Existence of pulses for a reaction-diffusion system of blood coagulation

TL;DR

This paper investigates the existence of pulse solutions in a reaction-diffusion model of blood coagulation, establishing a link between pulse existence and the positivity of wave speed using topological methods.

Contribution

It demonstrates that pulse solutions exist if and only if the associated traveling wave speed is positive, providing a new criterion for pulse existence in blood coagulation models.

Findings

Pulse solutions exist if and only if the wave speed is positive.

The proof employs Leray-Schauder topological degree and a priori estimates.

The study advances understanding of blood coagulation dynamics in reaction-diffusion systems.

Abstract

The paper is devoted to the investigation of a reaction-diffusion system of equations describing the process of blood coagulation. Existence of pulses solutions, that is, positive stationary solutions with zero limit at infinity is studied. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray-Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.

Equations

$$\frac{\partial \mathbf{v}}{\partial t}=D\frac{{\partial }^2\mathbf{v}}{\partial x^2}+\mathbf{F}(\mathbf{v}),$$

$$\mathbf{F}(\mathbf{v})=\{\begin{array}{l}{F}_1(\mathbf{v})=k_1{v}_3{v}_6-h_1{v}_1\\ F_2(\mathbf{v})=k_2{v}_4{v}_5-h_2{v}_2\\ F_3(\mathbf{v})=k_3{v}_8({\rho }_3-v_3)-h_3{v}_3\\ F_4(\mathbf{v})=k_4{v}_8({\rho }_4-v_4)-h_4{v}_4\\ F_5(\mathbf{v})=k_5{v}_7({\rho }_5-v_5)-h_5{v}_5\\ F_6(\mathbf{v})=(k_6{v}_5+\overline{k_6}{v}_2)({\rho }_6-v_6)-h_6{v}_6\\ F_7(\mathbf{v})=k_7{v}_8({\rho }_7-v_7)-h_7{v}_7\\ F_8(\mathbf{v})=(k_8{v}_6+\overline{k_8}{v}_1)({\rho }_8-v_8)-h_8{v}_8\end{array}.$$

$$\mathcal{C}=\{\mathbf{v}=(v_1,...,v_8)\in {\mathbb{R}}_{+}^8|v_i\le {\rho }_i\text{ for }i\in \{3,...,8\}\}.$$

$$\forall \mathbf{v}\in \mathcal{C},\frac{\partial F_i}{\partial v_j}(\mathbf{v})\ge 0,\forall i\ne j.$$

$$P(T)\equiv (k_8{\varphi }_6(T)+\overline{k_8}{\varphi }_1(T))(T^0-T)-h_{8}T=0.$$

$$\{\begin{array}{l}P\text{ possesses exactly three non negative zeros: }{T}^{+}=0<\bar{T}<T^{-}\\ P^{\prime }(T^{+})<0,P^{\prime }(\bar{T})>0,P^{\prime }(T^{-})<0.\end{array}$$

$$Q(0)=d<0,Q^{\prime }(\bar{T})>0,Q^{\prime }(T^{-})<0.$$

$$\{\begin{array}{l}D{\mathbf{u}}^{\prime \prime }+c{\mathbf{u}}^{\prime }+\mathbf{F}(\mathbf{u})=0,\\ \mathbf{u}(\pm \infty )={\mathbf{w}}^{\pm }.\end{array}$$

$$\{\begin{array}{l}D{\mathbf{w}}^{\prime \prime }+\mathbf{F}(\mathbf{w})=0,\\ {\mathbf{w}}^{\prime }(0)=0,\mathbf{w}(+\infty )=0,\\ {\mathbf{w}}^{\prime }<0\text{ for }x>0.\end{array}$$

$$g(s)\ge 0\text{ for }s\ge 0,\text{ Supp }g\subset (\bar{T},T^{-}).$$

$$F_8^{\tau }(\mathbf{v})=(k_8{v}_6+\overline{k_8}{v}_1)(T^0-T)-h_{8}T+\tau g(T).$$

$$F_8^{\tau }(\mathbf{v})=(k_8({\alpha }^{\tau }{v}_6+{\beta }^{\tau }{\varphi }_6(T))+\overline{k_8}({\alpha }^{\tau }{v}_1+{\beta }^{\tau }{\varphi }_1(T))(T^0-T)-h_{8}T+{\tau }_{1}g(T),$$

$${\alpha }^{\tau }=\frac{1-\tau }{1-{\tau }_1},{\beta }^{\tau }=\frac{\tau -{\tau }_1}{1-{\tau }_1}.$$

$$\begin{array}[]{c c c}\begin{cases}\alpha^{\tau}=\frac{1-\tau}{1-\tau_{1}}\text{ if }\tau\geq\tau_{1}\\ \alpha^{\tau}=1\text{ if }\tau<\tau_{1}\end{cases},&\begin{cases}\beta^{\tau}=\frac{\tau-\tau_{1}}{1-\tau_{1}}\text{ if }\tau\geq\tau_{1}\\ \beta^{\tau}=0\text{ if }\tau<\tau_{1}\end{cases},&\begin{cases}\gamma^{\tau}=\tau_{1}\text{ if }\tau\geq\tau_{1}\\ \gamma^{\tau}=\tau\text{ if }\tau<\tau_{1}\\ \end{cases}.\end{array}$$

$$F_8^{\tau }(\mathbf{v})={\alpha }^{\tau }{F}_8(\mathbf{v})+{\beta }^{\tau }P(T)+{\gamma }^{\tau }g(T).$$

$$\forall \mathbf{v}\in \mathcal{C},\frac{\partial F_i^{\tau }}{\partial v_j}(\mathbf{v})\ge 0,\forall i\ne j.$$

$$\frac{\partial F_8^{\tau }}{\partial v_j}(\mathbf{v})={\alpha }^{\tau }\frac{\partial F_8}{\partial v_j}(\mathbf{v})\ge 0,\text{ for }j\in \{1,...,7\}.$$

$$F_8^{\tau }(\mathbf{\varphi }(T),T)={\alpha }^{\tau }{F}_8(\mathbf{\varphi }(T),T)+{\beta }^{\tau }P(T)+{\gamma }^{\tau }g(T)=P(T)+{\gamma }^{\tau }g(T)=P^{\tau }(T).$$

$${\mathbf{F}}^{\tau }(\mathbf{v})=0⟺\mathbf{v}=(\varphi (T),T)\text{ and }{P}^{\tau }(T)=0.$$

$$F_i(\mathbf{\varphi }(T),T)=0$$

$$\frac{d{F}_i}{dT}(\varphi (T),T)=\sum _{j=1}^7\frac{\partial F_i}{\partial v_j}(\varphi (T),T){\varphi }_j^{\prime }(T)+\frac{\partial F_i}{\partial v_8}(\varphi (T),T)=0.$$

$$({\mathcal{J}}^{\tau }({\mathbf{w}}^{\ast })\mathbf{q}{)}_i=\sum _{j=1}^7\frac{\partial F_i}{\partial v_j}({\mathbf{w}}^{\ast })q_j+\frac{\partial F_i}{\partial v_8}({\mathbf{w}}^{\ast })q_8=0.$$

$$\frac{\partial F_3}{\partial v_3}({\mathbf{w}}^{\ast })q_3+\frac{\partial F_3}{\partial T}({\mathbf{w}}^{\ast })q_8=0.$$

$$q_3=-\frac{\partial F_3}{\partial v_3}({\mathbf{w}}^{\ast })/\frac{\partial F_3}{\partial T}({\mathbf{w}}^{\ast })q_8.$$

$$\frac{d{F}_3}{dT}({\mathbf{w}}^{\ast })=\frac{\partial F_3}{\partial v_3}({\mathbf{w}}^{\ast }){\varphi }_3^{\prime }(T)+\frac{\partial F_3}{\partial v_8}({\mathbf{w}}^{\ast }).$$

$$({\mathcal{J}}^{\tau }({\mathbf{w}}^{\ast })\mathbf{q}{)}_3=0⟺q_3={\varphi }_3^{\prime }(T^{\ast })q_8.$$

$$({\mathcal{J}}_8^{\tau }({\mathbf{w}}^{\ast })\mathbf{q}{)}_8=\frac{\partial F_8^{\tau }}{\partial v_8}({\mathbf{w}}^{\ast })q_8+\frac{\partial F_8^{\tau }}{\partial v_1}({\mathbf{w}}^{\ast })q_1+\frac{\partial F_8^{\tau }}{\partial v_6}({\mathbf{w}}^{\ast })q_6=$$

$$(\frac{\partial F_8^{\tau }}{\partial v_8}({\mathbf{w}}^{\ast })+\frac{\partial F_8^{\tau }}{\partial v_1}({\mathbf{w}}^{\ast }){\varphi }_1^{\prime }(T^{\ast })+\frac{\partial F_8^{\tau }}{\partial v_6}({\mathbf{w}}^{\ast }){\varphi }_6^{\prime }(T^{\ast }))q_8=P^{\prime }(T^{\ast })q_8.$$

$$P^{\prime }(T^{\ast }),-h_1,-h_2,-k_3{T}^{\ast }-h_3,-k_4{T}^{\ast }-h_4,-h_5,-h_6,-k_7{T}^{\ast }-h_7,$$

$$\{\begin{array}{l}{E}^1=\{\mathbf{w}\in {\mathcal{C}}^{2+\alpha }({\mathbb{R}}_{+}),{\mathbf{w}}^{\prime }(0)=0\},\\ E^2={\mathcal{C}}^{\alpha }({\mathbb{R}}_{+}).\end{array}$$

Full text

Existence of pulses for a reaction-diffusion

system of blood coagulation

N. Ratto1, M. Marion1, V. Volpert2,3,4,5

1Institut Camille Jordan, UMR 5585 CNRS, Ecole Centrale de Lyon

69134 Ecully, France

2Institut Camille Jordan, UMR 5585 CNRS, University Lyon 1

69622 Villeurbanne, France

3 INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan

43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France

4 Peoples’ Friendship University of Russia (RUDN University)

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

5 Poncelet Center, UMI 2615 CNRS, 11 Bolshoy Vlasyevskiy, 119002 Moscow

Russian Federation

Abstract. The paper is devoted to the investigation of a reaction-diffusion system of equations describing the process of blood coagulation. Existence of pulses solutions, that is, positive stationary solutions with zero limit at infinity is studied. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray-Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.

