dx+}

Z

Γ

ψ(ew_{) dσ}

≤ −_δ1

k

Z

Γ

(u1w1₋u2w2) ψ0(eu_{) +ψ}0_(ew₎

dσ.

(4.11)

Integration in time gives

Z

Ω

δΩψ(eu₍

·, t)) dx+

Z

Γ ψ(ew₍

·, t)) dσ_{≤ −} 1 δk

Z t

0 Z

Γ

(u1w1₋u2w2) ψ0(eu_{) +ψ}0_(ew₎

dσ dt,

(4.12)

aseu₍

·,0) = 0andew₍

·,0) = 0and we have chosenψsuch thatψ(0) = 0. Defining the function sgn(η) =

      

1 ifη >0 0 ifη= 0

−1 ifη <0,

we replaceψby a sequence of smooth functionsψksuch that

ψk(x)→ |x|, ψk0(x)→sgn(x), x∈R, pointwise and pass to the limit (k_{→ ∞}), which yields

Z

Ω

δΩ

eu(·, t) dx+

Z

Γ ew(·, t)

dσ≤ −

1 δk Z t 0 Z Γ

(u1w1₋u2w2) sgn(eu_{) + sgn(e}w₎

dσ dt.

(4.13)

Fora1, b1, a2, b2_∈R+it is easily verified that

(a1b1−a2b2)(sgn(a1−a2) + sgn(b1−b2))≥0,

hence the right hand side of (4.13) is non-positive Thus for a.e.,t_∈(0, T)

Z

Ω

δΩ|eu| dx+

Z

Γ| ew| dσ

= 0,

which completes the proof of uniqueness and hence the proof of the theorem.

In the subsequent sections we will consider the limit problems obtained on sendingδΩ, δΓandδk to zero in (1.1). To this end we derive some estimates on the solution pair(u, w)of (4.1), which we will

use in the subsequent sections to deduce the existence of convergent subsequences which converge to solutions of the limit problems. We note that the bounds hold for constants which are independent of

δk, δΓandδΩ.

4.3.Lemma(Estimates for the solution of (4.1) and (4.2)). The solution pair(u, w)to (4.1) and (4.2) satisfy the following estimates,

δΩkuk2L∞_((0,T);L2_(Ω))+ 2k∇uk

2

L2_((0,T);L2_(Ω))≤δΩ

Z

Ω

u20 dx+CD

kwk2L∞_((0,T);L2_(Γ))+ 2δΓk∇Γwk2L2_((0,T);L2_(Γ))≤

Z

Γ w02 dσ,

whereCD ∈R+depends on the Dirichlet boundary datauDandCD = 0in the case of the Neumann

boundary condition. Furthermore, we have an estimate on the nonlinearity:

1

δk k

uw_kL1_((0,T)×Γ)≤ kw0k_L1_(Γ).

(4.15)

The following estimate on time translates ofuandwalong with Lemma 3.5 will be used to deduce the necessary compactness

δΩ Z T−τ

0 Z

Ω

u(_·, t+τ)₋u(_·, t)2

dx dt+

Z T−τ

0 Z

Γ

w(_·, t+τ)₋w(_·, t)2

dσ dt_≤Cτ,

(4.16)

where the constantCis independent ofτ, δΩ, δΓandδk.

Proof. The first estimate (4.14) follows from a straightforward energy argument due to the non negativity ofuandw. Specifically, test with(u−Du, w)whereDusatisfies∆Du = 0inΩ,Du = 0onΓand

Du=uDon∂0Ωin the Dirichlet case (4.1) or simply with(u, w)in the Neumann case (4.2). For the estimate (4.15) we have using the non-negativity ofu, w

1

δk k

uw_kL1_(Γ×(0,T))=

1 δk Z T 0 Z Γ

wu dσ dt

=

Z T

0 Z

Γ−

∂tw dσ dt

=

Z

Γ−

w(_·, T) +w0₍ ·) dσ ≤

w 0

_L1_(Γ),

where we have used the non-negativity ofwin the last step.

