WELL-POSEDNESS OF A MATHEMATICAL MODEL OF DIABETIC ATHEROSCLEROSIS

Abstract

Atherosclerosis is a leading cause of death in the United States and worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 21 million Americans have diabetes, a disease where patients’ cells cannot efficiently take in dietary sugar, causing it to build up in the blood. In part because diabetes increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. Past work has shown that hyperglycemia and insulin resistance alter function of multiple cell types, including endothelium, smooth muscle cells and platelets, indicating the extent of vascular disarray in this disease. Although the pathophysiology of diabetic vascular disease is generally understood, there is no mathematical model to date that includes the effect of diabetes on plaque growth. In this paper, we propose a mathematical model for diabetic atherosclerosis; the model is given by a system of partial differential equations with a free boundary. We establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.

1. Introduction

Atherosclerosis is a leading cause of death in the United States and worldwide. Atherosclerosis emerges as a result of multiple dynamical cell processes [1, 2, 3]. Damage to the endothelial layer of the artery wall triggers an inflammatory response in which monocytes, T-lymphocytes and other immune cells are recruited to the region of damage. These cells penetrate the endothelial layer, reaching the “tunica intima”, along with low density lipoprotein (LDL) and high-density lipoprotein (HDL)-particles. Stimulated by the presence of interferon gamma (IFN-c) and macrophage colony stimulating factor (M-CSF), monocytes differentiate into macrophages once they have entered the artery wall. While embedded within the “tunica intima”, both LDL and HDL become oxidized by free oxygen radicals. Macrophages will phagocytose oxidized LDL, but not oxidized HDL. Macrophages heavily loaded with oxidized LDL transform into foam cells that eventually undergo apoptosis. The resulting mass of debris embedded in the “tunica intima” is known as an atheroma. Foam cells, along with endothelial cells, secrete monocyte chemoattractant protein-1 (MCP-1) to recruit more monocytes to the site of inflammation. Naive T cells contained within the artery wall differentiate into individual T cell types that can secrete IFN-c. Smooth muscle cells (SMCs) are also recruited into the ‘tunica intima’, where they undergo apoptosis and contribute to the formation of a plaque in the artery wall. This accumulation of cells and debris can cause a swelling of the artery wall that restricts blood flow, leading to stenosis. The atherosclerosis inflammatory processes have been modeled mathematically [4,5]. The governing equations used in LDL oxidation were either ordinary differential equation (ODE) [6.7] or partial differential equation (PDE) [8,9,10]. The dynamics of cells were modeled by advection- diffusion equations that describe monocyte recruitment and chemoattractants, monocyte to macrophage differentiation, foam cell formation, T-cell recruitment and proliferation of SMCs [8,9,10,11,12,13,14]. The formation of a plaque was then modeled by a free boundary problem [9,10]. The results obtained from previous models are consistent with the guidelines issued by the American Heart Association periodically regarding to the risk of a heart attack associated with high level of cholesterol [9,10].

Nearly twenty one million Americans have diabetes, a disease where patients’ cells cannot efficiently take in dietary sugar, causing it to build up in the blood. When insulin secretion by the pancreas is insufficient or absent, due to (autoimmune) destruction of β-cells, the clinical picture of Type 1 Diabetes Mellitus (T1DM) results; when insulin is secreted in normal, or supernormal amounts, but it is in-effective in lowering glycemia to normal levels, Type 2 Diabetes Mellitus (T2DM) is said to be present. Diabetic patients have an increased risk of cardiovascular disease (CVD), and have a more than twofold increase in the risk of dying from CVD.

Myeloid cells are involved in diabetic atherosclerosis. Diabetes and atherosclerosis are chronic inflammatory conditions. Myeloid cells (neutrophils, monocytes, and macrophages) are involved in both atherosclerosis and diabetes. The migration of circulating monocytes into the vessel wall is critical for the development of diabetic atherosclerosis. Moreover, intercellular cell adhesion molecule-1(ICAM-1), chemoattractant protein-1(MCP-1), and macrophage migration inhibitory factor (MIF), which regulate the adhesion of monocytes, are dysregulated in hyperglycemia-induced atherosclerosis. Increased foam cells derived from macrophages promote the acceleration of atherosclerotic lesions in diabetic mice [15, 16]. A more inflammatory monocyte/macrophage phenotype with secretion of higher levels of proinflammatory cytokines was detected in both animal models and patients with diabetes mellitus [17]. In addition, T-cell function is closely related to atherosclerosis in the diabetic environment, and inflammatory monocytes have been shown to activate Th17 cells under diabetic conditions [18]. Diabetes cause endothelial cell dysfunction: Endothelial dysfunction due to inflammation and oxidative stress is a crucial characteristic in diabetes mellitus-linked atherosclerosis. Endothelial dysfunction is associated with decreased nitric oxide (NO) availability, either through loss of NO production or NO biological activity [19, 20]. The excess generation of free oxygen radicals leads to apoptosis in endothelial cells [20]. In hyperglycemia, chronic inflammation increases vascular permeability, promotes the generation of adhesion molecules and chemokines, and stimulates accumulation of monocytes in the artery wall.

Diabetes cause vascular smooth muscle cell dysfunction. Proliferation and accumulation of smooth muscle cells are detected in both type 1 and type 2 diabetes mellitus. The changes in smooth muscle cells are a result of the diabetic environment directly [21,22,23]. Diabetes stimulates atherogenic activities of SMCs. SMCs cultured from patients with type 2 diabetes demonstrate enhanced migration [29]. Advanced atherosclerotic lesions in diabetic patients have fewer SMCs compared with those of controls [29].

Diabetes impair platelet function. Platelets from patients with diabetes have been shown to have decreased sensitivity to antiaggregation agents, such as prostacyclin (PGI2) and NO [24]. Glycated low-density lipoprotein- (GlyLDL-) and hyperinsulinemia-induced impairment of calcium homeostasis, activation of protein kinase C (PKC), increased generation of reactive oxygen species (ROS), and decreased NO bioactivity result in hyperactivation of platelets [25].

Animal studies of diabetic atherosclerosis exist for mouse, rabbit and pig [15,38]. As atherosclerosis is a cardiovascular condition that affects critical circulatory systems, studying human atheroma poses logistical and ethical problems, as access to live atherosclerotic tissue is limited and disturbances risk triggering plaque rupture. As a result, it is compelling to study diabetic atherosclerosis by mathematical model which is biologically relevant and can be used to simulate pathway dynamics of the disease. Although mathematical modeling of the glucose-insulin feedback system has been established [26,27,28], there are no mathematical model to date for the dynamical processes where hyperglycemia and insulin resistance cause arterial dysfunction. To remedy this gap, we propose a model that will establish governing equations for the mechanism of how diabetes impairs endothelium-dependent (nitric oxide-mediated) vasodilation before the formation of atheroma and mechanism of how arteries affected by diabetes have altered vascular smooth muscle cell function.Such mathematical model has never been reported before. We will establish governing equations for the mechanism whereby excess glucose metabolites inhibits production of NO by blocking eNOS synthase activation and increase the production of ROS. Governing equations will be established for the mechanism that insulin resistance decrease endothelium-derived NO and increase the production of ROS. Governing equations will also be established for the mechanism where diabetes stimulates atherogenic activity of vascular smooth muscle cells.

In order to guide the numerical simulation of the diabetic atherosclerosis based on the proposed model, it is important to rigorously study several mathematical issues about the proposed model. In this paper we will address the local existence and uniqueness of solution to the proposed model. The methodology is to use Hanzawa transformation [53] to reduce the free boundary to a fixed boundary and reduce the system of PDEs to an abstract system of evolution equations in Banach spaces, and then apply the analytic semigroup theory [55].

The values of the parameters in the model are important for numerical simulations of the model, but the mathematical analysis such as well-posedness of the model does not require the exact values of the parameters, so we do not give the estimation of the parameters in this paper. The parameter estimation and numerical simulation of the model will be presented in a forthcoming paper.