Key words: reaction-diffusion system, blood coagulation, existence of pulses, Leray-Schauder method

AMS subject classification: 35K57

1. Introduction

Hemostasis is a physiological process which aims to prevent bleeding in the case of blood vessel damage. It includes vasoconstriction, platelet plug formation and blood coagulation in plasma with the formation of fibrin clot. In this work we will focus on the blood coagulation process. A malfunction in this process can lead to thrombosis or to various bleeding disorders. The process of blood coagulation has three main stages: initiation, amplification and clot growth arrest. They are determined by chemical reactions in plasma between different proteins (blood factors), among which the most important role is played by thrombin. Thrombin is an enzyme catalyzing the conversion of the fibrinogen into the fibrin polymer which forms the clot.

In the process of blood coagulation initiated by the vessel wall damage (extrinsic pathway), the initial quantity of thrombin is produced due to the interaction of the tissue factor (TF) with the activated factors VIIa. During the amplification phase, there is a positive feedback loop of thrombin production through the activation of the factors V, VIII, IX, X and XI. It can be noted that hemophilia is characterized by the lack of factor VIII, IX or XI. Finally, clot growth is stopped by the reaction of antithrombin with thrombin, by the protein C pathway, and due to the blood flow removing blood factors from the clot.

The amplification phase of blood coagulation can start only if the amount of thrombin produced during the initiation phase exceeds certain threshold. In this work we will show that this critical thrombin concentration is determined by a particular solution (pulse) of the reaction-diffusion system of equations describing the coagulation cascade. On the other hand, the amplification phase can be described as a reaction-diffusion wave [2], [4], [5], [9]. The main result of this work affirms that the pulse solution exists if and only if the wave speed is positive. Thus, we obtain two conditions of blood coagulation: the wave speed should be positive providing the existence of the pulse solution; the distributions of the concentrations of blood factors after the initiation phase should be greater than those for the pulse solution.

We consider the reaction-diffusion system of equations [1]:

[TABLE]

where $\mathbf{v}=(v_{1},...,v_{8})$ and $\mathbf{F}=(F_{1},...,F_{8})$,

[TABLE]

Parameters $(k_{i})_{i}$, $(h_{i})_{i}$ and $(\overline{k_{i}})_{i}$ and $(\rho_{i})_{i}$ are positive constants. The matrix $D=diag(D_{i})$ is a diagonal matrix with positive diagonal elements $D_{i}$.

In this system, $v_{3},v_{4},v_{5},v_{6}$ and $v_{7}$ denote, respectively, the concentrations of the activated Factors $Va,VIIIa,IXa,Xa$ and $XIa$, $v_{8}$ is the concentration of activated Factor $IIa$ (thrombin), $v_{1}$ and $v_{2}$ are the concentrations of prothrombinase and intrinsic tenase complexes. The constants $k_{i}$ and $\bar{k_{i}}$ are the activation rates of the corresponding factors by other factors or complexes, while the constants $h_{i}$ are the rates their inhibition. The constants $D_{i}$ are the diffusion coefficients of each factor. The thrombin concentration, $v_{8}$, has a major role in the coagulation process. It will also have a particular importance in the mathematical study. We will use notation $T=v_{8}$ and $T^{0}=\rho_{8}$.

Let us introduce the set $\mathcal{C}$:

[TABLE]

Then the system (1.1) is a monotone system on $\mathcal{C}$ , that is:

[TABLE]

Note that $\mathbf{v}=\mathbf{0}$ is a zero of $\mathbf{F}$. In order to determine other zeros, let us express $v_{i}$ through $T$ from the equations $F_{i}(\mathbf{v})=0$, $i=1,...,7$: $v_{i}=\phi_{i}(T)$ (see Appendix A for the explicit form of these functions). Substituting them in the equation $F_{8}(\mathbf{v})=0$, we obtain:

[TABLE]

It can be directly verified that $P(T)$ is a rational fraction that takes the form $P(T)=\frac{TQ(T)}{S(T)}$, where $Q(T)$ and $S(T)$ are third-order polynomials. Moreover $Q(T)=aT^{3}+bT^{2}+cT+d$ with $a<0$, and $S(T)>0$ for $T\geq 0$ (the reader is referred to the appendix).

Hereafter, we will assume that $P$ satisfies the following properties:

[TABLE]

In view of the form of the rational fraction $P(T)$, the above conditions mean that the third order polynomial $Q$ with negative leading coefficient $a$ has exactly two positive roots. Also it is easy to check that the conditions on the derivatives in (1.6) also read

[TABLE]

In particular ${Q}(T)>0$ for some $T>0$. This assumption is biologically justified, and we will return to it in the discussion (Section 5).

The non negative roots $T^{+}$, $\bar{T}$ and $T^{-}$ correspond to three zeros of $\mathbf{F}$: $\mathbf{w}^{+}=(\phi(T^{+}),T^{+})=\mathbf{0}$, $\bar{\mathbf{w}}=(\phi(\bar{T}),\bar{T})$ and $\mathbf{w}^{-}=(\phi(T^{-}),T^{-})$. Moreover we have $\mathbf{w}^{+}=\mathbf{0}<\bar{\mathbf{w}}<\mathbf{w}^{-}$, where the inequalities between the vectors are understood component-wise.

As stated before, thrombin propagation in blood plasma is described by a traveling wave solution of the system (1.1). A wave solution of (1.1) is a solution that can be written as $\mathbf{v}(x,t)=\mathbf{u}(x-ct)$ where the wave speed $c\in{\mathbb{R}}$ is unknown. Hence we look for a function $\mathbf{u}$ and a constant $c$ that are solutions of the problem:

[TABLE]

Under the conditions (1.4) and (1.6), the problem (1.8) possesses a unique solution (up to some translation in space for $u$). This solution is a monotonically decreasing vector-function. This results are presented in [8].

Biologically, it has been noted that the amplification of thrombin generation occurs if the amount of thrombin produced during the initiation phase reaches a certain threshold. We will show that this threshold is a pulse of the stationary system, that is a function $\mathbf{w}:{\mathbb{R}}\rightarrow{\mathbb{R}}^{8}$ that satisfies the following problem:

[TABLE]

The link between the solutions of the wave problem (1.8) and the pulse problem (1.9) is given by the main result of this work:

Theorem 1.1**.**

Under the condition (1.6) the problem (1.9) has a solution if and only if the wave speed $c$ in the problem (1.8) is positive.

The proof of the theorem mainly relies on the Leray-Schauder method and some homotopy arguments. Therefore, we will introduce some appropriate homotopy deformation in Section 2. In Section 3 we obtain a priori estimates of solutions in some weighed Hölder spaces using positivity of the wave speed. These estimates are independent of the parameter of the homotopy. Hence, the value of the topological degree is preserved along the homotopy providing the existence of solutions. We finish the proof of the existence of solution of problem (1.9) in Section 4. Also in this section we will show that the problem has no solution if $c\leq 0$.

2. Homotopy

In order to prove the existence part of Theorem 1.1 we introduce in this section a homotopy deformation and highlight some its properties.

2.1. Description of the homotopy

The homotopy aims to modify continuously the function $\mathbf{F}$. We will only modify the last component $F_{8}(\mathbf{v})=(k_{8}v_{6}+\overline{k_{8}}v_{1})(T^{0}-T)-h_{8}T$ of (1.2) which depends on $T,v_{6}$ and $v_{1}$ into a new function depending only on $T$. Hence, the last equation will be independent from the other equations. In Lemma 4.2 (Section 4) we will show that this equation possesses a unique solution. For notation purpose the initial function, $\mathbf{F}$, corresponds to $\tau=0$ and is written $\mathbf{F}^{0}$. The homotopy functions reads $\mathbf{F}^{\tau}$, $\tau\in[0,1]$. The homotopy is defined in two steps, and we introduce some $\tau_{1}$ in $(0,1)$.