For the estimate (4.16) we argue as follows. For a fixedτ ∈(0, T)and fort∈[0, T −τ)introducing

the notation_∂¯_{τf(t) :=f(t+τ)−f(t)we have using (4.1)}

Z

Γ

w(·, t+τ)−w(·, t)2

dσ=

Z τ

0 Z

Γ

∂tw(·, t+s) ¯∂τw(·, t) dσ ds

=

Z τ

0 Z

Γ−

δΓ∇Γw(·, t+s)· ∇Γ∂¯τw(·, t)−

1

δk

[uw](·, t+s) ¯∂τw(·, t) dσ ds. Integrating in time gives

Z T−τ

0 Z

Γ

w(_·, t+τ)₋w(_·, t)2

dσ dt

(4.17)

=

Z τ

0 Z T−τ

0 Z

Γ−

δΓ_∇Γw(_·, t+s)_{· ∇}Γ∂¯τw(·, t)−

1

δk

[uw](_·, t+s) ¯∂τw(·, t) dσ dt ds

≤ Z τ

0

2δΓ_k∇Γw_k2_L2_(Γ×(0,T))+

∂¯τw

_L∞_(Γ×(0,T))

1

δkk

uw_kL1_(Γ×(0,T)) ds,

where we have used Young’s inequality in the last step. Applying the estimates (4.8), (4.14) and (4.15) in (4.17) yields the desired estimate for the second term in (4.16). For the bound on the first term in (4.16), we note that as_∂¯_τu∈H1

e0(Ω)

δΩ Z

Ω

¯

∂τu(·, t)2 dx

=

Z τ

0 Z

Ω−∇

u(·, t+s)· ∇∂¯τu(·, t)−

1

δk

FIGURE2. Sketch of the functionβc.f., (5.2)

from which the desired bound follows from an analogous calculation to (4.17) together with the estimates

(4.8), (4.14) and (4.15).

5. FAST REACTION LIMIT PROBLEM(δk= 0)

We now show that for fixedδΩ, δΓ>0asδk →0the solution to (1.1) converges to a (weak) solution to the following constrained parabolic limit problem. For convenience we work withv = −wand set v0_=−w0_.

5.1.Problem(Problem for instantaneous reaction rate). Find¯u: Ω_×[0, T)_→R+,v¯: Γ×[0, T)→R−

such that

δΩ∂tu¯−∆¯u= 0 inΩ×(0, T)

(5.1a)

∇u¯_·ν+∂t¯v−δΓ∆Γv¯= 0 and v¯∈β(¯u) onΓ×(0, T)

(5.1b)

¯

u=uD or ∇u¯·νΩ= 0 on∂0Ω×(0, T)

(5.1c)

¯

u(·,0) =u0(·)≥0 inΩ

(5.1d)

¯

v(·,0) =v0(·)≤0 onΓ.

(5.1e)

Hereβ:R→ {0,1}Ris the set valued function (c.f., Figure 2)

(5.2) β(r) =



  

  

∅ ifr <0

[−∞,0] ifr= 0

{0_} ifr >0.

We consider (5.1) as a parabolic equation with dynamic boundary conditions interpreted as a differential inclusion.

In order to define a weak solution to (5.1) we define the Bochner spaces

Ve0(Ω) =

(

v∈L20, T;He10(Ω)

:∂tv∈L2

0, T;He10(Ω)

and

V(Γ) =

(

v_∈L20, T;H1(Γ):∂tv∈L2

0, T;H1(Γ)0

) .

We will make use of the following function space

Ve0(Ω,Γ) :=

(

v_{∈ V}e0(Ω) :v|Γ ∈ V(Γ)

)

.

We note that similar spaces have been introduced for the weak formulation of a parabolic problems with dynamic boundary conditions, [see, for example, Calatroni and Colli, 2013].

5.2.Definition(Weak solution of Problem 5.1). We say that a pair(¯u,¯v)withu∈L2_{(0, T;H}1

e_uD(Ω))∩

L∞_{(0, T;L}2_(Ω))and¯v

∈L2_{(0, T;H}1_(Γ))

∩L∞_{(0, T;L}2_(Γ))withu¯

≥0andv¯_≤0is a weak solution

of Problem 5.1 if for allη_{∈ V}e0(Ω,Γ)withη(·, T) = 0, we have

(5.3)

Z T

0 −

δΩ

H1

e₀(Ω)

0h∂_tη,u¯i

H1

euD(Ω)+ Z

Ω∇

¯

u_{· ∇}η dx_−(H1_(Γ))0h∂tη,v¯iH1_(Γ)

+

Z

Γ

δΓ_∇Γv¯· ∇Γη dσ !

dt=

Z

Ω

δΩu0η(_·,0) dx+

Z

Γ

v0η(_·,0) dσ

and

¯

v_∈β(¯u) a.e. onΓ_×(0, T).