2. Mathematical model

We will use a system of partial differential equations to develop a mathematical model of plaque formation in diabetic atherosclerosis. The mathematical model will be based on the network shown in Figure 1. The variables included in the model are listed in Table 1. We assume that all cells are moving with a common velocity u [9,10]; the velocity is the result of movement of macrophages, T-cells and SMCs into the intima. We also assume that all species are diffusing with appropriate diffusion coefficients. The equation for each species of cells X has the form

$$\frac{dX}{dt}+\nabla \cdot (uX)-D_X\Delta X=F_X.$$ (1)

where the expression on the left side includes advection and diffusion, and F_X accounts for various growth factors, bio-chemical reactions, chemotaxis and haptotaxis. The equation for the chemical species are the same but without the advection term. In this work, we consider only two dimensional plaque for simplicity. Figure 2 shows a 2D cross section of a blood vessel with plaque Ω.

Figure 1.

Glucose production and uptake depend on blood glucose and insulin levels. Insulin is secreted by pancreatic β cells, cleared by the liver, kidneys, and insulin receptors. β cells replication and death vary nonlinearly with glucose. Glucose metabolites inhibits production of NO and increase the production of ROS. Insulin resistance decrease endothelium-derived NO and increase the production of ROS. Diabetes stimulates atherogenic activity of vascular smooth muscle cells. LDL and HDL are oxidized by free radicals, Ox- LDL recruits macrophages to intima. By ingesting ox-LDL, macrophages are transformed to foam cells. In this figure, LDL and oxidized LDL are merged.

Table 1.

The variables of the model: Concentrations and densities are in units of g/cm³

L:	concentration of HDL	H:	concentration of HDL
G:	concentration of Glucose	I:	concentration of Insulin
R:	concentration of free radicals	M:	density of macrophages
S:	density of SMCs	β:	density of β-cells
F:	density of foam cells
u:	fluid velocity (in cm/day)	σ:	pressure (in gcm2/day)

Figure 2.

A 2D cross section of a plaque. Γ_M is the boundary of the intima in contact with the media, and Γ_I is the boundary of the intima in contact with the lumen.

Equations for glucose, insulin, and β-cells :

The distribution of G and I and β density are modeled by

$$\frac{dG}{dt}-D_G\Delta G={\tilde{G}}_0-\left(E_{G0}+S_{I}I\right)G,$$ (2)

$$\frac{d\beta }{dt}+\nabla \cdot (u\beta )-D_{\beta }\Delta \beta =\left(-d_0+r_{1}G-r_2{G}^2\right)\beta ,$$ (3)

$$\frac{dI}{dt}-D_I\Delta I=\frac{\sigma G^2\beta }{\alpha +G^2}-kI-{\lambda }_{RI}R;$$ (4)

Equation (2) describes the dynamics of glucose and insulin, and equation (3) describes the formation and loss of β-cells [35]. The first term on the right hand side of (4) is secretion of insulin [35], the second term is clearance of insulin. The third term on the right side of (4) models that ROS, in excess and over time, cause chronic oxidation stress, which in turn result in reduction of insulin secretion as well as increased apoptosis [36,37], where λ_RI is the reduction rate of insulin due to ROS.

Equations for lipoproteins [LDL (L), HDL (H)]:

The distribution of LDL, HDL and free radicals in the intima are described using reaction diffusion equations [7,9,10,39]

$$\frac{dL}{dt}-D_L\Delta L=-k_{L}RL,$$ (5)

$$\frac{dH}{dt}-D_H\Delta H=-k_{H}RH,$$ (6)

where k_L and k_H are reaction rates of oxidization. Eqs. (5) and (6) model the evolution of LDL and HDL concentrations. It is assumed that LDL and HDL are lost by reaction of oxidation with free radicals.

Equation for free radicals (R):

The equation for the concentration of free radicals is modeled by

$$\frac{dR}{dt}-D_R\Delta R=R_0-R\left(k_{L}L+k_{H}H\right)+{\lambda }_{GR}G+{\lambda }_{IR}I,$$ (7)

where R₀ is the base line growth, the third term on the right hand of the equation models the mechanism whereby excess glucose metabolites inhibits production of NO by blocking eNOS synthase activation and increase the production of ROS [31,32]; where λ_GR is the growth rate of ROS due to excess glucose. The fourth term on the right hand of the equation models the mechanism that insulin resistance decrease endothelium-derived NO and increase the production of ROS [33,34]; where λ_IR is the growth rate of ROS due to insulin resistance.

Equation for SMCs (S):

The equation of the SMCs density is given by

$$\frac{dS}{dt}+\nabla \cdot (uS)-D_S\Delta S=-\nabla \cdot \left(S{\chi }_G\nabla G\right)-{\lambda }_{GL}GL,$$ (8)

Diabetes stimulates atherogenic activities of SMCs. SMCs cultured from patients with type 2 diabetes demonstrate enhanced migration [29]. Advanced atherosclerotic lesions in diabetic patients have fewer SMCs compared with those of controls [29]. The first term in the right hand side models haptotaxis by glucose. The second term models that oxidized glycated LDL can induce apoptosis of SMCs [29].

Equation for macrophages (M):

The evolution of macrophage density is modeled by

$$\frac{dM}{dt}+\nabla \cdot (uM)-D_M\Delta M=-{\lambda }_{ML}\frac{ML}{K_{ML}+L}+{\lambda }_{HF}\frac{HF}{K_{HF}+H}-d_{M}M,$$ (9)

Here the first and the second term on right-hand side accounts for the transition between M and L: when L molecules are ingested by a macrophage, the macrophage becomes a foam cell; when H combines with membrane protein on a foam cell in a process that clears it from the bad cholesterol, the foam cell turns into a macrophage. The third term accounts for the death of macrophages.

Equation for foam cells (F):

The equation for foam cells is given by

$$\frac{dF}{dt}+\nabla \cdot (uF)-D_F\Delta F={\lambda }_{ML}\frac{ML}{K_{ML}+L}-{\lambda }_{HF}\frac{HF}{K_{HF}+H}-d_{F}F,$$ (10)

where on the right side of the equation, a gain of foam cells from macrophages M ingesting L, and a loss of foam cells triggered by H. The death rate of foam cells is d_F.

Equations pressure (σ):

We assume that the intima has the constituency of a porous medium. Then, by Darcy’s law, the velocity u of the cells is given by

$$u=-\nabla \sigma ,$$ (11)

where σ is the pressure. We also assume that the total density of all the cells is constant. This constant should be smaller than the average density of a plaque, 1.22 ± 0.03g/cm³[9], because plaques contain some debris, which are not included in our model. We take the constant to be 1 g/cm3, i.e.,

$$M+F+\beta +S=1$$ (12)

We assume that all cells are approximately of the same volume and surface area, so that the diffusion coefficients of the all cells have the same coefficient. By adding Eqs. (3), (8), (9) and (10) and using (12), we get

$$-\Delta \sigma =\left(-d_0+r_{1}G-r_2{G}^2\right)\beta -\nabla \cdot \left(S{\chi }_G\nabla G\right)-{\lambda }_{GL}GL-d_{F}F-d_{M}M.$$ (13)

Boundary Conditions:

For simplicity, we consider only 2-dimensional plaques as in Fig. 2. Then the boundary of the plaque consists of i) Γ_M, in contact with media; ii) a free boundary Γ_I, inside the lumen.

Boundary conditions on Γ_I:

We assume flux boundary conditions of the form

$$\frac{\partial X}{\partial n}+{\alpha }_X\left(X-X_0\right)=0,\frac{\partial Y}{\partial n}=0$$ (14)

for X = G, I, L, H, β, and non-flux boundary conditions for all other variables Y = R, S, F. The flux boundary condition for M is determined by M = 1−S−β−F. Note that G₀, I₀, L₀ and H₀ are the glucose, insulin, LDL and HDL concentrations in the blood, so we shall be interested to see how these concentrations determine whether a small plaque will grow or shrink.