For the first step we introduce a smooth function $g$ that will be chosen below. We will construct the homotopy in such a way that the zeros of the function $\mathbf{F}^{\tau}$ do not change and coincide with the zeros of $\mathbf{F}$. Hence, assuming that $P$ satisfies the conditions (1.6), we impose that $g$ satisfies the following condition:

[TABLE]

Next, for $\tau\in(0,\tau_{1})$ we define the homotopy by the equality:

[TABLE]

At the second step of the homotopy we will deal with the variables $v_{1}$ and $v_{6}$. We will replace them by the functions $\phi_{1}(T)$ and $\phi_{6}(T)$ given by (A.1) and (A.6) (see also (1.5)) without modifying the zeros of $\mathbf{F}^{\tau}$. Hence, for $\tau\in(\tau_{1},1)$ we set:

[TABLE]

where

[TABLE]

It will be convenient to introduce notation:

[TABLE]

Then equalities (2.2) and (2.3) can be put together as follows:

[TABLE]

2.2. Preservation of some properties

Along the homotopy, and independently of the choice of $\tau_{1}$, some properties of $\mathbf{F}$ are preserved: the monotony, the stationary points and their stability. For the monotony we have the following result.

Lemma 2.1**.**

For all $\tau$ in $[0,1]$, the function $\mathbf{F}^{\tau}$ satisfies the monotony property on $\mathcal{C}$:

[TABLE]

Proof.

Since only the last component of $\mathbf{F}$ differs during the homotopy, we only need to verify the result for $F_{8}^{\tau}$. Using the expression (2.6) and the monotony of $\mathbf{F}$ given by (1.4) we have:

[TABLE]

∎

We will now show that the homotopy does not have any impact on the zeros of $\mathbf{F}^{\tau}$.

Lemma 2.2**.**

For all $\tau$ in $[0,1]$, the zeros of $\mathbf{F}^{\tau}$ are exactly the same as the zeros of $\mathbf{F}$.

Proof.

As before, since the homotopy only modifies the last component of $\mathbf{F}^{\tau}$, we only need to prove that the zeros of $F_{8}^{\tau}$ remain unchanged as $\tau$ varies. In order to find the zeros of $\mathbf{F}^{\tau}$ we use the same method as for the zeros of $\mathbf{F}$ in the Appendix A. Let $\mathbf{v}=(v_{1},...,v_{7},T)$ denote some zero of $\mathbf{F}^{\tau}$. Since $F_{i}^{\tau}=F_{i}$ for $1\leq i\leq 7$, the relations (A.1)-(A.7) remain unchanged. Also, in view of (1.5) and (2.6) we have:

[TABLE]

Hence the zeros of $\mathbf{F}^{\tau}$ are given by:

[TABLE]

Since the zeros of $P^{\tau}$ and $P$ coincide, we have $P^{\tau}(T)=0\iff P(T)=0$. Hence $\mathbf{F}^{\tau}(\mathbf{w})=\mathbf{0}\iff\mathbf{F}(\mathbf{w})=\mathbf{0}$. ∎

Therefore the zeros of $\mathbf{F}^{\tau}$ are $\mathbf{w}^{+},\bar{\mathbf{w}}$ and $\mathbf{w}^{-}$. Let us now investigate their stability. Let $\mathbf{w}^{*}$ refer to either one them. For this purpose we will need the following lemma.

Lemma 2.3**.**

For all $T\geq 0$ and $i=1,...,7$ we have the equalities:

[TABLE]

and

[TABLE]

The first equality follows from the definition of the functions $\phi_{i}$, $1\leq i\leq 7$ given in the Appendix A. The second one is obtained by differentiating the first one.

The next result concerns the stability of these zeros.

Proposition 2.4**.**

*The sign of the principal eigenvalue of the Jacobian $(\mathbf{F}^{\tau})^{\prime}(\mathbf{w}^{*})$ does not depend on $\tau\in[0,1]$. *

Proof.

Since the function $\mathbf{F}^{\tau}$ is monotone on $\mathcal{C}$ for every $\tau\in[0,1]$, then the Perron-Frobenius theorem guarantees that the principal eigenvalue, that is, the eigenvalue with maximal real part of the Jacobian matrix is real.

Let $\mathbf{w}^{*}=(w_{1}^{*},...,w_{7}^{*},T^{*})$, $T^{*}\geq 0$ be one of the zeros of $\mathbf{F}^{\tau}$. The stability of $\mathbf{w}^{*}$ is preserved during the homotopy if the principal eigenvalue of the Jacobian matrix does not change sign as $\tau$ varies. Let us denote by $\mathcal{J}^{\tau}=(\mathbf{F}^{\tau})^{\prime}$ the Jacobian matrix of $\mathbf{F}^{\tau}$. To prove that the principal eigenvalue does not change sign as $\tau$ varies, we will show that the determinant of $\mathcal{J}^{\tau}(\mathbf{w}^{*})$ is different from zero for all values of $\tau\in[0,1]$. To this end, let us check that $\text{Ker}\mathcal{J}^{\tau}(\mathbf{w}^{*})=\{\mathbf{0}\}$.

Let $\mathbf{q}\in\text{Ker}\mathcal{J}^{\tau}(\mathbf{w}^{*})$. Then we have:

[TABLE]

For $i=3$ we get:

[TABLE]

Hence,

[TABLE]

From Lemma 2.3 applied for $i=3$ it follows that

[TABLE]

Hence, applying (2.14) and (2.13) we have:

[TABLE]

We proceed with the other components in the following order: $i=4,7,5,2,$ $6,1$. This successively provides: $q_{i}=\phi_{i}^{\prime}(T^{*})q_{8},\;\;i=1,...,7.$ Then the last component of $\mathcal{J}^{\tau}(\mathbf{w}^{*}).q$ is given by the equality:

[TABLE]

[TABLE]

By virtue of (1.6), $P^{\prime}(T^{*})\neq 0$. Hence $q_{8}=0$, implying $\mathbf{q}=\mathbf{0}$ and concluding the proof. ∎

Proposition 2.4 affirms that stability of the stationary points $\mathbf{w}^{-}$, $\bar{\mathbf{w}}$ and $\mathbf{w}^{+}$ does not depend on $\tau$. Hence their stability is same for $\mathbf{F}$ and $\mathbf{F}^{\tau}$ for all $\tau\in[0,1]$. Consider $\tau=1$. The Jacobian matrix $\mathcal{J}^{1}$ can be reduced to a triangular matrix by taking the component in the order $i=8,3,4,7,5,2,6,1$. Then the eigenvalue of matrix $\mathcal{J}^{1}(\mathbf{w}^{*})$ are given by its diagonal elements:

[TABLE]

where $T^{*}$ refers to either $T^{+}$, $\bar{T}$ or $T^{-}$. The last seven eigenvalues are negative for any of the points $\mathbf{w}^{-}$, $\bar{\mathbf{w}}$ and $\mathbf{w}^{+}$. Consequently, their stability depends only on the sign of $P^{\prime}(T^{*})$: if $P^{\prime}(T^{*})<0$ the point is stable, if $P^{\prime}(T^{*})>0$ the point is unstable. Thus, the points $\mathbf{w^{-}}$ and $\mathbf{w^{+}}$ are stable, while $\bar{\mathbf{w}}$ is unstable.

3. Functional spaces and a priori estimates

In this section we will introduce functional spaces and will obtain a priori estimates of solutions.

3.1. Hölder spaces

We introduce Hölder space $\mathcal{C}^{k+\alpha}({\mathbb{R}}_{+})$, $\alpha\in(0,1)$ consisting of vector-functions from $\mathcal{C}^{k}$ bounded on ${\mathbb{R}}_{+}$ together with their derivatives up to the order $k$, and the derivative of order $k$ satisfies Hölder condition. This space is equipped with the usual Hölder norm. We set:

[TABLE]

We now introduce the weighted spaces $E^{1}_{\mu}$ and $E^{2}_{\mu}$ where $\mu$ is the weight function, $\mu(x)=\sqrt{1+x^{2}}$. The norm in these spaces is defined by the equality:

[TABLE]

Thus, we consider the operator $A^{\tau}:E^{1}_{\mu}\to E^{2}_{\mu}$,

[TABLE]

We are looking for positive monotone solutions of the equation $A^{\tau}(\mathbf{w})=\mathbf{0}$ such that $\mathbf{w}\in E^{1}_{\mu}$.

3.2. Bounded solutions

We will obtain a priori estimates of solutions of the equation

[TABLE]

with the boundary conditions:

[TABLE]

assuming that

[TABLE]

Clearly, a solution of (3.4)-(3.5) with condition (3.6) is positive. The corresponding wave problem (1.8) becomes as follows:

[TABLE]

We begin with $L^{\infty}$ of solutions of problem (3.4), (3.5). We first prove the following lemma.

Lemma 3.1**.**

There exists a function $\mathbf{\Psi}(s)=(\psi_{1},...,\psi_{8})$ defined for $s\geq 0$ such that:

[TABLE]

where $w_{i,\ell}=\underset{s\rightarrow+\infty}{\text{lim}}\phi_{i}(s)$ for $1\leq i\leq 7$.

Proof.

We set $\psi^{0}_{i}(s)=\phi_{i}(T^{-}+s)$, $i=1,...,7$, $\psi^{0}_{8}(s)=T^{-}+s$, and $\mathbf{\Psi}^{0}(s)=(\psi^{0}_{1}(s),...,\psi^{0}_{8}(s))$. Then we have $\mathbf{\Psi}^{0}(0)=w^{-}$. Since the functions $\phi_{i}$ are increasing, then $\mathbf{\Psi}^{0}(s)>\mathbf{w}^{-}$ for $s>0$. Finally, $\underset{s\rightarrow\infty}{\text{lim}}\psi^{0}_{i}(s)=w_{i,\ell}$ by definition of these functions.