(5.4)

We make the corresponding modifications to the function spaces for the Neumann boundary condition.

5.3.Theorem(Convergence of the solution of (1.1) to a solution of (5.1)). Asδk →0the solution pair

(u, w)to (4.1) converge (up to a subsequence) to a pair(¯u,w¯)in the following topologies

u *u¯ inL2(0, T;He1_uD(Ω)) (5.5)

(u *u¯ inL2_{(0, T;H}1_{(Ω)), in the Neumann case)} w *w¯ inL2_{(0, T;H}1_(Γ)),

(5.6)

u_→u¯ inL2_(Ω

×(0, T)),

(5.7)

w_→w¯ inL2_(Γ

×(0, T)).

(5.8)

Moreover, the pair(¯u,v¯), withv¯=−w¯are a weak solution to Problem 5.1.

Proof. In the interests of brevity we give the details for the Dirichlet boundary condition case. The Neumann case is handled similarly.

From standard weak compactness arguments (3.5) together with the estimate (4.14), we can extract a subsequence which we will still denote(u, w)such that

u *u¯ inL2(0, T;He1_uD(Ω)),

w *w¯ inL2_{(0, T;H}1_(Γ)).

From the Aubin-Lions-Simon compactness theory (Lemma 3.5), the estimate on time translates (4.16) means we can extract a subsequence which we will still denote(u, w)such that

We now show the pair(¯u,¯v), withv¯=₋w¯, are a weak solution to Problem 5.1. We start by noting that for allη_{∈ V}e0(Ω,Γ)withη(·, T) = 0, we have

Z T

0 −

δΩh∂tη, uiH1

euD(Ω)+ Z

Ω∇

u· ∇η dx dt−δΩ Z

Ω

u0η(·,0) dx=

Z T 0 Z Γ− 1 δk

uwη dσ dt

=

Z T

0 − h

∂tη, wiH1_(Γ)+δΓ

Z

Γ∇

Γw_{· ∇}Γη dσ dt₋ Z

Γ

w0η(_·,0) dσ.

Lettingδk→0the convergence results (5.5)—(5.8) give

Z T

0

−δΩ_h∂tη,u¯i_H1

euD(Ω)+ Z

Ω∇

¯

u_{· ∇}η dx

dt₋δΩ Z

Ω

u0η(_·,0) dx

=

Z T

0

− h∂tη,w¯i_H1_(Γ)+δΓ

Z

Γ∇

Γw¯· ∇Γη dσ

dt₋ Z

Γ

w0η(_·,0) dσ,

and hence withv¯=₋w¯

(5.9) Z T

0

−δΩ_h∂tη,u¯iH1

euD(Ω)+ Z

Ω∇

¯

u_{· ∇}η dx_{− h}∂tη,v¯iH1_(Γ)+δΓ

Z

Γ∇

Γ¯v· ∇Γη dσ

dt

=δΩ Z

Ω u0_η(

·,0) dx+

Z

Γ v0_η(

·,0) dσ.

It remains to show that¯v∈β(¯u). Asu, w≥0for allδk, we haveu¯ ≥0andv¯=−w¯ ≤0. Moreover from (4.15) we have

Z T

0 Z

Γ

uw_≤δkkw0kL1_(Γ),

and hence the strong convergence results (5.7) and (5.8) imply

¯

u¯v=₋u¯w¯= 0 a.e. inΓ_×(0, T).

Thus the limit pair(¯u,¯v)are a weak solution to Problem 5.1 in the sense of Definition 5.2.

5.4.Remark(Uniqueness of the solution to Problem 5.1.). Theorem 5.3 ensures existence of a solution to

Problem 5.1. However we are unable at present to prove uniqueness. In particular, the strategy employed for the proof of uniqueness to the limiting problems 6.1 and 7.1 does not seem applicable in this case.