As in [46,47,48], we assume that the free boundary Γ_I is held together by cell-to-cell adhesion forces so that

$$\sigma =\gamma \kappa {\text{ on Γ}}_I,$$ (15)

where κ is the mean curvature of the surface Γ_I. (If Γ_I is circular, then κ is the reciprocal of the radius) Furthermore, the continuity condition u_n = V_n, where n is the outward normal and V_n is the velocity of the free boundary Γ_I in the direction n, yields the relation

$$V_n=-\frac{\partial \sigma }{\partial n}{\text{ on Γ}}_I.$$ (16)

Boundary conditions on Γ_M:

We assume non-flux boundary conditions for all variables except S and M on Γ_M: For S, we have

$$\frac{\partial S}{\partial n}+{\alpha }_S\left(S-S_0\right)=0{\text{ on Γ}}_M,$$ (17)

where S₀ is SMCs density in the media. The flux condition for M is determined by M = 1 − S − β − F.

Parameter estimation:

Most parameters on the right hand side of (2)–(4) are given in [35] and most estimates of parameters on the right hand side of (5)–(20) can be found given in [9,10]. Since rigorous mathematical analysis in this paper does not require the exact values of the parameters, we omit the details.

2.1. The slow model.

Models for the glucose-insulin control system, typically span the short time (minutes) along which perturbation experiments (IVGTT,OGTT or clamps) are performed and blood concentration of metabolites are obtained. Long time spans (months, years)are used when assessing the response to antidiabetic therapy during clinical trials. As in [27, 51], we may substitute the fast-scale time t with a slow scale time τ in the glucose-insulin system (2) and (4) by setting τ = ϵt where ϵ is a very small parameter. Letting ϵ → 0, then we can replace the glucose-insulin system (2) and (4) by the following slow equations for the glucose and insulin:

$$-D_G\Delta G={\tilde{G}}_0-\left(E_{G0}+S_{I}I\right)G,$$ (18)

$$-D_I\Delta I=\frac{\sigma G^2\beta }{\alpha +G^2}-kI-{\lambda }_{RI}R;$$ (19)

Using (18), we can write (13) as

$$-\Delta \sigma +\nabla S\cdot \left({\chi }_G\nabla G\right)=F_1(G,S,\beta ,I,L,F),$$ (20)

where

$$F_1(G,S,\beta ,I,L,F)=\left(-d_0+r_{1}G-r_2{G}^2\right)\beta -{\lambda }_{GL}GL-{\chi }_G{D}_G^{-1}\left[{\tilde{G}}_0-\left(E_{G0}+S_{I}I\right)G\right]-d_{F}F-d_M(1-\beta -S-F).$$ (21)

Using (20) and (11), we can write (3) as

$$\frac{d\beta }{dt}-\nabla \sigma \cdot \nabla \beta +\beta \nabla S\cdot \left({\chi }_G\nabla G\right)-D_{\beta }\Delta \beta =F_2(G,S,\beta ,I,L,F);$$ (22)

where

$$F_2(G,S,\beta ,I,L,F)=\left(-d_0+r_{1}G-r_2{G}^2\right)\beta -\beta F_1(G,S,\beta ,I,L,F).$$ (23)

(7) can be written as

$$\frac{dR}{dt}-D_R\Delta R=F_3(G,I,L,H,R),$$ (24)

where

$$F_3(G,I,L,H,R)=R_0-R\left(k_{L}L+k_{H}H\right)+{\lambda }_{GR}G+{\lambda }_{IR}I.$$ (25)

Using (19) and (11), we can write (8) as

$$\frac{dS}{dt}+\left[\left({\chi }_G+S\right)\nabla G-\nabla \sigma \right]\cdot \nabla S-D_S\Delta S=F_4(G,S,\beta ,I,L,F),$$ (26)

where

$$F_4(G,S,\beta ,I,L,F)=-S{F}_1(G,S,\beta ,I,L,F)+{\chi }_{G}S{D}_G^{-1}\left[{\tilde{G}}_0-\left(E_{G0}+S_{I}I\right)G\right]-{\lambda }_{GL}GL.$$ (27)

Using (20) and (11), we can write (10) as

$$\frac{dF}{dt}-\nabla \sigma \cdot \nabla F+F\nabla S\cdot \left({\chi }_G\nabla G\right)-D_F\Delta F=F_5(G,S,\beta ,I,L,F);$$ (28)

where

$$F_5(G,S,\beta ,I,L,F)={\lambda }_{ML}\frac{ML}{K_{ML}+L}-{\lambda }_{HF}\frac{HF}{K_{HF}+H}-d_{F}F-F{F}_1(G,S,\beta ,I,L,F).$$ (29)

Now the slow model consists of functions {L, H, G, I, β, R, S, F, σ} which satisfy equations (5), (6), (23)–(34) in the region Ω(t) bounded by a free boundary Γ_I(t) an a fixed boundary Γ_M and the following boundary conditions:

On the free boundary Γ_I(t),

$$\begin{gathered} \frac{\partial L}{\partial n}+{\alpha }_L\left(L-L_0\right)=0,\frac{\partial H}{\partial n}+{\alpha }_H\left(H-H_0\right)=0, \\ \frac{\partial G}{\partial n}+{\alpha }_G\left(G-G_0\right)=0,\frac{\partial I}{\partial n}+{\alpha }_I\left(I-I_0\right)=0, \\ \frac{\partial \beta }{\partial n}+{\alpha }_{\beta }\left(\beta -{\beta }_0\right)=0,\frac{\partial R}{\partial n}=\frac{\partial S}{\partial n}=\frac{\partial F}{\partial n}=0, \\ \sigma =\gamma \kappa ,V_n=-\frac{\partial \sigma }{\partial n} \end{gathered}$$ (30)

and on the fixed boundary Γ_M:

$$\begin{gathered} \frac{\partial L}{\partial n}=\frac{\partial H}{\partial n}=\frac{\partial G}{\partial n}=0 \\ \frac{\partial R}{\partial n}=\frac{\partial \beta }{\partial n}=\frac{\partial I}{\partial n}=\frac{\partial F}{\partial n}=0 \\ \frac{\partial S}{\partial n}+{\alpha }_S\left(S-L_0\right)=0 \\ \sigma =0. \end{gathered}$$ (31)

We impose the following initial conditions: for x ∈ Ω₀,

$$\begin{gathered} L(0,x)=L_0(x),H(0,x)=H_0(x),\beta (0,x)={\beta }_0(x), \\ R(0,x)=R_0(x),S(0,x)=S_0(x),F(0,x)=F_0(x),{{\Gamma }_I|}_{t=0}={\Gamma }_0, \end{gathered}$$ (32)

where the initial free boundary Γ₀ is assumed to not intersect the fixed boundary Γ_M and Ω₀ is the region between Γ₀ and Γ_M.

3. Well posedness of the mathematical model

In this section we are interested in the question of well-posedness of the free boundary problem for the slow model proposed in the previous section. To answer the question, we will first use the Hanzawa transformation [53] to reduce the domain Ω(t) to a fixed domain Ω₀, the free boundary Γ_I(t) is described by a function ρ(ω, t) which is defined on the fixed boundary Γ₀. The free boundary problem will be changed into a system of PDEs in a fixed domain domain Ω₀ with boundary conditions on the fixed boundaries Γ₀ and Γ_M. Then we will further reduce the fixed boundary problem to an abstract evolution equation in Banach spaces, and we will obtain existence uniqueness of solution employing the analytic semigroup theory [55, 56].

3.1. Hanzawa diffeomorphism and reduction to a fixed region.

We assume that Γ₀ ∈ C^m,α, where Γ₀ ∈ C^m,α is the classical Hölder space with norm ∥·∥_m,α; m ≥ 4 is an integer, and 0 < α < 1.