We note that $F^{\tau}_{i}(\mathbf{\Psi}^{0}(s))=0$, $i=1,...,7$, $F_{8}^{\tau}(\mathbf{\Psi}^{0}(s))=P^{\tau}(T^{-}+s)<0$ for $s>0$. Next, set $\psi_{i}(s)=\phi_{i}(T^{-}+\kappa_{i}s)$, where $\kappa_{i}$ are some numbers, $\mathbf{\Psi}(s)=(\psi_{1}(s),...,\psi_{8}(s))$. It can be directly verified that $F^{\tau}_{i}(\mathbf{\Psi}(s))<0$, $i=1,...7$ for $s>0$ if the following conditions are satisfied:

[TABLE]

Consider now $F^{\tau}_{8}(\mathbf{\Psi}(s))$. Let $\kappa_{i}=1+\varepsilon\lambda_{i}$, where $\lambda_{i}>0$ are chosen to satisfy (3.10). Since $F_{8}^{\tau}(\mathbf{\Psi}^{0}(s))=P^{\tau}(T^{-}+s)<0$ for $s>0$, then $F^{\tau}_{8}(\mathbf{\Psi}(s))<0$ for $s\geq s_{0}>0$ with any $s_{0}$ and all $\varepsilon$ sufficiently small. It remains to verify this inequality for $0<s\leq s_{0}$. Since $(P^{\tau})^{\prime}(T^{-})<0$, then $(F^{\tau}_{8})^{\prime}(\mathbf{\Psi}(0))<0$ for all $\varepsilon$ sufficiently small. Therefore, $F^{\tau}_{8}(\mathbf{\Psi}(s))<0$ for $0<s\leq s_{0}$ and $s_{0}$ sufficiently small.

∎

We can now obtain a uniform estimate of solutions.

Proposition 3.2**.**

Consider a solution $\widehat{\mathbf{w}}$ of problem (3.4), (3.5) with condition (3.6). Then

[TABLE]

Proof.

Since $\widehat{\mathbf{w}}$ is decreasing, we only need to verify this result for $\hat{\mathbf{w}}(0)$. Let us set $\mathbf{w}_{\ell}=(w_{1,\ell},...,w_{7,\ell},T^{-})$ where $w_{i,\ell}=\underset{s\rightarrow+\infty}{\text{lim}}\phi_{i}(s)$ for $1\leq i\leq 7$. We claim that:

[TABLE]

We will first show that $\widehat{w}_{i}$ is bounded by $w_{i,\ell}$ for $i=1,...,7$. Suppose that this is not the case and $\hat{w}_{i}(0)\geq w_{i,\ell}$ for some $i$. Since $\underset{s\rightarrow+\infty}{\text{lim}}\phi_{i}(s)=w_{i,\ell}$, and $\phi_{i}(s)$ is increasing for $s>0$, then $\widehat{w}_{i}(0)>\phi_{i}(s)$ for $s>0$.

Let us now proceed in the same order as for the computation of the functions $\phi_{i}$, and show that the former inequality leads to a contradiction. Suppose that for $i=3$ we have the inequality $w_{3}(0)>\phi_{3}(s)$ for all $s>0$. It follows that $F_{3}(\widehat{\mathbf{w}}(0))<0$. From the equation for the component $\widehat{w}_{3}(x)$ it follows that $\widehat{w}_{3}^{\prime\prime}(0)>0$. This contradicts the monotonicity of the solution. The same approach applied the others components proves inequality (3.11). Next, assume that $\widehat{w}_{8}(0)\geq T^{-}$. Since $\widehat{w}_{i}(0)<w_{i,\ell}$, it follows that $F_{8}^{\tau}(\widehat{\mathbf{w}}(0))<0$. As before, inequality $\widehat{w}_{8}^{\prime\prime}(0)>0$ leads to the contradiction.

In order to prove Proposition 3.2, we consider the function $\Psi(s)$ from Lemma 3.1. From (3.11) we have the inequality $\underset{s\rightarrow+\infty}{\text{lim}}\Psi(s)=\mathbf{w}_{\ell}>\hat{\mathbf{w}}(0)$. Hence it exists $\sigma$ such that $\Psi(\sigma)\geq\widehat{\mathbf{w}}(0)$. Denote $\mathbf{w}^{\sigma}=\Psi(\sigma)$. From Lemma 3.1 it follows that $\mathbf{F}^{\tau}(\mathbf{w}^{\sigma})<0$.

Consider the domain

[TABLE]

and its boundaries $\Gamma_{i}(\mathbf{w}^{\sigma})=\{\mathbf{w}|w_{i}=w_{i}^{\sigma},w_{j}^{+}\leq w_{j}\leq w_{j}^{\sigma},j\neq i\}$ for $i=1,...,7$.

Since $\widehat{\mathbf{w}}(0)<\mathbf{w}^{\sigma}$ and $\widehat{\mathbf{w}}(x)$ is decreasing, it follows that $\widehat{\mathbf{w}}\in D(w^{\sigma})$. Thanks to the monotony of $\mathbf{F}^{\tau}$ given in (2.7), for every $\mathbf{w}\in\Gamma_{i}(\mathbf{w}^{\sigma})$ we have $F_{i}^{\tau}(\mathbf{w})\leq F_{i}^{\tau}(\mathbf{w}^{\sigma})<0$.

We now decrease the value of $\sigma$. If the lemma is not true, then it exists $\sigma$ and a component $\widehat{w_{i}}(0)>w_{i}^{-}$ with $\widehat{w_{i}}\in\Gamma_{i}(\mathbf{w}^{\sigma})$. We fix $\sigma$, and the corresponding $\mathbf{w}^{\sigma}$, the minimal value for which $\widehat{w_{i}}\in\Gamma_{i}(\mathbf{w}^{\sigma})$ holds. Then $\widehat{w_{i}}(0)=w_{i}^{\sigma}$ and $\widehat{w_{j}}(0)\leq w_{j}^{\sigma}$ for all $j\neq i$, $j<8$. The properties of $D(\mathbf{w}^{\sigma})$ assure that $F_{i}^{\tau}(\widehat{\mathbf{w}}(0))<0$. Since $\widehat{\mathbf{w}}$ is a solution (3.4), it leads to $\widehat{w_{i}}^{\prime\prime}(0)>0$ which is impossible since $\hat{w_{i}}^{\prime}(0)=0$ and $\hat{w_{i}}^{\prime}(x)<0$ for $x>0$.

∎

We will now obtain estimates of solutions in Hölder spaces.

Theorem 3.3**.**

Let condition (1.6) be satisfied and the speed $c^{\tau}$ in problem (3.7), (3.8) be positive for all $\tau$ in $[0,1]$. Then there exists a constant $R$ such that for all $\tau$ and all solutions $\mathbf{w}$ of problem (3.4),(3.5) with condition (3.6) the following estimate holds:

[TABLE]

Proof.

From the uniform estimate of solution given by Proposition 3.2 it easily follows that solutions of problem (3.4), (3.5) with condition (3.6) are uniformly bounded in the Hölder space without weight. Hence to prove the theorem it is sufficient to prove that $\sup_{x}|\mathbf{w}(x)\mu(x)|$ is uniformly bounded.

The solutions decay exponentially at infinity. Therefore, the weighted norm $\sup_{x}|\mathbf{w}(x)\mu(x)|$ is bounded for each solution. Suppose that the solutions are not uniformly bounded in the weighted norm. Then there exists a sequence $\mathbf{w}^{k}$ of solutions of (3.4) such that:

[TABLE]

These solutions can correspond to different values of $\tau$.

Let $\varepsilon>0$ be small enough, so that exponential decay of the solutions gives the existence of a constant $M$, independent of $k$, such that the estimate

[TABLE]

follows from the inequality $|\mathbf{w}^{k}(x)|\leq\varepsilon$. We choose $\varepsilon$ small enough, such $|\mathbf{w}^{k}(0)|>\varepsilon$ (see inequality (3.22) below). Then there exists $x_{k}>0$ such that $|\mathbf{w}^{k}(x_{k})|=\varepsilon$. Then we have:

[TABLE]

If the sequence of $x_{k}$ is bounded, then $|\mathbf{w}^{k}(x)\mu(x)|$ is uniformly bounded. Indeed, for $0\leq x\leq x_{k}\leq\sup_{k}(x_{k})<+\infty$ we have $\mathbf{w}^{k}<w^{-}$ from Lemma 3.2, and the estimate (3.16) holds for $x\geq\sup_{k}(x_{k})$.

Suppose now that $\sup_{k}(x_{k})=+\infty$. Then we consider a subsequence of $x_{k}$, still written $x_{k}$, such that $x_{k}\rightarrow+\infty$. Consider the sequence of functions $\mathbf{z}^{k}(x)=\mathbf{w}^{k}(x+x_{k})$. We can extract a subsequence that converges to a function $\mathbf{z}^{0}$ in $\mathcal{C}^{2}_{loc}({\mathbb{R}})$. Then the function $\mathbf{z}^{0}$ is monotonically decreasing, it is defined on ${\mathbb{R}}$, and it satisfies the equation

[TABLE]

for some value $\tau_{0}\in[0,1]$. The estimate (3.16) still holds, hence, $\mathbf{z}^{0}(+\infty)=\mathbf{w}^{+}$. Moreover we have $\mathbf{z}^{0}(-\infty)=\mathbf{z}^{-}$, where $\mathbf{F}^{\tau_{0}}(\mathbf{z}^{-})=0$ and $\mathbf{z}^{-}\neq\mathbf{w}^{+}$ since $|\mathbf{z}^{0}(0)|=\varepsilon$. Then $\mathbf{z}^{0}$ is solution of (3.7) for $c^{\tau_{0}}=0$. Since $\mathbf{F}^{\tau_{0}}$ possesses exactly three stationary points, $\mathbf{z}^{-}=\mathbf{w}^{-}\text{ or }\bar{\mathbf{w}}$. Both cases lead to a contradiction.