6. PARABOLIC LIMIT PROBLEM WITH DYNAMIC BOUNDARY CONDITION(δk=δΓ = 0) We now present a rigorous derivation of the parabolic problem with dynamic boundary conditions presented in_§2.1 as a limit of (1.1). Specifically we show that for fixedδΩ>0, in the limitδk =δΓ →0 the unique solution of the problem (1.1) converges to the unique solution of the following problem. 6.1.Problem. Findu˜: Ω_×[0, T)→R+andv˜: Γ×[0, T)→R−such that

δΩ∂tu˜−∆˜u= 0 inΩ×(0, T)

(6.1a)

∇u˜_·ν+∂tv˜= 0 onΓ×(0, T)

(6.1b)

˜

v_∈β(˜u) onΓ_×(0, T)

(6.1c)

˜

u=uD or ∇u˜·ν= 0 on∂0Ω×(0, T)

(6.1d)

˜

u(·,0) =u0(·)≥0 onΩ

(6.1e)

˜

v(_·,0) =v0(_·)_≤0 onΓ,

(6.1f)

In order to define a weak solution of Problem 6.1 we introduce the space

H10, T;He10(Ω)

:=

v_∈L20, T;He10(Ω)

:∂tv∈L2

0, T;He10(Ω)

.

6.2.Definition(Weak solution of (6.1)). We say a function pair(˜u,v˜)withu˜ _∈L2_{(0, T;H}1

e_uD(Ω))∩

L∞_{(0, T;L}2_(Ω))andv˜

∈L∞_{(0, T;L}2_{(Γ))is a weak solution of(6.1), if for allη}

∈H1 _{0, T;H}1

e0(Ω)

withη(_·, T) = 0, we have

(6.2)

Z T

0 Z

Ω−

δΩu∂˜ tη+∇u˜· ∇η dx+

Z

Γ−

˜

v∂tη dσ

dt=δΩ Z

Ω

u0η(_·,0) dx+

Z

Γ

v0η(_·,0) dσ

and ˜v_∈β(˜u)a.e. inΓ_×(0, T).

We make the obvious modifications for the Neumann case.

6.3.Theorem(Convergence of the solution of (1.1) to a solution of (6.1)). Asδk =δΓ→0the solution

pair(u, w)to (4.1) converge to a pair(˜u,w˜)in the following topologies

u *u˜ inL2(0, T;He1_uD(Ω)) (6.3)

(u *u˜ inL2(0, T;H1(Ω))in the Neumann case)

w *w˜ inL2(0, T;L2(Γ)),

(6.4)

u→u˜ inL2(Ω×(0, T)),

(6.5)

u|Γ→u˜|Γ inL2(Γ×(0, T)).

(6.6)

Moreover, the pairu,˜ v˜, with˜v =₋w˜ are the unique weak solution to(6.1)in the sense of Definition (6.2).

Proof. As in the proof of Theorem 5.3, the uniform estimates of Lemma 4.3 together with the compact-ness results of Lemma 3.5 and Lemma 3.9 imply the weak and strong convergence results given in the theorem.

We now show that the limit pair(˜u,v˜), with˜v=₋w˜are a weak solution of (6.1). We start by noting

that for allη_∈C∞_(Ω

×(0, T))withη= 0on∂0Ω_×(0, T)andη(_·, T) = 0, we have

Z T

0 Z

Ω−

δΩu∂tη+∇u· ∇η dx dt−δΩ

Z

Ω u0_η(

·,0) dx=

Z T 0 Z Γ− 1 δk

uwη dσ dt

=

Z T

0 Z

Γ−

w∂tη+δΓ∇Γw· ∇Γη dσ dt−

Z

Γ

w0η(_·,0) dσ

=

Z T

0 Z

Γ−

w∂tη−δΓw∆Γη dσ dt−

Z

Γ

w0η(_·,0) dσ

Lettingδk =δΓ→0, the convergence results (6.3)—(6.5) give

Z T

0 Z

Ω−

δΩu∂˜ tη+∇u˜· ∇η dx dt−δΩ

Z

Ω

u0η(_·,0) dx=

Z T

0 Z

Γ−

˜

w∂tη dσ dt−

Z

Γ

w0η(_·,0) dσ,

and hence withv¯=₋w¯, we infer that

Z T

0 Z

Ω−

δΩu∂˜ tη+∇u˜· ∇η dx dt−δΩ

Z

Ω

u0η(·,0) dx− Z T

0 Z

Γ

˜

v∂tη dσ dt−

Z

Γ

v0η(·,0) dσ= 0.