Let ω be a variable point of Γ₀ and n(ω) the unit exterior normal to Γ₀ at ω. Let γ₀ be a given sufficiently small positive number such that the mapping

$$x:{\Gamma }_0\times \left[-{\gamma }_0,{\gamma }_0\right]\to N_0\subset R^2,x(\omega ,\lambda )=\omega +\lambda n(\omega )$$

is a C^m−1,α diffeomorphism.

We define Γ_0,T = Γ₀×[0, T] and Ω_0,T = Ω₀×[0, T]. For ρ(ω, t) ∈ C([0, T], C^m,α(Γ₀)) with ∥ρ∥_m,α≤ γ₀. Let

$${\Gamma }_{\rho ,T}=\left\{\omega +\rho (\omega ,t)n(\omega );(\omega ,t)\in {\Gamma }_{0,T}\right\}$$

and Ω_ρ,T be the domain bounded by Γ_ρ,T and Γ_M.

Let δ₀ be sufficiently small so that 4δ₀ < γ₀. Choose a function χ(λ) ∈ C^∞(R) so that

1. χ(λ) = 1, for |λ| ≤ δ₀.

2. χ(λ) = 0, for λ > 3δ₀.

3. $\left|{\chi }^{\prime }(\lambda )\right|\le \frac{3}{4{\delta }_0}$ for λ ∈ R.

For ρ(ω, t) ∈ C([0, T], C^m,α(Γ₀)) ∩ C¹([0, T], C^m−1,α(Γ₀)), with |ρ|_m+α ≤ δ₀, we define Hanzawa diffeomorphism:

$$\begin{gathered} e_{\rho }:R_y^2\times [0,T]\to R_x^2\times [0,T] \\ {e}_{\rho }(y,t)=(x(\omega ,\lambda +\chi (\lambda )\rho (\omega ,t)),t) \\ \text{for }(y,t)=(x(\omega ,\lambda ),t)\in N_0\times [0,T] \\ {e}_{\rho }(y,t)=(y,t)\text{ for }(y,t)\in \left(R^2-N_0\right)\times [0,T] \end{gathered}$$ (33)

Note that e_ρ(Ω₀, T) = Ω_ρ,T, e_ρ(Γ₀, T) = Γ_ρ,T.

Define a function η : R × [−δ₀, δ₀] → R by η(λ, μ) + χ₀(η(λ, μ))μ = λ for (λ, μ) ∈ R × [−δ₀, δ₀]. Since $\left|{\chi }^{\prime }(\lambda )\right|\le \frac{3}{4{\delta }_0}$, the function η is well defined and C^∞. And we have

$$\begin{gathered} e_{\rho }^{-1}:R_x^2\times [0,T]\to R_y^2\times [0,T] \\ {e}_{\rho }^{-1}(x(\omega ,\lambda ),t)=(x(\omega ,\eta (\lambda ,\rho (\omega ,t)),t) \\ \text{for }(x,t)=(x(\omega ,\lambda ),t)\in N_0\times [0,T] \\ {e}_{\rho }^{-1}(x,t)=(x,t)\text{ for }(x,t)\in \left(R^2-N_0\right)\times [0,T] \end{gathered}$$ (34)

We define the following operators: for $v\in C^2\left(\overline{{\Omega }_{0T}}\right)$

$$\begin{gathered} {\mathcal{L}}_{\rho }v\equiv \Delta \left(v\circ e_{\rho }^{-1}\right)\circ e_{\rho }=\sum _{k,j=1}^2{A}_{\rho ,kj}(y,t){\partial }_{y_k{y}_j}^{2}v+\sum _{k=1}^2{A}_{\rho ,k}{\partial }_{y_k}v, \\ {A}_{\rho ,kj}(y,t)=A_{kj}\left(y,\rho (\omega (y),t);{\partial }_{\omega }\rho (\omega (y),t)\right), \\ \text{for }k,j=1,2,\text{ and }(y,t)\in N_0\times [0,T]\text{,} \\ {A}_{\rho ,kj}(y,t)={\delta }_{kj}\text{ for }k,j=1,2,(y,t)\in \left(R^n-N_0\right)\times [0,T]. \end{gathered}$$ (35)

$$A_{\rho ,k}(y,t)=A_k\left(y,\rho (\omega (y),t),{\partial }_{\omega }\rho (\omega (y),t),{\partial }_{\omega }^2\rho (\omega (y),t)\right)k=1,2;$$ (36)

$$S_{\rho }=1+{\left|{\partial }_{\omega }\rho {\nabla }_x\omega (x(\omega ,\rho (\omega ,t)))\right|}^2.$$ (37)

For $v\in C^1\left(\overline{{\Omega }_{0T}}\right)$ we define the operator

$${\nabla }_{\rho }v=\nabla \left(v\circ e_{\rho }^{-1}\right)\circ e_{\rho }.$$ (38)

For given ρ(ω, t) ∈ C¹([0, T] × Γ₀), we define the function

$${\psi }_{\rho }:N_0\to R,{\psi }_{\rho }(x)=\lambda (x)-\rho (\omega (x),t),\text{ for }x\in N_0.$$ (39)

So Γ_ρ = {x ∈ N₀ : ψ_ρ(x) = 0}, the unit outward nomal vector on Γ_ρ is given by

$$n_{\rho }(x)=\frac{\nabla {\psi }_{\rho }(x)}{|\nabla {\psi }_{\rho }(x)|}\text{ for }x\in {\Gamma }_{\rho }.$$ (40)

The normal velocity V_n is given by

$$V_n=\frac{{\partial }_t\rho (\omega (x),t)}{|\nabla {\psi }_{\rho }(x)|}\text{ for }x\in {\Gamma }_{\rho }.$$ (41)

For $v\in C^1\left(\overline{{\Omega }_{0T}}\right)$ we define the operator

$${\mathcal{D}}_{\rho }v=\left({{\nabla \left(v\circ e_{\rho }^{-1}\right)|}_{{\Gamma }_{\rho }}\cdot \nabla {\psi }_{\rho }|}_{{\Gamma }_{\rho }}\right)\circ e_{\rho }.$$ (42)

We define

$${\mathcal{M}}_{\rho }(y)=\{\begin{array}{ll}\chi (\lambda ){\partial }_{\lambda }x(\omega (y),\lambda +\chi (\lambda )\rho (\omega (y),t)),\hfill & \hfill \text{for }(y,t)=(x(\omega ,\lambda ),t)\in N_0\times [0,T],\\ 0,\hfill & \hfill \text{for }(y,t)=(x(\omega ,\lambda ),t)\in {\Omega }_{0T}/N_0\times [0,T].\end{array}$$ (43)

Set

$$\begin{gathered} \tilde{G}=G\circ e_{\rho },\tilde{I}=I\circ e_{\rho },\tilde{\sigma }=\sigma \circ e_{\rho },\tilde{\beta }=\beta \circ e_{\rho },\tilde{R}=R\circ e_{\rho }, \\ \tilde{S}=S\circ e_{\rho },\tilde{L}=I\circ e_{\rho },\tilde{H}=H\circ e_{\rho },\tilde{F}=F\circ e_{\rho }. \end{gathered}$$