Indeed, if $\mathbf{z}^{-}$ is unstable ($\mathbf{z}^{-}=\bar{\mathbf{w}}$), then a solution of (3.7) exists if and only if $c^{\tau}<0$ [8]. If $\mathbf{z}^{-}$ is stable ($\mathbf{z}^{-}=\mathbf{w}^{-}$), then $\mathbf{z}^{0}$ is a solution of (3.7) and (3.8) with $c^{\tau_{0}}=0$, which contradicts the assumption of Theorem 3.3, that is, $c^{\tau}>0$ for all $\tau$. Hence the function $\mathbf{z}^{0}$ can not exist, and $\sup_{k}(x_{k})<+\infty$, completing the proof of the uniform estimate in the weighted space. Estimate (3.13) can now be easily proved by conventional methods.

∎

3.3. Separation of monotone solutions

Problem (3.4), (3.5) can have monotone solutions, satisfying condition (3.6) and called pulses, and non monotone solutions. Since we are looking for monotone solutions, we need to assure that the Leray-Schauder method can be applied to this kind of solutions. The idea is to construct an open subset of $E^{1}_{\mu}$ containing all pulses but such that the non monotone solutions are not in its closure. We will prove that the monotone and non monotone solutions are separated in the function space.

Theorem 3.4**.**

Suppose that all monotone solutions of (3.4), (3.5) are uniformly bounded in $E^{1}_{\mu}$. Then there exists $r>0$ such that for all monotone solution $\mathbf{w}^{M}$ and all non monotone solution $\mathbf{w}^{N}$ the estimate

[TABLE]

holds.

Remark*.*

The existence of the uniform bound in $E^{1}_{\mu}$ is guaranteed by Theorem 3.3.

To prove Theorem 3.4, we need beforehand two lemmas. First, let us prove that a non negative solution of (3.4), (3.5) is either positive or identically zero.

Lemma 3.5**.**

Let $\widehat{\mathbf{w}}(x)$ be a non negative solution of problem (3.4), (3.5) for some $\tau\in[0,1]$. Then one of the two following conclusions holds:

• •

$\widehat{\mathbf{w}}(x)\equiv 0$* ,*

• •

$\widehat{\mathbf{w}}(x)>0$, $x\geq 0$ .

Proof.

Let $\widehat{\mathbf{w}}$ be a non negative solution. Suppose that there exists $\sigma\geq 0$ and a component of the solution such that $\widehat{w}_{i}(\sigma)=0$. Since the solution $\widehat{\mathbf{w}}$ is non negative we have $\widehat{w}_{i}^{\prime}(\sigma)=0$ and $\widehat{w}_{i}^{\prime\prime}(\sigma)\geq 0$. Then from the corresponding equation of system (3.4) we conclude that $F_{i}^{\tau}(\widehat{\mathbf{w}}(\sigma))\leq 0$. Using the expression of $\mathbf{F}^{\tau}$ given by (1.2), (2.8) and (2.6) we see that $F_{i}^{\tau}(\mathbf{w})=G_{i}^{\tau}(\mathbf{w})-\lambda_{i}(\mathbf{w})w_{i}$, for some positive function $\lambda_{i}$ which does not depends on $w_{i}$ and where $G_{i}^{\tau}(\mathbf{w})\geq 0$ for $\mathbf{w}\geq\mathbf{0}$. Since $\widehat{w}_{i}(\sigma)=0$, it follows $F_{i}^{\tau}(\widehat{\mathbf{w}}(\sigma))=0$.

Suppose, for example, that $i=1$. Then from the equality $F_{i}^{\tau}(\widehat{\mathbf{w}}(\sigma))=0$ it follows that $\widehat{w}_{3}(\sigma)=0$ or $\widehat{w}_{6}(\sigma)=0$. We repeat the same arguments, respectively, for $i=3$ or $i=6$ and conclude that some other components vanish at $x=\sigma$. Finally, we get $\widehat{\mathbf{w}}(\sigma)=0$. From the uniqueness of solution of the Cauchy problem we conclude that $\widehat{\mathbf{w}}(x)\equiv\mathbf{0}$. The proof remains similar for all other values of $i$ for which the corresponding component of the solution vanish.

∎

The next result concerns the Jacobian matrix $(\mathbf{F}^{\tau})^{\prime}$.

Lemma 3.6**.**

There exists a constant vector $\mathbf{q}>\mathbf{0}$ such that $(\mathbf{F}^{\tau})^{\prime}(\mathbf{0}).\mathbf{q}<\mathbf{0}$ $\forall\tau\in[0,1]$.

Proof.

The expression of the Jacobian matrix $(\mathbf{F}^{\tau})$ is given in the Appendix C. We notice that for $(\mathbf{F}^{\tau})^{\prime}(\mathbf{0})$ we have $\theta_{1,3}=\theta_{1,6}=\theta_{2,4}=\theta_{2,5}=0$. The other elements of this matrix, $\theta_{i,j}$ and $d_{i}$ are strictly positive, while $H^{\tau}(0)=\beta^{\tau}P^{\prime}(0)-\alpha^{\tau}h_{8}<0$. Direct calculations give:

[TABLE]

Inequality $(\mathbf{F}^{\tau})^{\prime}(\mathbf{0}).\mathbf{q}<\mathbf{0}$ is satisfied if the following conditions hold:

[TABLE]

[TABLE]

The first four inequalities are independent and the last four inequalities lead to the condition:

[TABLE]

It can be satisfied if the following estimate holds:

[TABLE]

It is equivalent to the condition:

[TABLE]

The right-hand side of this inequality equals $-\alpha^{\tau}\frac{d}{h_{3}h_{4}}$ (see (B.3)). Since $d<0$ due to conditions (1.7), then this expression is positive. To verify (3.21), it remains to note that $P^{\prime}(0)<0$. ∎

We now return to the proof of Theorem 3.4. Let us consider a sequence of monotone solutions $(\mathbf{w}^{M,k})_{k\geq 0}$ of problem (3.4), (3.5) and a sequence of non monotone solutions $(\mathbf{w}^{N,k})_{k\geq 0}$. First, we will prove that the set of monotone solutions is closed. After that we will show that a squence of non monotone solutions cannot converge to a monotone solution.

Let us start with the first step. Since $(\mathbf{w}^{M,k})_{k\geq 0}$ is bounded in $E_{\mu}^{1}$, then it is compact in $E^{1}$. Therefore, we can extract a subsequence of $(\mathbf{w}^{M,k})_{k\geq 0}$, still written $(\mathbf{w}^{M,k})_{k\geq 0}$, that converges in $E^{1}$. We call $\widehat{\mathbf{w}}$ its limit. Thus we have

[TABLE]

for some $\hat{\tau}\in[0,1]$. Moreover, $\widehat{\mathbf{w}}\geq 0$, $\widehat{\mathbf{w}}^{\prime}\leq 0$. We will show that the solution $\widehat{\mathbf{w}}$ is a pulse, that is, it satisfies problem (3.4), (3.5) with condition (3.6). First, let us prove that $\widehat{\mathbf{w}}$ remains positive.

Lemma 3.7**.**

The inequality $\widehat{\mathbf{w}}(x)>0$ holds for all $x\geq 0$.

Proof.

To prove this lemma, it is sufficient to show that $\widehat{\mathbf{w}}(0)>0$ and to apply the Lemma 3.5. We remark that if at least one component of the vector $\widehat{\mathbf{w}}(0)$ equals zero, then the whole vector vanish, $\widehat{\mathbf{w}}(0)=\mathbf{0}$. If this is the case, then $\mathbf{w}^{M,k}(0)\rightarrow\mathbf{0}$. Lemma 3.6 assures the existence of a positive vector $\mathbf{q}$ such that for all $\tau\in[0,1]$, $(\mathbf{F}^{\tau})^{\prime}(0)\mathbf{q}<\mathbf{0}$. Let $\varepsilon>0$. Then the following expansion holds:

[TABLE]

with $\mathbf{F}^{\hat{\tau}}(0)=\mathbf{0}$ and $(\mathbf{F}^{\hat{\tau}})^{\prime}(0).\mathbf{q}<\mathbf{0}$. Hence $\mathbf{F}^{\hat{\tau}}(\varepsilon\mathbf{q})<\mathbf{0}$ for $\varepsilon$ small enough.

Due to the continuity of the function $\mathbf{F}^{\tau}$ with respect to $\tau$, the inequality $\mathbf{F}^{\tau}(\varepsilon\mathbf{q})<\mathbf{0}$ holds for all $\tau$ close to $\hat{\tau}$. Furthermore the monotony property (2.7) assures that for any $\mathbf{w}\in B_{\varepsilon}=[\mathbf{0},\varepsilon\mathbf{q}]$, $\mathbf{w}\neq 0$, at least one component of the vector $\mathbf{F}^{\tau}(\mathbf{w})$ is negative.