(6.7)

A density argument yields that the above holds for all test functionsηin the spaces of Definition 6.2. As

u, w _≥0we haveu˜ _≥0,v˜ =₋w˜ _≤0. To check˜v _∈ β(˜u)it remains to show thatR

follows since Z T 0 Z Γ ˜

u˜v dσ dt=

Z T

0 Z

Γ−

˜

uw˜ dσ dt

=₋ lim

δk,δΓ→0

Z T

0 Z

Γ

(˜u₋u) ˜w+u( ˜w₋w) +uw dσ dt= 0,

where we have used that the first term on the right hand side is zero sinceu→u˜andw˜∈L2(0, T;L2(Γ))

(6.6), (4.14), the second term is zero sincew * w˜ anduis bounded in L2_{(0, T;L}2_{(Γ))(6.4) and the}

final term is zero from the estimate (4.15).

To prove that the solution is unique we argue as follows. Let(˜u1,˜v1)and(˜u2,˜v2)be solutions of (6.1) in the sense of Definition 6.2. We defineθu˜₍

·, t) := (˜u1(_·, t)₋u2˜ (_·, t)), θv˜₍

·, t) := (˜v1(_·, t)₋˜v2(_·, t)). The pair(θu˜_{, θ}v˜_)satisfy

Z T

0 Z

Ω− δΩθu˜_∂

tη+∇θu˜· ∇η dx dt−

Z T

0 Z

Γ θv˜_∂

tη dσ dt= 0, (6.8)

for allη _∈ H1 _{0, T;H}1

e0(Ω)

withη(_·, T) = 0. Fort _∈ (0, T)we defineθz˜₍

·, t) = RT

t θ

˜

u₍

·, s) ds.

Noting thatθz˜_{is an admissible test function, we setη=θ}z˜_{in (6.8) which gives} δΩ

Z T

0 Z

Ω

(θu˜)2 dx dt− Z T 0 1 2 d dt Z Ω ∇θ ˜ z 2

dx dt+

Z T

0 Z

Γ

θ˜vθu˜ dσ dt= 0.

Asθ˜z₍

·, T) = 0we have

δΩ Z T

0 Z

Ω

(θu˜₎2 _{dx dt+}1

2 Z T 0 Z Ω ∇θ ˜ u 2

dx dt+

Z T

0 Z

Γ

(˜v1₋˜v2) (˜u1₋u2˜ ) dσ dt= 0.

Recalling thatv˜i∈β(˜ui), i= 1,2, the monotonicity ofβgives

θ ˜ u 2

L2_((0,T);H1_(Ω))= 0.

Finally, (6.8) and the above bound yield

Z T

0 Z

Γ

θv˜∂tη dσ dt= 0

for allηthat are admissible test functions in the sense of Definition 6.2. For anyφ_∈L2_{(0, T;H}1/2_(Γ))

we define Dφsuch that Dφ = φ on Γ, ∆Dφ = 0 in Ωand Dφ = 0 on ∂0Ω. Then we may take

η(_·, t) =RT

t Dφ(·, s) dsas a test function in the above which gives

Z T

0 Z

Γ

θv˜_{φ dσ dt= 0,} for allφ_∈L2_{(0, T;H}1/2_{(Γ)). Hence}

kθv˜k_L2_0,T;(H1/2_(Γ))0= 0

which completes the proof of the theorem.

7.1.Problem. Finduˆ: Ω_×(0, T)_→R+andˆv: Γ×[0, T)→R−such that −∆ˆu= 0inΩ_×(0, T)

(7.1a)

∇uˆ_·ν+∂tvˆ= 0onΓ×(0, T) (7.1b)

ˆ

v∈β(ˆu)onΓ×(0, T)

(7.1c)

ˆ

u=uDon∂0Ω×(0, T) (7.1d)

ˆ

v(_·,0) =v0(_·)_≤0onΓ,

(7.1e)

whereβ:R→ {0,1}Ris the set valued function defined in (5.2).