Then the system (5)–(6) and (23)–(34) become

$$-D_G{\mathcal{L}}_{\rho }\tilde{G}={\tilde{G}}_0-\left(E_{G0}+S_I\tilde{I}\right)\tilde{G},$$ (44)

$$-D_I{\mathcal{L}}_{\rho }\tilde{I}=\frac{{\sigma }_0{\tilde{G}}^2\tilde{\beta }}{\alpha +{\tilde{G}}^2}-k\tilde{I}-{\lambda }_{RI}\tilde{R};$$ (45)

$$-{\mathcal{L}}_{\rho }\tilde{\sigma }+{\nabla }_{\rho }\tilde{S}\cdot \left({\chi }_G{\nabla }_{\rho }\tilde{G}\right)={\tilde{F}}_1(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F}),$$ (46)

where

$${\tilde{F}}_1(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F})=\left(-d_0+r_1\tilde{G}-r_2{\tilde{G}}^2\right)\tilde{\beta }-{\lambda }_{GL}\tilde{G}\tilde{L}-{\chi }_G{D}_G^{-1}\left[{\tilde{G}}_0-\left(E_{G0}+S_I\tilde{I}\right)\tilde{G}\right]-d_F\tilde{F}-d_M(1-\tilde{\beta }-\tilde{S}-\tilde{F}).$$ (47)

$$\frac{d\tilde{\beta }}{dt}-\left({\nabla }_{\rho }\tilde{\sigma }+\left({\mathcal{D}}_{\rho }\tilde{\sigma }\right)\mathcal{M}(y)\right)\cdot {\nabla }_{\rho }\tilde{\beta }+\beta {\nabla }_{\rho }\tilde{S}\cdot \left({\chi }_G{\nabla }_{\rho }\tilde{G}\right)-D_{\beta }{\mathcal{L}}_{\rho }\tilde{\beta }={\tilde{F}}_2(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F});$$ (48)

where

$${\tilde{F}}_2(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F})=\left(-d_0+r_1\tilde{G}-r_2{\tilde{G}}^2\right)\tilde{\beta }-\tilde{\beta }{\tilde{F}}_1(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F}).$$ (49)

$$\frac{d\tilde{L}}{dt}-D_L{\mathcal{L}}_{\rho }\tilde{L}-\left({\mathcal{D}}_{\rho }\tilde{\sigma }\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{L}=-k_L\tilde{R}\tilde{L},$$ (50)

$$\frac{d\tilde{H}}{dt}-D_H{\mathcal{L}}_{\rho }\tilde{H}-\left({\mathcal{D}}_{\rho }\tilde{\sigma }\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{H}=-k_L\tilde{R}\tilde{H},$$ (51)

$$\frac{d\tilde{R}}{dt}-D_R{\mathcal{L}}_{\rho }\tilde{R}-\left({\mathcal{D}}_{\rho }\tilde{\sigma }\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{R}={\tilde{F}}_3(\tilde{G},\tilde{I},\tilde{L},\tilde{H},\tilde{R}),$$ (52)

where

$${\tilde{F}}_3(\tilde{G},\tilde{I},\tilde{L},\tilde{H},\tilde{R})=R_0-\tilde{R}\left(k_L\tilde{L}+k_H\tilde{H}\right)+{\lambda }_{GR}\tilde{G}+{\lambda }_{IR}\tilde{I}.$$ (53)

$$\frac{d\tilde{S}}{dt}+\left[\left({\chi }_G+\tilde{S}\right){\nabla }_{\rho }\tilde{G}-{\nabla }_{\rho }\tilde{\sigma }-\left({\mathcal{D}}_{\rho }\tilde{\sigma }\right){\mathcal{M}}_{\rho }(y)\right]\cdot {\nabla }_{\rho }\tilde{S}-D_S{\mathcal{L}}_{\rho }\tilde{S}={\tilde{F}}_4(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F}),$$ (54)

where

$${\tilde{F}}_4(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F})=-\tilde{S}{\tilde{F}}_1(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F})+{\chi }_G\tilde{S}{D}_G^{-1}\left[{\tilde{G}}_0-\left(E_{G0}+S_I\tilde{I}\right)\tilde{G}\right]-{\lambda }_{GL}\tilde{G}\tilde{L}.$$ (55)

$$\frac{d\tilde{F}}{dt}-\left({\nabla }_{\rho }\tilde{\sigma }+\left({\mathcal{D}}_{\rho }\tilde{\sigma }\right){\mathcal{M}}_{\rho }(y)\right)\cdot {\nabla }_{\rho }\tilde{F}+\tilde{F}{\nabla }_{\rho }\tilde{S}\cdot \left({\chi }_G{\nabla }_{\rho }\tilde{G}\right)-D_F{\mathcal{L}}_{\rho }\tilde{F}={\tilde{F}}_5(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{H},\tilde{F});$$ (56)

where

$${\tilde{F}}_5(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{H},\tilde{F})={\lambda }_{ML}\frac{\tilde{M}\tilde{L}}{K_{ML}+\tilde{L}}-{\lambda }_{HF}\frac{\tilde{H}\tilde{F}}{K_{HF}+\tilde{H}}-d_F\tilde{F}-\tilde{F}{\tilde{F}}_1(\tilde{G},\tilde{S},\tilde{\beta },\tilde{I},\tilde{L},\tilde{F}).$$ (57)

The boundary conditions (35) becomes the following conditions on Γ × [0, T]:

$${\partial }_t\rho +{\mathcal{D}}_{\rho }\tilde{\sigma }=0;$$ (58)

$$\tilde{\sigma }=\gamma {\kappa }_{\rho }(\omega ,t){\text{ on Γ}}_{0T};$$ (59)

where

$${\kappa }_{\rho }(\omega ,t)=\nabla \left({|\nabla {\psi }_{\rho }|}^{-1}\nabla {\psi }_{\rho }\right)(\omega ,t).$$ (60)

$$\begin{gathered} {\mathcal{D}}_{\rho }\tilde{L}+S_{\rho }{\alpha }_L\left(\tilde{L}-L_0\right)=0,{\mathcal{D}}_{\rho }\tilde{H}+S_{\rho }{\alpha }_H\left(\tilde{H}-H_0\right)=0, \\ {\mathcal{D}}_{\rho }\tilde{G}+S_{\rho }{\alpha }_G\left(\tilde{G}-G_0\right)=0,{\mathcal{D}}_{\rho }\tilde{I}+S_{\rho }{\alpha }_I\left(\tilde{I}-I_0\right)=0, \\ {\mathcal{D}}_{\rho }\tilde{\beta }+S_{\rho }{\alpha }_{\beta }\left(\tilde{\beta }-{\beta }_0\right)=0,{\mathcal{D}}_{\rho }\tilde{R}={\mathcal{D}}_{\rho }\tilde{S}={\mathcal{D}}_{\rho }\tilde{F}=0. \end{gathered}$$ (61)

and on the fixed boundary Γ_M:

$$\begin{gathered} \frac{\partial \tilde{L}}{\partial n}=\frac{\partial \tilde{H}}{\partial n}=\frac{\partial \tilde{G}}{\partial n}=0 \\ \frac{\partial \tilde{R}}{\partial n}=\frac{\partial \tilde{\beta }}{\partial n}=\frac{\partial \tilde{I}}{\partial n}=\frac{\partial \tilde{F}}{\partial n}=0 \\ \frac{\partial \tilde{S}}{\partial n}+{\alpha }_S\left(\tilde{S}-S_0\right)=0 \\ \sigma =0. \end{gathered}$$ (62)

The initial conditions are: for x ∈ Ω₀,

$$\begin{gathered} \tilde{L}(0,x)=L_0(x),\tilde{H}(0,x)=H_0(x),\tilde{\beta }(0,x)={\beta }_0(x), \\ \tilde{R}(0,x)=R_0(x),\tilde{S}(0,x)=S_0(x),\tilde{F}(0,x)=F_0(x),{\rho (\omega ,t)|}_{t=0}=0. \end{gathered}$$ (63)

We denote by h^m,α the closure of C^∞(Ω₀) in the usual Hölder space C^m,α(Ω₀) with norm ∥·∥_m,α. Similarly we define h^m,α(Γ₀) and $h_{{\delta }_0}^{m,\alpha }\left({\Gamma }_0\right)=\left\{\rho \in h^{m,\alpha }\left({\Gamma }_0\right)\mid \parallel \rho {\parallel }_{m,\alpha }<{\delta }_0\right\}$. We are going to prove the following main theorem:

Theorem 3.1. Assume that m ≥ 4, 0 < α < 1, Ω₀ is of class C^m,α. Given that all initial data in (68) are in h^m,α(Ω₀) and satisfy the corresponding boundary conditions (66) and (67). Then there exist T > 0 such that the system (49)–(68) has a unique solution ρ ∈ L^∞([0, T]), h^m,α(Γ₀)) ∩ C([0, T]), h^m−3,α(Γ₀)) ∩ C((0, T], h^m,α(Γ₀)) ∩ C¹((0, T]), h^m−3,α(Γ₀)), $\tilde{G}$, $\tilde{I}$, $\tilde{L}$, $\tilde{H}$, $\tilde{S}$, $\tilde{\beta }$, $\tilde{\sigma }$, $\tilde{R}$, and $\tilde{F}$ all belong to L^∞([0, T]), h^m−1,α(Ω₀)) ∩ C([0, T]), h^m−3,α(Ω₀)) ∩ C((0, T], h^m−1,α(Ω₀)) ∩ C¹((0, T]), h^m−3,α(Ω₀)).