Thus, for some component $i$ and for some element $k$ of the sequence, we have the inequality $F_{i}^{\tau_{k}}(\mathbf{w}^{M,k})<0$. Then from (3.4), we get $w_{i}^{M,k^{\prime\prime}}(0)>0$ implying that the function $w_{i}^{M,k}(x)$ is not decreasing (since $\mathbf{w}^{M,k^{\prime}}(0)=\mathbf{0}$), which contradicts (3.6). Thus, $\widehat{\mathbf{w}}(0)\neq\mathbf{0}$. Lemma 3.5 guarantees the positiveness of $\widehat{\mathbf{w}}(x)$ for all $x\geq 0$. ∎

Let us note that this lemma implies that pulse solutions are separated from the trivial solution $\mathbf{w}\equiv 0$:

[TABLE]

Lemma 3.8**.**

The inequality $\widehat{\mathbf{w}}^{\prime}(x)<\mathbf{0}$ holds for all $x>0$.

Proof.

Suppose that there exits a component $\widehat{\mathbf{w}}$ of the solution and a value $x_{0}$ such that $\widehat{w}^{\prime}_{i}(x_{0})=0$. Denote $v=-\widehat{w}_{i}^{\prime}$. Then $v\geq 0$ since $\widehat{\mathbf{w}}$ is a monotone solution of problem (3.4), (3.5). Differentiation of the $i$th of system (3.4) gives the following equality:

[TABLE]

We introduce the operator

[TABLE]

Since $\mathbf{F}^{\hat{\tau}}$ satisfies the monotonicity condition and, according to Lemma 3.2 $\mathbf{\widehat{w}}$ is in $\mathcal{C}$ (defined in (1.3)), then $\frac{\partial F_{j}^{\hat{\tau}}}{\partial v_{i}}(\mathbf{\widehat{w}}(x))<0$ for all $j\neq i$. Moreover, the solution $\widehat{\mathbf{w}}$ is decreasing. Consequently, the right-hand side of (3.23) is positive. Thus, $Lv\geq 0$. The operator $L$ is elliptic and $v$ reaches its minimum at $x_{0}$. Hence, the maximum principle states that $v\equiv 0$. Since $\widehat{\mathbf{w}}$ vanishes at infinity, it follows that $\widehat{w}_{i}\equiv 0$, which is impossible. ∎

Lemmas 3.7 and 3.8 prove that $\widehat{\mathbf{w}}$ is a pulse, that is, a solution of problem (3.4), (3.5) with condition (3.6). Hence the set of pulse solutions is closed in $E^{1}_{\mu}$. We will now conclude the proof of Theorem 3.4 and prove that a sequence of non monotone solutions cannot converge to a monotone solution.

Proof.

Let us assume that the separation between monotone and non monotone solutions does not hold. Then we can find a sequence of monotone solutions $(\mathbf{w}^{M,k})$ and a sequence of non monotone solutions $(\mathbf{w}^{N,k})$ such that $\|\mathbf{w}^{M,k}-\mathbf{w}^{N,k}\|_{E^{1}_{\mu}}\rightarrow 0$. As it is shown in Lemmas 3.7 and 3.8, we can extract a subsequence from $(\mathbf{w}^{M,k})$, still denoted by $(\mathbf{w}^{M,k})$, that converges to some pulse solution $\widehat{\mathbf{w}}$. Then we have the convergence $\mathbf{w}^{N,k}\rightarrow\widehat{\mathbf{w}}$, possibly for a subsequence. Next, we can extract a subsequence, for which we keep the same notation, for which some given component $i$ is non monotone. The solution $\mathbf{w}^{N,k}$ belongs to $\mathcal{C}^{1}$. Hence there exists a sequence $x_{k}>0$ such that $\mathbf{w}^{N,k^{\prime}}(x_{k})=\mathbf{0}$. Then there exists a subsequence, still written as $x_{k}$, such that $x_{k}\rightarrow x_{0}$. There are three possible values for the limit $x_{0}$:

• •

$x_{0}\in]0,+\infty[$,

• •

$x_{0}=+\infty$,

• •

$x_{0}=0$.

We will show that none of them is possible. If $x_{0}\in]0,+\infty[$, then $\widehat{w}_{i}^{\prime}(x_{0})=0$. We obtain a contradiction with Lemma 3.8.

Next, we consider the case $x_{0}=+\infty$. We claim that for $k$ and $y$ large enough the non monotone solution is decreasing: $\mathbf{w}^{N,k^{\prime}}(x)<\mathbf{0}$ for $x\geq y$. Consequently, we will have $x_{0}<+\infty$ reducing this case to the previous one. According to Lemma 3.6, there exits a vector $\mathbf{q}>\mathbf{0}$ such that for all $\tau\in[0,1]$ we have $(\mathbf{F}^{\tau})^{\prime}(0).\mathbf{q}<\mathbf{0}$. Since the function $(\mathbf{F}^{\tau})^{\prime}$ is continuous with respect to $\tau$, then there exist $\boldsymbol{\delta}>0$ and $\varepsilon>0$ such that for $\tau$ and $\mathbf{w}\in{\mathbb{R}}^{8}$ satisfying $|\tau-\hat{\tau}|<\varepsilon$ and $\|\mathbf{w}\|_{{\mathbb{R}}^{8}}<\boldsymbol{\delta}$, we have the inequality $(\mathbf{F}^{\tau})^{\prime}(\mathbf{w}).\mathbf{q}<\mathbf{0}$.

The solution $\widehat{\mathbf{w}}$ is decreasing (Lemma 3.8) and converges to [math] at infinity. Hence there exists $y>0$ such that for all $x\geq y$ we have the estimate $\|\widehat{\mathbf{w}}(x)\|_{{\mathbb{R}}^{8}}<\boldsymbol{\delta}$. The sequence $(\mathbf{w}^{N,k})$ converges to $\widehat{\mathbf{w}}$ in $E_{\mu}^{1}$. Then for $k_{1}$ large enough and $k\geq k_{1}$ we have $|\tau_{k}-\hat{\tau}|<\varepsilon$ and $|\mathbf{w}^{N,k}(x)\|_{{\mathbb{R}}^{8}}<\boldsymbol{\delta}$ for $x\geq y$. Moreover, $(\mathbf{w}^{N,k})^{\prime}(y)<\mathbf{0}$.

Let us show that $(\mathbf{w}^{N,k})^{\prime}(x)$ remains negative for $x\geq y$. We differentiate (3.4) and denote $\mathbf{z}^{k}(x)=-(\mathbf{w}^{N,k})^{\prime}(x)$. Then

[TABLE]

Suppose that there exist $k\geq k_{1}$ and $x>y$ such that ${z_{i}}^{k}(x)<{0}$ for some component of the function $\mathbf{z}^{k}(x)$. Since $\mathbf{z}^{k}(y)>\mathbf{0}$ and $\mathbf{z}^{k}(\infty)=\mathbf{0}$, then there exists a constant $a$ such that $\boldsymbol{\zeta}^{k}(x)=\mathbf{z}^{k}(x)+a\mathbf{q}\geq 0$, and a value $s>y$ such that ${\zeta_{i}}^{k}(s)=0$. Then the function ${\zeta_{i}}^{k}(x)$ satisfies the equation:

[TABLE]

where $a_{k}=-a\sum_{j\neq i}{F_{j}}^{\tau_{k}^{\prime}}(\mathbf{w}^{N,k}){q_{j}}>\mathbf{0}$ since $k>k_{1}$ and $x>y$. Thus $({\zeta_{i}}^{k})^{\prime\prime}(s)<0$, which contradict the fact that $s$ is a minimum.

Finally, we consider the case $x_{0}=0$. Since $\widehat{\mathbf{w}}$ is a solution of problem (3.4), (3.5) with conditio (3.6), it follows that $\mathbf{F}^{\widehat{\tau}}(\widehat{\mathbf{w}}(0))\geq\mathbf{0}$. First, let us prove that

[TABLE]

Suppose that this is not true. Then there exists a component of the solution such that $F_{i}(\widehat{\mathbf{w}}(0))=0$. Set $z(x)=-\widehat{w}_{i}^{\prime}(x)$, then we differentiate (3.7) in order to obtain:

[TABLE]

where $a(x)=-\sum_{j\neq i}\frac{\partial F_{i}^{\hat{\tau}}}{\partial v_{j}}(\widehat{\mathbf{w}}(x))\widehat{w}_{j}^{\prime}(x)\geq 0$. Since $z(0)=0$ and $z^{\prime}(0)=0$, it leads to a contradiction with the Hopf lemma.

The functions $\mathbf{w}^{N,k}(x)$ converge to $\widehat{\mathbf{w}}(x)$. Consequently for $k$ large enough we have $\mathbf{F}^{\tau_{k}}(\mathbf{w}^{N,k}(0))>\mathbf{0}$. Hence, there exist $\varepsilon>0$ small enough and a constant $k_{1}$ such that for $x\in]0,\varepsilon[$ and $k\geq k_{1}$ we have $\mathbf{F}^{\tau_{k}}(\mathbf{w}^{N,k}(x))>\mathbf{0}$. Hence $(\mathbf{w}^{N,k})^{\prime}(x)<\mathbf{0}$ in the interval $]0,\varepsilon[$, and the convergence $x_{k}\rightarrow 0$ does not hold.

∎

In this section we obtained a priori estimates of monotone solutions and proved that they are separated from non monotone solutions. In the next section we will prove Theorem 1.1.