7.2.Definition(Weak solution of (7.1)). We say a function pair(ˆu,vˆ)withuˆ_∈L2_{(0, T;H}1

e_uD(Ω))and

ˆ

v _∈L∞_{(0, T;L}2_{(Γ))is a weak solution of (7.1), if for allη}

∈H1 _{0, T;H}1

e0(Ω)

withη(_·, T) = 0on

Γ, we have

(7.2)

Z T

0 Z

Ω∇

ˆ

u_{· ∇}η dx₋ Z

Γ

ˆ

v∂tη dσ

dt₋ Z

Γ v0_η(

·,0) dσ= 0,

and ˆv∈β(ˆu)a.e. inΓ×(0, T).

The strategy of passing to the limit follows that of§6.

7.3.Theorem(Convergence of the solution of (1.1) to a solution of (7.1)). AsδΩ =δΓ =δk → 0the

solution pair(u, w)to (4.1) converge to a pair(ˆu,wˆ)in the following topologies

u *uˆ inL2(0, T;He1_uD(Ω)) (7.3)

w *wˆ inL2_{(0, T;L}2_(Γ)),

(7.4)

w→wˆ inL2(0, T;H−1/2(Γ)),

(7.5)

Moreover, the pairu,ˆ vˆ, withvˆ=₋wˆare the unique solution to Problem (7.1) in the sense of Definition (7.2).

Proof. As in the proof of Theorems 5.3 and 6.3, the estimates of Lemma 4.3, specifically (4.14) together with the compactness results recalled in (3.5) imply the convergence results (7.3) and (7.4). The strong convergence result (7.5) follows due to the Lions-Aubin-Simon compactness theory (Lemma 3.5) together with the estimate on the time translates ofw(4.16) and the compact embedding ofL2_(Γ)intoH−1/2_(Γ)

shown in Lemma 3.8.

The fact that the limitsu,ˆ ˆv=₋wˆsatisfy

Z T

0 Z

Ω∇

ˆ

u· ∇η dx− Z

Γ

ˆ

v∂tη dσ

dt− Z

Γ

v0η(·,0) dσ= 0,

for allη as in Definition 7.2, follows from the weak convergence results (7.3) and (7.4) together with an analogous density argument to that used in the proof of Theorem 6.3. It remains to checkˆv _∈β(ˆu). As previously we haveuˆ _≥0andvˆ_≤ 0. The fact thatu,ˆ vˆ_∈ L2_(Γ

×(0, T)), the strong convergence result (7.5), the weak convergence result (7.3) which implies weak convergence of the trace of uin

L2_{(0, T;H}1/2_{(Γ))and the estimate (4.15) imply} Z T

0 Z

Γ

ˆ

uˆv dσ dt=

Z T

0 h

ˆ

v,uˆ_iH1/2_(Γ) dt= 0,

and henceˆv_∈β(ˆu).

Similarly the uniqueness argument mirrors that used in the proof of Theorem 6.3. Letting(ˆu1,v1ˆ )

and(ˆu2,ˆv2)be two solutions of (7.1) in the sense of Definition 7.2 and settingθuˆ₍

ˆ

u2(_·, t)), θvˆ₍

·, t) := (ˆv1(_·, t)₋ˆv2(_·, t)). The pair(θuˆ_{, θ}ˆv_)satisfy

Z T

0 Z

Ω∇ θuˆ

· ∇η dx dt− Z T

0 Z

Γ θvˆ_∂

tη dσ dt= 0, (7.6)

for allη ∈H1 _{0, T;H}1

e0(Ω)

withη(·, T) = 0onΓ. Fort∈(0, T)we defineθˆz₍

·, t) =RT

t θ

ˆ

u₍

·, s) ds.

Notingθzˆ_{is an admissible test function, we setη=θ}zˆ_{in (7.6) which gives, using the fact thatθ}z˜₍ ·, T) = 0,

1 2

Z T

0 Z

Ω ∇θ

ˆ

u 2

dx dt+

Z T

0 Z

Γ

(ˆv1−ˆv2) (ˆu1−u2ˆ ) dσ dt= 0.