3.2. Proof of the main theorem.

To prove the main theorem, we are going to use similar steps to that in [56]. We first rewrite the system (49)–(68) as a quasilinear abstract evolution equation in Banach spaces and then apply the analytic semigroup theory [55].

Lemma 3.2. If ρ ∈ h^m,α(Γ₀), $\tilde{R}$, $\tilde{\beta }\in h^{k-2,\alpha }\left({\Omega }_0\right)$, then the problem

$$\{\begin{array}{l}-D_G{\mathcal{L}}_{\rho }\tilde{G}={\tilde{G}}_0-\left(E_{G0}+S_I\tilde{I}\right)\tilde{G}in{\text{ Ω}}_0,\hfill \\ -D_I{\mathcal{L}}_{\rho }\tilde{I}=\frac{{\sigma }_0{\tilde{G}}^2\tilde{\beta }}{\alpha +{\tilde{G}}^2}-k\tilde{I}-{\lambda }_{RI}\tilde{R}in{\text{ Ω}}_0;\hfill \\ {\mathcal{D}}_{\rho }\tilde{G}+S_{\rho }{\alpha }_G\left(\tilde{G}-G_0\right)=0,{\mathcal{D}}_{\rho }\tilde{I}+S_{\rho }{\alpha }_I\left(\tilde{I}-I_0\right)=0on{\text{ Γ}}_0,\hfill \\ \frac{\partial \tilde{G}}{\partial n}=\frac{\partial \tilde{I}}{\partial n}=0on{\text{ Γ}}_M\hfill \end{array}$$ (64)

has a unique solution $\tilde{G}$, $\tilde{I}\in h^{k,\alpha }\left({\Omega }_0\right)$. Moreover there are operators ${\mathcal{U}}_j(\rho )$, ${\mathcal{V}}_j(\rho )$, j = 1, 2 such that

$${\mathcal{U}}_j(\rho )\in \underset{2\le k\le m}{\cap }L\left(h^{k-2,\alpha }\left({\Omega }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right),{\mathcal{V}}_j(\rho )\in \cap L\left(h^{k-2,\alpha }\left({\Omega }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right)$$

and

$$\tilde{G}={\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta },\tilde{I}={\mathcal{U}}_2(\rho )\tilde{R}+{\mathcal{V}}_2(\rho )\tilde{\beta }.$$ (65)

Furthermore we have

$$\begin{gathered} {\mathcal{U}}_j\in C^{\infty }\left(h^{m,\alpha }\left({\Gamma }_0\right),L\left(h^{k-2,\alpha }\left({\Omega }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right)\right), \\ {\mathcal{V}}_j\in C^{\infty }\left(h^{m,\alpha }\left({\Gamma }_0\right),L\left(h^{k-2,\alpha }\left({\Omega }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right)\right). \end{gathered}$$

Proof. The proof follows from Theorem 5.1 of chapter 8 in [54]. □

Lemma 3.3. If ρ ∈ h^m,α(Γ₀), $\tilde{h}\in h^{k-2,\alpha }\left({\Omega }_0\right)$, ϕ ∈ h^m,α(Γ₀), then the problem

$$\{\begin{array}{ll}{\mathcal{L}}_{\rho }\tilde{\sigma }=\tilde{h}\hfill & in{\text{ Ω}}_0\hfill \\ \tilde{\sigma }=\tilde{\varphi }\hfill & on{\text{ Γ}}_0\hfill \\ \tilde{\sigma }=0\hfill & on{\text{ Γ}}_M\hfill \end{array}$$ (66)

has a unique solution $\tilde{\sigma }\in h^{k,\alpha }\left({\Omega }_0\right)$. Moreover there are operators $\mathcal{U}(\rho )$, $\mathcal{V}(\rho )$, j = 1, 2 such that

$$\mathcal{U}(\rho )\in \underset{2\le k\le m}{\cap }L\left(h^{k-2,\alpha }\left({\Omega }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right),\mathcal{V}(\rho )\in \cap L\left(h^{k,\alpha }\left({\Gamma }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right)$$

and

$$\tilde{\sigma }=\mathcal{U}(\rho )\tilde{h}+\mathcal{V}(\rho )\tilde{\varphi }.$$ (67)

Furthermore we have

$$\begin{gathered} \mathcal{U}\in C^{\infty }\left(h^{m,\alpha }\left({\Gamma }_0\right),L\left(h^{k-2,\alpha }\left({\Omega }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right)\right), \\ \mathcal{V}\in C^{\infty }\left(h^{m,\alpha }\left({\Gamma }_0\right),L\left(h^{k,\alpha }\left({\Gamma }_0\right),h^{k,\alpha }\left({\Omega }_0\right)\right)\right). \end{gathered}$$

Lemma 3.4. The curvature κ_ρcan be written as

$${\kappa }_{\rho }(\omega )=\mathcal{L}(\rho )\rho +\mathcal{K}(\rho )\rho ,$$ (68)

where $\mathcal{L}(\rho )$ is a second order elliptic operator with coefficients being functions of ρ and its first order derivative and $\mathcal{K}(\rho )$ is a first order nonlinear operator. Moreover

$$\begin{gathered} \mathcal{L}\in C^{\infty }\left(h^{m,\alpha }\left({\Gamma }_0\right),L\left(h^{k,\alpha }\left({\Gamma }_0\right),h^{k-2,\alpha }\left({\Gamma }_0\right)\right)\right), \\ \mathcal{K}\in C^{\infty }\left(h^{m,\alpha }\left({\Gamma }_0\right),L\left(h^{k,\alpha }\left({\Gamma }_0\right),h^{m-1,\alpha }\left({\Gamma }_0\right)\right)\right). \end{gathered}$$

Proof. The proof follows from (65) and (44). □

Using (51), (64), (70),(72) and (73), we have

$$\tilde{\sigma }=\gamma \mathcal{V}(\rho )\mathcal{L}(\rho )\rho +{\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F});$$ (69)

where

$${\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})=\gamma \mathcal{V}(\rho )\mathcal{K}(\rho )\rho -\mathcal{U}(\rho )\left({\nabla }_{\rho }\tilde{S}\cdot \left({\chi }_G{\nabla }_{\rho }\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta }\right)\right)\right)+\mathcal{U}(\rho ){\tilde{F}}_1\left(\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta }\right),\tilde{S},\tilde{\beta },\left({\mathcal{U}}_2(\rho )\tilde{R}+{\mathcal{V}}_2(\rho )\tilde{\beta }\right),\tilde{L},\tilde{F}\right).$$ (70)