4. Proof of the main theorem

In this section the main Theoreme 1.1 will be proved. The first part of the proof will be devoted to the existence of pulse solutions if the wave speed is positive. The function $g$ defined at the first step of the homotopy (2.2) will be established in order to preserve the positiveness of the wave speed along the homotopy. Then we will prove that the pulse problem (3.4), (3.5) with condition (3.6) possesses a solution for $\tau=1$. Finally, we will find the value of the topological degree of the operator $A^{1}$, and we will show that it does not change along the homotopy. We will use here Theorems 3.3 and 3.4 presented in Section 3. The second part of the proof will be devoted to the non existence of solution of problem (1.9) if the wave speed $c$ in problem (1.8) is non positive.

4.1. Positiveness of the wave speed

Let us start with the preservation of the positiveness of the wave speed. We assume that the wave speed $c$, also denoted as $c^{0}=c$, in problem (1.8) is positive:

[TABLE]

We will construct a function $g$ that satisfies (2.1) and which will provide a positive wave speed along the homotopy.

We begin with $\tau\in]0,\tau_{1}]$. The definition of the homotopy (2.2) leads to the inequalilty $F_{8}^{\tau}>F_{8}^{0}$ independently of the choice of $g$. Consequently, using the result of [7] (pages 111-134), we have $c^{\tau}\geq c^{0}$. Hence

[TABLE]

Let us consider $\tau\in]\tau_{1},1]$. The homotopy on this interval is defined by equality (2.3). For $\mathbf{w}\in\mathcal{C}$ and for $\tau\in[\tau_{1},1]$ we have

[TABLE]

The condition (2.1) on $g(T)$ leads to $G(0)=0$ and $G^{\prime}(0)=-h_{8}<0$. Then we can find a function $g(T)$ such that $G(T)$ satisfies the following conditions:

[TABLE]

We can now establish the condition providing the positiveness of the wave speed along the homotopy:

Proposition 4.1**.**

Suppose that function $g(T)$ satisfies conditions (2.1), (4.4). Furthermore, assume that

[TABLE]

Then $c^{\tau}>0$ for $\tau\in[\tau_{1},1]$.

Proof.

Let us consider the scalar parabolic equation

[TABLE]

on the whole axis with the initial condition $\theta(x,0)=\theta^{0}(x)$, where $\theta^{0}(x)$ is a monotonically decreasing function, $\theta^{0}(+\infty)=T^{+}=0$ and $\theta^{0}(-\infty)=T_{1}$. Since the function $G(T)$ satisfies the condition (4.4), the solution of (4.6) converges to a traveling wave $\theta_{1}(x-c_{1}t)$. The wave speed $c_{1}$ has the sign of the integral (4.5). Hence, by hypothesis, $c_{1}>0$.

Next, we claim that

[TABLE]

In order to prove this inequality, we consider the parabolic problem:

[TABLE]

Let us assume that $\mathbf{v}^{0}(-\infty)=\mathbf{w}^{-}$ and $\mathbf{v}^{0}(+\infty)=w^{+}$, then the solution of (4.8) converges to the wave solution $\mathbf{u}^{\tau}(x-c^{\tau}t)$ with the wave speed $c^{\tau}$.

Similarly to (4.8) we consider the problem

[TABLE]

where $H_{i}(\mathbf{v})=F_{i}^{\tau}(\mathbf{v})$ for $i=1,...,7$ and $H_{8}(\mathbf{v})=G(v_{8})$. Furthermore, we assume that $z_{8}^{0}(x)=\theta^{0}(x)$ and $z_{i}^{0}(x)\equiv 0$ for $i=1,...,7$.

Let us assume that $v_{8}^{0}(x)\geq z_{8}^{0}(x)$ on the whole axis. Taking into account that $\mathbf{F}^{\tau}(\mathbf{v})\geq\mathbf{H}^{\tau}(\mathbf{v})$, and $\mathbf{v}^{0}\geq\mathbf{z}^{0}$, we conclude that $\mathbf{v}^{\tau}(x,t)\geq\mathbf{z}^{\tau}(x,t)$ for all $x\in{\mathbb{R}}$ and $t>0$. Since $\mathbf{v}^{\tau}$ converges to the wave with the speed $c^{\tau}$ and $\mathbf{z}^{\tau}$ to the wave with the speed $c_{1}$, then $c^{\tau}\geq c_{1}>0$. ∎

We have shown that it is possible to find a function $g(T)$ satisfying (2.1) such that the wave speed $c^{\tau}$ remains positive along the homotopy. Let us fix such a function $g(T)$ and proceed to the resolution of the pulse problem for $\tau=1$.

4.2. Scalar equation: the case $\tau=1$

Consider problem (3.4), (3.5) with condition (3.6) for $\tau=1$. We have the equation for the last component $T$ independent of the other equations:

[TABLE]

Lemma 4.2**.**

Let $P(T)$ satisfy (1.6) and $g(T)$ satisfy (2.1), (4.4), (4.5). Then problem (4.10) possesses a unique solution.

Proof.

The second-order equation in (4.10) can be written as the system of first-order equations:

[TABLE]

where $f(w)=({P(w)+\tau_{1}g(w)})/{{D_{8}}}$. From (4.11) it follows that:

[TABLE]

Integrating this equation, and taking into account that $p(0)=0$, we get:

[TABLE]

The function $f$ only changes sign once on $]0,T^{-}[$ since $P$ and $g$ satisfy respectively (1.6) and (2.1). Moreover, (4.5) assures that the integral of $f$ on $[0,T^{-}]$ is strictly positive, and Lemma (3.2) guarantees that $w(0)<T^{-}$, hence it exists a unique value $w_{0}=w(0)$ satisfying:

[TABLE]

Hence, the right-hand side of equation (4.13) is non-negative. Its solution provides a solution of system (4.11).

∎

Thus scalar equation possesses a pulse solution. We now consider system (3.4) for $\tau=1$. We claim the following result.

Proposition 4.3**.**

Let $P(T)$ satisfy (1.6) and $g(T)$ satisfy (2.1), (4.4) and (4.5). Then problem (3.4), (3.5) with condition (3.6) possesses a unique solution for $\tau=1$.

Proof.

The last equation of (3.4) for $\tau=1$, that is the equation for $T$, is independent from the other equations. Lemma 4.2 assures the existence and uniqueness of a monotone solution $T(x)$ satisfying (4.10). Let us fix this function $T(x)$.

In the rest of the proof, we will first show the existence and uniqueness of a positive solution of problem (3.4), (3.5), and then we will prove that this solution is monotone. In both steps we will proceed in the same order of components of solution as for the computation of the functions $\phi_{i}$.

First, let us focus on the existence of a positive solution of problem (3.4), (3.5). We start with the third component, and we have the following problem:

[TABLE]

We introduce the operator

[TABLE]

and we are looking for a solution $v_{3}\in E^{1}_{\mu}$ of the equation $Lv_{3}=-k_{3}\rho_{3}T$. Then we have $Lv_{3}<0$ and $a_{3}(x)=k_{3}T(x)+h_{3}>h_{3}>0$. Hence, the operator $L$ is invertible, and there exists $\varepsilon>0$ such that for all $\lambda\in\text{ Spect }\{L\}$, we have the estimate $\Re\lambda<-\varepsilon$. Consequently, there exists a unique solution $w_{3}$ of problem (4.15), and the solution satisfies the inequality $w_{3}(x)>0$ for all $x\geq 0$.

Proceeding with the same method for the other components (and in the same order as for the computation of $\phi_{i}$) we prove the existence and uniqueness of a solution $\mathbf{w}(x)$ to (3.4) with the boundary condition $\mathbf{w}^{\prime}(0)=0$.

Let us now prove that this solution is monotonically decreasing. Lemma 4.2 already states that $w_{8}^{\prime}<0$. Once again we proceed in the same order for the components of solution, starting with the third component. Let $v(x)=-w_{3}(x)$. Differentiation of equation (4.15) leads to the equation:

[TABLE]

Setting $f_{3}=-k_{3}\rho_{3}T^{\prime}-k_{3}w_{3}T^{\prime}$ we note that $v$ satisfies the equation $Lv=f_{3}>0$. Hence, as for the existence of $w_{3}$, we get $v>0$. Consequently it follows that ${w_{3}}^{\prime}(x)<0$ for $x>0$. We repeat the same argument for the other components of solution.

Let us investigate the behavior of $\mathbf{w}(x)$ at infinity. Since $\mathbf{w}(x)$ is positive and decreasing, then there is a limit at $+\infty$. Consequently $\mathbf{w}^{\prime\prime}$ has also a limit, and this limit is $\mathbf{0}$. Then we have $\mathbf{F}^{1}(\mathbf{w(+\infty)})=\mathbf{0}$. Moreover, Lemma 4.2 assures that $w_{8}(+\infty)=0$. Hence, $\mathbf{w(+\infty)}=\mathbf{0}$. ∎

We have proved that the problem (3.4), (3.5) with condition (3.6) possesses exactly one solution for $\tau=1$. Now let us show that the topological degree is different from [math] and that it is preserved along the homotopy.