Recalling that vˆi ∈ β(ˆui), i = 1,2, the monotonicity of β, together with the Poincare inequality as

θˆz

∈H1

e0(Ω)gives

θ

˜

u

2

L2_((0,T);H1_(Ω))= 0.

Finally, via the same argument used in the proof of Theorem 6.3, (7.6) and the above bound yield

θ

ˆ

v

_L2_((0,T);H−1/2_(Γ))= 0,

which completes the Proof of the Theorem.

8. DEGENERATE PARABOLIC EQUATIONS

In this Section we give alternative formulations of the limiting problems of§5-7. Solutions to the problems 8.1, 8.2 and 8.3 introduced in this section are solutions of problems 5.1, 6.1 and 7.1 respectively. The structure of the equations is revealed by writing them as abstract degenerate parabolic equations holding on the surfaceΓ. Doing this, one observes that the problems are the analogues of the Hele-Shaw and steady one phase Stefan problems with the half-Laplacian replacing the usual Laplacian(₋∆)(see [Crowley, 1979; Elliott and Ockendon, 1982] for further details on the formulation of the Hele-Shaw and one phase Stefan problems).

First, we define a parabolic extension operator

PδΩ_:L2_{(0, T;H}1/2_(Γ))→L2_{(0, T;H}1

e_uD(Ω)), or

PδΩ_:L2_{(0, T;H}1/2_(Γ))

→L2(0, T;H1(Ω))

in the Neumann case. We fixη∈L2_{(0, T;H}1/2_{(Γ))we defineP}δΩ_ηto be the unique solution of

(8.1)

δΩ∂t(PδΩη)−∆(PδΩη) = 0 inΩ×(0, T)

PδΩ_{η =η} on_{Γ×(0, T)}

PδΩ_{η= 0}or

∇(PδΩ_η)

·νΩ= 0 on∂0Ω×(0, T)

(PδΩ_{η)(·,0) = 0} in_Ω.

This allows us to define a parabolic Dirichlet to Neumann (DtN) mapAδΩ: _L2_{(0, T;H}1/2_(Γ)) →

L2

(0, T);H1/2_(Γ)0

by

(8.2) AδΩ_η:=∇(PδΩ_η)·ν for_η∈L2_{(0, T;H}1/2_(Γ)).

Next, we define a new elliptic extension operator P0_{: L}2_{(0, T;H}1/2_{(Γ)) → L}2_{(0, T;H}1

solution of

(8.3)

−∆(P0η) = 0 inΩ×(0, T)

P0η=η onΓ_×(0, T)

P0η= 0 on∂0Ω_×(0, T).

This allows us to define the elliptic DtN mapA0_:L2_{(0, T;H}1/2_(Γ)) → L2

(0, T);H1/2_(Γ)0

by

(8.4) A0_η:=

∇(P0η)·ν forη ∈L2(0, T;H1/2_(Γ)).

We note that the operatorA0 _{may also be viewed as the half-Laplacian(}

−∆Γ)1/2 for functions onΓ

[Caffarelli and Silvestre, 2007].

It is also convenient to introduce extensions of the data. First we introduceUδΩ

D as the solution of the parabolic problem

(8.5)

δΩ∂tUDδΩ−∆U δΩ

D = 0 inΩ×(0, T)

UδΩ

D = 0 onΓ×(0, T)

UδΩ

D =uDor∇(UDδΩ)·νΩ= 0 on∂0Ω×(0, T)

UδΩ

D (·,0) = 0 inΩ. Second we haveUDas the solution of an elliptic problem

(8.6)

−∆UD= 0 inΩ

UD= 0 onΓ

UD=uD or∇UD·νΩ = 0 on∂0Ω.