We define the following operators:

$$\mathcal{B}(\rho )v=\gamma {\mathcal{D}}_{\rho }\mathcal{V}(\rho )\mathcal{L}(\rho )v\text{ for }v\in h^{m,\alpha }\left({\Gamma }_0\right);$$ (71)

$$\mathcal{C}(\rho ,u)v=(\mathcal{B}(\rho )v)M_{\rho }(y)\cdot {\nabla }_{\rho }u\text{ for }u\in h^{k,\alpha }\left({\Omega }_0\right),v\in h^{m,\alpha }\left({\Gamma }_0\right);$$ (72)

$$\mathcal{E}(\rho ,u)v=\left(\mathcal{B}(\rho )v+\gamma {\nabla }_{\rho }\mathcal{V}(\rho )\mathcal{L}(\rho )v\right)M_{\rho }(y)\cdot {\nabla }_{\rho }u\text{ for }u\in h^{k,\alpha }\left({\Omega }_0\right),v\in h^{m,\alpha }\left({\Gamma }_0\right).$$ (73)

Equations (53)–(62) can be written as

$$\frac{d\tilde{L}}{dt}-D_L{\mathcal{L}}_{\rho }\tilde{L}-\mathcal{C}(\rho ,\tilde{L})\rho ={\mathcal{G}}_1(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F});$$ (74)

where

$${\mathcal{G}}_1(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})=\left({\mathcal{D}}_{\rho }{\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{L}-k_L\tilde{R}\tilde{L};$$ (75)

$$\frac{d\tilde{H}}{dt}-D_H{\mathcal{L}}_{\rho }\tilde{H}-\mathcal{C}(\rho ,\tilde{H})\rho ={\mathcal{G}}_2(\tilde{R},\tilde{\beta },\tilde{L},\tilde{H},\tilde{S},\tilde{F});$$ (76)

where

$${\mathcal{G}}_2(\tilde{R},\tilde{\beta },\tilde{L},\tilde{H},\tilde{S},\tilde{F})=\left({\mathcal{D}}_{\rho }{\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{H}-k_L\tilde{R}\tilde{H};$$ (77)

$$\frac{d\tilde{R}}{dt}-D_R{\mathcal{L}}_{\rho }\tilde{R}-\mathcal{C}(\rho ,\tilde{R})\rho ={\mathcal{G}}_3(\tilde{R},\tilde{\beta },\tilde{L},\tilde{H},\tilde{S},\tilde{F});$$ (78)

where

$${\mathcal{G}}_3(\tilde{R},\tilde{\beta },\tilde{L},\tilde{H},\tilde{S},\tilde{F})=\left({\mathcal{D}}_{\rho }{\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{R}+{\tilde{F}}_3\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta },{\mathcal{U}}_2(\rho )\tilde{R}+{\mathcal{V}}_2(\rho )\tilde{\beta },\tilde{L},\tilde{H},\tilde{R}\right);$$ (79)

$$\frac{d\tilde{\beta }}{dt}-D_{\beta }{\mathcal{L}}_{\rho }\tilde{\beta }-\mathcal{E}(\rho ,\tilde{\beta })\rho ={\mathcal{G}}_4(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F});$$ (80)

where

$${\mathcal{G}}_4(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})=\left(\left({\mathcal{D}}_{\rho }+{\nabla }_{\rho }\right){\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{\beta }+{\tilde{F}}_2\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta },{\mathcal{U}}_2(\rho )\tilde{R}+{\mathcal{V}}_2(\rho )\tilde{\beta },\tilde{L},\tilde{R}\right)-\tilde{\beta }{\nabla }_{\rho }\tilde{S}\cdot \left({\chi }_G{\nabla }_{\rho }\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta }\right)\right);$$ (81)

$$\frac{d\tilde{S}}{dt}-D_S{\mathcal{L}}_{\rho }\tilde{S}-\mathcal{E}(\rho ,\tilde{\beta })\rho ={\mathcal{G}}_5(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F});$$ (82)

where

$${\mathcal{G}}_5(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})=\left(\left({\mathcal{D}}_{\rho }+{\nabla }_{\rho }\right){\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{S}+{\tilde{F}}_4\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta },{\mathcal{U}}_2(\rho )\tilde{R}+{\mathcal{V}}_2(\rho )\tilde{\beta },\tilde{L},\tilde{R}\right)-\left({\chi }_G+\tilde{S}\right){\nabla }_{\rho }\tilde{S}\cdot \left({\chi }_G{\nabla }_{\rho }\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta }\right)\right);$$ (83)

$$\frac{d\tilde{F}}{dt}-D_F{\mathcal{L}}_{\rho }\tilde{F}-\mathcal{E}(\rho ,\tilde{F})\rho ={\mathcal{G}}_6(\tilde{R},\tilde{\beta },\tilde{L},\tilde{H},\tilde{S},\tilde{F});$$ (84)

where

$${\mathcal{G}}_6(\tilde{R},\tilde{\beta },\tilde{L},\tilde{H},\tilde{S},\tilde{F})=\left(\left({\mathcal{D}}_{\rho }+{\nabla }_{\rho }\right){\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})\right){\mathcal{M}}_{\rho }(y)\cdot {\nabla }_{\rho }\tilde{F}+{\tilde{F}}_5\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta },{\mathcal{U}}_2(\rho )\tilde{R}+{\mathcal{V}}_2(\rho )\tilde{\beta },\tilde{L},\tilde{H},\tilde{R}\right)-\tilde{F}{\nabla }_{\rho }\tilde{S}\cdot \left({\chi }_G{\nabla }_{\rho }\left({\mathcal{U}}_1(\rho )\tilde{R}+{\mathcal{V}}_1(\rho )\tilde{\beta }\right)\right);$$ (85)

$$\frac{d\tilde{\rho }}{dt}+\mathcal{B}(\rho )\rho ={\mathcal{G}}_7(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F});$$ (86)

where

$${\mathcal{G}}_7(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F})=-{\mathcal{D}}_{\rho }{\mathcal{A}}_{1,\rho }(\tilde{R},\tilde{\beta },\tilde{L},\tilde{S},\tilde{F}).$$ (87)

Let $U=(\tilde{L},\tilde{H},\tilde{R},\tilde{\beta },\tilde{S},\tilde{F},\rho )$, $\mathcal{E}(U)=\left({\mathcal{E}}_1,{\mathcal{E}}_2,{\mathcal{E}}_3,{\mathcal{E}}_4,{\mathcal{E}}_5,{\mathcal{E}}_6,{\mathcal{E}}_7\right)$ and

$$\mathcal{A}(U)=\left(\begin{array}{ccccccc}-D_L{\mathcal{L}}_{\rho }& 0& 0& 0& 0& 0& -\mathcal{C}(\rho ,\tilde{L})\\ 0& -D_H{\mathcal{L}}_{\rho }& 0& 0& 0& 0& -\mathcal{C}(\rho ,\tilde{H})\\ 0& 0& -D_R{\mathcal{L}}_{\rho }& 0& 0& 0& -\mathcal{C}(\rho ,\tilde{R})\\ 0& 0& 0& -D_{\beta }{\mathcal{L}}_{\rho }& 0& 0& -\mathcal{E}(\rho ,\tilde{\beta })\\ 0& 0& 0& 0& -D_S{\mathcal{L}}_{\rho }& 0& -\mathcal{E}(\rho ,\tilde{S})\\ 0& 0& 0& 0& 0& -D_F{\mathcal{L}}_{\rho }& -\mathcal{E}(\rho ,\tilde{F})\\ 0& 0& 0& 0& 0& 0& -\mathcal{B}(\rho )\end{array}\right)$$ (88)

Then

$$\{\begin{array}{l}\frac{d{U}^T}{dt}+\mathcal{A}(U)U^T=\mathcal{E}{(U)}^T,t>0\hfill \\ {U|}_{t=0}=U_0;\hfill \end{array}$$ (89)

where U₀ = (L₀(x), H₀(x), R₀(x), β₀(x), S₀(x), F₀(x), 0).