In the Neumann case we haveUδΩ

D =UD= 0. Third, we introduceUδΩ

I as the solution of the parabolic problem

(8.7)

δΩ∂tUIδΩ−∆U δΩ

I = 0 inΩ×(0, T)

UδΩ

I = 0 onΓ×(0, T)

UδΩ

I = 0or∇(U δΩ

I )·νΩ= 0 on∂0Ω×(0, T)

UδΩ

I (·,0) =u0 inΩ. Note that asδΩ_→0thatUδΩ

I →0andU δΩ

D →UDinL2(0, T;H1(Ω)). Finally, we writeLfor−∆Γas

an operatorL2_{(0, T;H}1_(Γ))

→L2_{0, T; H}1_(Γ)0

8.1.Problem(Fast reaction limit,δk = 0). Findu¯ ≥ 0andv¯ ≤ 0withu¯ ∈ L2(0, T;H1/2(Γ))and

¯

v∈L2_{(0, T;H}1_(Γ))with∂

t¯v∈L2

0, T; H1_(Γ)0

such that

(8.8)

∂tv¯+δΓLv¯+AδΩu¯+∇(UDδΩ+U δΩ

I )·ν= 0 inL

2

0, T;H1(Γ)0

¯

v_∈β(¯u) onΓ_×(0, T) ¯

v(_·,0) =v0 inΩ.

8.2.Problem(Bulk parabolic limit equation with dynamic boundary condition, δk = δΓ = 0). Find

˜

u≥0and˜v≤0withu˜∈L2_{(0, T;H}1/2_{(Γ))andv˜∈L}2_{(0, T;L}2_(Γ))with∂

t˜v∈L2

such that

(8.9)

∂tv˜+AδΩu˜+∇(UDδΩ+U δΩ

I )·ν= 0 inL

2

0, T;H1(Γ)0

˜

v_∈β(˜u) onΓ_×(0, T) ˜

v(_·,0) =v0 inΩ.

8.3.Problem(Elliptic equation with dynamic boundary condition,δk =δΓ =δΩ= 0). Finduˆ≥0and

ˆ

v≤0withuˆ∈L2_{(0, T;H}1/2_{(Γ))andˆv∈L}2_{(0, T;L}2_(Γ))with∂

tvˆ∈L2

0, T; H1_(Γ)0

such that

(8.10)

∂tvˆ+A0uˆ+∇UD·ν= 0 inL2

0, T;H1(Γ)0

ˆ

v_∈β(ˆu) onΓ_×(0, T) ˆ

v(_·,0) =v0 inΩ.

9. VARIATIONAL INEQUALITY FORMULATION

Similarly to the Hele-Shaw and one phase Stefan problems, that may be reformulated as variational inequalities via an integration in time [Duvaut, 1973; Elliott, 1980; Elliott and Janovsk`y, 1981; Rodrigues, 1987], via integrating in time, the systems (6.1) and (7.1) and Problems 8.2 and 8.3 may be reformulated, respectively, as parabolic and elliptic variational inequalities of obstacle type. The obstacle problem lies on the surfaceΓand is a consequence of the complementarity which is maintained after an integration with respect to time and noting that this integration commutes with the operatorsAδΩ and_A0.

We set

(9.1) z(_·, t) =

Z t

0

ˆ

u(_·, s) ds,

whereuˆsatisfies (7.1). We find it convenient to introduceZδΩ

D as

(9.2) ZδΩ

D(·, t) =tUD.

Proceeding formally, we claim that if the pair(ˆu,vˆ)satisfy 6.1 (or (7.1) withδΩ = 0) then the pair

(z,vˆ)satisfy the following problem

9.1.Problem. For eacht∈(0, T), findz(t)∈H1_{(Ω)andˆv(t)∈L}2_{(Γ)such that}

δΩ∂tz−δΩu0−∆z= 0 inΩ

(9.3a)

∇z·ν+ ˆv−v0= 0 onΓ

(9.3b)

ˆ

v_∈β(z) onΓ

(9.3c)

z=ZD on∂0Ω.

(9.3d)

We check the condition vˆ _∈ β(z)onΓ_×(0, T). The remaining conditions follow formally from

integration in time of (7.1). LetχBdenote the characteristic function of the setB, then we have

Z

Γ

ˆ

vχz>0 dσ= Z

Γ

ˆ

v(χz>0−χˆu>0) dσ+ Z

Γ

ˆ

vχˆu>0 dσ.

Noting thatχz>0≥χˆu>0asuˆ≥0and recallingvˆ≤0we have Z

Γ

ˆ

vχz>0 dσ≥ Z

Γ

ˆ

vχˆu>0 dσ= 0.