Now we define Banach spaces by

$${\mathcal{H}}_1^m=\left\{\tilde{L}\in h^{m,\alpha }\left({\Omega }_0\right)\mid {\mathcal{D}}_{\rho }\tilde{L}+S_{\rho }{\alpha }_L\left(\tilde{L}-L_0\right)=0{\text{ on Γ}}_0,\frac{\partial \tilde{L}}{\partial n}=0{\text{ on Γ}}_M\right\};$$

$${\mathcal{H}}_2^m=\left\{\tilde{H}\in h^{m,\alpha }\left({\Omega }_0\right)\mid {\mathcal{D}}_{\rho }\tilde{H}+S_{\rho }{\alpha }_H\left(\tilde{H}-H_0\right)=0{\text{ on Γ}}_0,\frac{\partial \tilde{L}}{\partial n}=0{\text{ on Γ}}_M\right\};$$

$${\mathcal{H}}_3^m=\left\{\tilde{R}\in h^{m,\alpha }\left({\Omega }_0\right)\mid {\mathcal{D}}_{\rho }\tilde{R}=0{\text{ on Γ}}_0,\frac{\partial \tilde{R}}{\partial n}=0{\text{ on Γ}}_M\right\};$$

$${\mathcal{H}}_4^m=\left\{\tilde{\beta }\in h^{m,\alpha }\left({\Omega }_0\right)\mid {\mathcal{D}}_{\rho }\tilde{\beta }+S_{\rho }{\alpha }_{\beta }\left(\tilde{\beta }-{\beta }_0\right)=0{\text{ on Γ}}_0,\frac{\partial \tilde{\beta }}{\partial n}=0{\text{ on Γ}}_M\right\};$$

$${\mathcal{H}}_5^m=\left\{\tilde{S}\in h^{m,\alpha }\left({\Omega }_0\right)\mid {\mathcal{D}}_{\rho }\tilde{S}=0{\text{ on Γ}}_0,\frac{\partial \tilde{S}}{\partial n}+{\alpha }_S\left(\tilde{S}-S_0\right)=0{\text{ on Γ}}_M\right\};$$

$${\mathcal{H}}_6^m=\left\{\tilde{F}\in h^{m,\alpha }\left({\Omega }_0\right)\mid {\mathcal{D}}_{\rho }\tilde{F}=0{\text{ on Γ}}_0,\frac{\partial \tilde{F}}{\partial n}=0{\text{ on Γ}}_M\right\}.$$

$$\mathcal{X}=\left(\underset{j=1}{\overset{6}{\otimes }}{\mathcal{H}}_j^{m-3}\right)\otimes h^{m-3,\alpha }\left({\Gamma }_0\right);$$

$${\mathcal{X}}_0=\left(\underset{j=1}{\overset{6}{\otimes }}{\mathcal{H}}_j^{m-1}\right)\otimes h^{m,\alpha }\left({\Gamma }_0\right);$$

$$\mathcal{Y}=\left(\underset{j=1}{\overset{6}{\otimes }}{\mathcal{H}}_j^{m-2}\right)\otimes h^{m-2,\alpha }\left({\Gamma }_0\right);$$

$${\mathcal{O}}_{{\delta }_0}=\left(\underset{j=1}{\overset{6}{\otimes }}{\mathcal{H}}_j^{m-1}\right)\otimes h_{{\delta }_0}^{m,\alpha }\left({\Gamma }_0\right).$$

Obviously we have ${\mathcal{X}}_0\hookrightarrow \mathcal{X}$.We denote by $\mathit{H}\left({\mathcal{X}}_0,\mathcal{X}\right)$ the set of all linear operators $\mathcal{A}$ such that $\text{dom}(\mathcal{A})={\mathcal{X}}_0$ and $-\mathcal{A}$ generates an analytic semigroup.

Lemma 3.5. Let$U\in {\mathcal{O}}_{{\delta }_0}$, then$\mathcal{A}(U)\in \mathit{H}\left({\mathcal{X}}_0,\mathcal{X}\right)$. Moreover

$$\mathcal{A}\in C^{\infty }\left({\mathcal{O}}_{{\delta }_0},H\left({\mathcal{X}}_0,\mathcal{X}\right)\right).$$

Proof. By adapting the proofs of Lemma 4.1 and Lemma 4.2 in [56], we have the proof of the lemma. □

Applying Theorem 8.1.1 and Theorem 8.3.4 in [55], we obtain the following theorem

Theorem 3.6. Let m ≥ 4, 0 < α < 1, given that $U_0\in {\mathcal{O}}_{{\delta }_0}$,then there exists T > 0 such that the problem (94) has a unique solution $U\in L^{\infty }\left([0,T],{\mathcal{X}}_0\right)\cap C([0,T],\mathcal{X})\cap C\left((0,T],{\mathcal{X}}_0\right)\cap C^1((0,T],\mathcal{X})$.

The main Theorem 3.1 follows easily from the above theorem.

4. Conclusion and discussion

Atherosclerosis is a leading cause of death in the United States and worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. As atherosclerosis is a cardiovascular condition that affects critical circulatory systems, studying human atheroma poses logistical and ethical problems, as access to live atherosclerotic tissue is limited and disturbances risk triggering plaque rupture. As a result, the appropriate framework to consider emergent dynamical behavior of this type of disease is mathematical and computational modeling. The existing mathematical models [9, 10, 52, 57] that describe the growth of a plaque in the artery recognize the critical role of low density lipoprotein (LDL) and high-density lipoprotein (HDL), in determining whether a plaque, once formed, will grow or shrink. Making matters worse, nearly 21 million Americans have diabetes. In part because diabetes increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. Past work has shown that hyperglycemia and insulin resistance alter function of multiple cell types, including endothelium, smooth muscle cells and platelets, indicating the extent of vascular disarray in this disease. Although the pathophysiology of diabetic vascular disease is generally understood, there is no mathematical model to date that includes the effect of diabetes on plaque growth. In this work we proposed a mathematical model for diabetic atherosclerosis. In the model, we established governing equations to model the mechanism of how diabetes impairs endothelium-dependent (nitric oxide-mediated) vasodilation before the formation of atheroma and mechanism of how arteries affected by diabetes have altered vascular smooth muscle cell function. The proposed model would enhance our understanding of diabetic atherosclerosis as a dynamical process. A comprehensive program of mathematical modeling and simulation will provide many benefits. Principally, it provides a framework for therapeutic hypothesis generation and for in silico drug target identification, with the potential to streamline the drug development pipeline. The mathematical model will also be used to confirm the recent guidelines issued by the American Heart Association and American Diabetes Association.

In order to guide numerical simulation of the proposed model, it is important to study the wellposedness of the model. We proved the local existence and uniqueness of solution to the proposed model. The methodology is to use Hanzawa transformation [53] to reduce the free boundary to a fixed boundary and reduce the system of PDEs to an abstract system of evolution equations, and then apply the analytic semigroup theory.

There are many questions to be answered regarding this model for diabetic atherosclerosis. The main question is: what is the risk to plague growth associated with hyperglycemia and high cholesterol? To address this question one can study how (G₀, I₀, L₀, H₀) affect the evolution of the free boundary of the plaque by rigorous mathematical analysis and numerical simulation of the mathematical model.

Although a limited set of variables were included in the model, these variables are most important in describing the dynamic process of diabetic atheroscleroses. More variables such as oxidized LDL and HDL,MCP-1, T-cells [10, 46] and PDGF [42] can be included. Animal studies of diabetic atherosclerosis do exist for mouse, rabbit and pig [15,38]. As atherosclerosis is a cardiovascular condition that affects critical circulatory systems, studying human atheroma poses logistical and ethical problems, as access to live atherosclerotic tissue is limited and disturbances risk triggering plaque rupture. Consequently, data are limited. As a result, establishing biologically relevant kinetic parameters that can be used to simulate pathway dynamics is challenging, and comprehensive parameterizations and validation of the mathematical model are difficult.