4.3. Leray-Schauder method and the existence of pulses

We us the construction of the topological degree for elliptic operators in unbounded domains in weighted Hölder spaces [6]. In order to calculate its value, we need to assure that the operator $A^{\tau}$ defined by (3.3) and linearized about this solution does not have a zero eigenvalue. We consider the eigenvalue problem for system (3.4) linearized about a pulse solution $\mathbf{w}$:

[TABLE]

on the half-axis $x>0$ with the boundary conditions:

[TABLE]

Here $v\in E^{1}_{\mu}$. We claim the following result.

Proposition 4.4**.**

All eigenvalues of the linearized problem (4.17), (4.18) are different from [math].

Proof.

Let us assume that the assertion of the proposition does not hold. Then there exists a nonzero function $\mathbf{v}$ such that

[TABLE]

and $\mathbf{v}^{\prime}(0)=\mathbf{0}$, $\mathbf{v}(+\infty)=\mathbf{0}$.

The last component of ${v_{8}}$ of the solution satisfies the equation

[TABLE]

independent of other equations. We will show that it leads to a contradiction. Let us note that $v_{8}(0)\neq 0$. Indeed, otherwise $v_{8}(x)\equiv 0$. In this case, it can be easily proved that all other components of the solution are also identical zeros. Without loss of generality we can assume that $v_{8}(0)>0$.

Next, we differentiate equation (4.10) and set $u=-T^{\prime}$, where $T$ is the solution of problem (4.10) given by Lemma 4.2. It satisfies the problem

[TABLE]

Moreover, $u(x)>0$ for all $x>0$.

Let us recall that $(F_{8}^{1})^{\prime}(T)=P^{\prime}(T)+\tau_{1}g^{\prime}(T)$. Since $g(T)$ satisfies (2.1) and $P(T)$ satisfies (1.6), it follows that $(F_{8}^{1})^{\prime}(0)<0$. Furthermore, the function $T(x)$ converges to [math] at infinity, so there exists $x_{*}>0$ such that $(F_{8}^{1})^{\prime}(T(x))<0$ for $x\geq x_{*}$. We need the following lemma to continue the proof of the proposition.

Lemma 4.5**.**

Suppose that solution $z(x)\not\equiv 0$ of the equation

[TABLE]

is such that $z(x_{0})\geq 0$ for some $x_{0}>0$. If $(F_{8}^{1})^{\prime}(T(x))<0$ for $x\geq x_{0}$, then $z(x)>0$ for $x\geq x_{0}$.

Proof.

Suppose that the assertion of the lemma does not hold. Then there exists $x_{1}>x_{0}$ such that $z(x_{1})\leq 0$. Since $z(\infty)=0$, then $z(x)$ admits a minimum $x_{2}\geq x_{1}$. If $z(x_{2})=0$, then $z(x_{2})=z^{\prime}(x_{2})=0$, and $z\equiv 0$, which is impossible. Hence $z(x_{2})<0$. In this case we have $(F_{8}^{1})^{\prime}(T(x_{2}))z(x_{2})>0$. Hence, $z^{\prime\prime}(x_{2})<0$ which contradicts the fact that $x_{2}$ is a minimum. ∎

We now return to the proof of the proposition. Let $v(x)$ be a solution of problem (4.19). Let us recall that $v(0)>0$. Consider, first, the case where $v(x)>0$ for all $x\geq 0$. We compare this solution with the solution $u(x)$ of problem (4.19). Set $u_{k}(x)=kv(x)-u(x)$, where $k$ is the minimal positive number such that $u_{k}(x)\geq 0$ for $0\leq x\leq x_{0}$, where $x_{0}$ is the same as in Lemma 4.5. Then $u_{k}(x_{*})=0$ for some $x_{*}\in(0,x_{0}]$ and $u_{k}(x)\geq 0$ for $0\leq x\leq x_{0}$. If $x_{*}\in(0,x_{0})$, then by virtue of the maximum principle, $u_{k}(x)\equiv 0$. Hence $kv(x)\equiv u(x)$. We obtain a contradiction with the boundary conditions since $v(0)>0$ and $u(0)=0$. If $x_{*}=x_{0}$, then it follows from Lemma 4.5 that $u_{k}(x)\geq 0$ for all $x\geq x_{0}$. Therefore, $u_{k}(x)\geq 0$ for all $x\geq 0$ and $u_{k}(x_{*})=0$. As above, we obtain a contradiction with the maximum principle.

Consider now the case where $v(x)$ changes sign. Therefore, there is a value $\hat{x}>0$ such that $v(\hat{x})=0$. Since $v(x)\not\equiv 0$ for $x\geq\hat{x}$, then, without loss of generality, we can assume that it has some positive values. Indeed, otherwise we multiply this solution by $-1$ since we do not use condition $v(0)>0$ anymore. As before, we compare solutions $u(x)$ and $v(x)$ on the half-axis $x\geq\hat{x}$ taking into account that $u(x)>0$ for $x\geq\hat{x}$. A similar contradiction completes the proof of the proposition.

∎

We can now complete the proof of the existence of solutions. Propositions 4.3 and 4.4 affirm that problem (3.4), (3.5) with condition (3.6) possesses a unique solution for $\tau=1$ and that the operator linearized about this solution does not have a zero eigenvalue. Then the topological degree for this problem is given by the expression [8]:

[TABLE]

where $K$ is the number of solutions and $\nu_{k}$ is the number of positive eigenvalues for the problem linearized around each solution together with their multiplicity. Here $K=1$ since we have exactly one solution, hence, it follows that $\gamma^{*}\neq 0$.

From Theorem 3.3 it follows that there exists a ball $B$ containing all the monotone solutions of the equation $A^{\tau}(\mathbf{w})=\mathbf{0}$. The operator $A^{\tau}$ is proper on closed bounded set with respect to both $\mathbf{w}$ and $\tau$, hence the set of monotone solutions of this equation is compact. The separation Theorem 3.4 implies that we can construct a compact domain $D$ that contains all monotone solutions of the equation $A^{\tau}(\mathbf{w})=\mathbf{0}$ for all $\tau\in[0,1]$, and it does not contain any non monotone solution. In fact, we can consider $D$ as the union of the ball of radius $r$ around each solution. For this domain the topological degree of the problem remains unchanged along the homotopy:

[TABLE]

Consequently $\gamma(A^{0},D)\neq 0$, and the problem (1.9) possesses a solution.

4.4. Non existence of pulses for $c\leq 0$

We will now prove that problem (1.9) does not possess any solution if the wave speed $c$ in problem (1.8) is non positive. We use a similar approach as presented in [3]. Let us first assume that the wave speed $c$ in problem (1.8) is negative. Suppose that there exists a solution $\mathbf{w}(x)$ of problem (1.9), and we extend this solution on ${\mathbb{R}}$ by parity. Let us consider the parabolic problem:

[TABLE]

where the initial condition satisfies the inequality:

[TABLE]

and the conditions at infinity:

[TABLE]

From the comparison theorem it follows that the solution $v(x,t)$ of this problem satisfies the inequality $\mathbf{v}(x,t)>\mathbf{w}(x)$ for all $x$ and $t\geq 0$. Furthermore, according to [8], the solution $\mathbf{v}(x,t)$ converges to the traveling wave $\mathbf{u}(x-ct+h)$:

[TABLE]

Here $h$ is some number.

Since $c<0$, then $\lim_{t\rightarrow+\infty}\mathbf{v}(x,t)=\mathbf{w}^{+}$ for any $x$. We obtain a contradiction with the inequality $\mathbf{v}(x,t)>\mathbf{w}(x)$ since $\mathbf{w}(x)>\mathbf{w}^{+}$. Thus, there is no pulse solution for $c<0$.

Let us now consider the case $c=0$. Problem (1.8) with $c=0$ has a solution $\mathbf{u}(x)$, and for any constant $h$, the function $\mathbf{u}^{h}(x)=\mathbf{u}(x-h)$ is also a solution.

First, let us show that if $\mathbf{u}^{h}(N)>\mathbf{w}(N)$ for some $N$ large enough, then

[TABLE]

Indeed, the difference $\mathbf{z}(x)=\mathbf{u}^{h}(x)>\mathbf{w}(x)$ satisfies the equation similar to equation (3.24). The required property was proved above for this equation.

Denote by $h_{0}$ the minimal value of $h$ for which $\mathbf{u}^{h}(x)>\mathbf{w}(x)$ for all $x\leq N$. Then $\mathbf{u}^{h_{0}}(x)\geq\mathbf{w}(x)$ for $x\leq N$, and ${u}_{i}^{h_{0}}(x_{0})={w}_{i}(x_{0})$ for some $x_{0}\leq N$ and for some component $i$ of the solution. By virtue of (4.28), $\mathbf{u}^{h_{0}}(x)\geq\mathbf{w}(x)$ for all $x\in\mathbb{R}$. Equality ${u}_{i}^{h_{0}}(x_{0})={w}_{i}(x_{0})$ gives a contradiction with the maximum principle. This contradiction proves that there are no pulse solutions for $c=0$.

5. Discussion

Blood coagulation propagates in plasma as a reaction-diffusion wave. Its properties and the convergence to this wave from some initial distribution have important physiological meaning. We will discuss here the biological interpretation of the mathematical results.

Let us begin with the number and stability of stationary points. As it was indicated in Section 1, they are determined by the number and stability of zeros of the polynomial $P(T)$. Depending on parameters, it can have from one to four non negative zeros including $T=0$ and, possibly, from one to three positive zeros. If there is only the trivial equilibrium $T=0$, then the reaction-diffusion wave does not exist, and blood coagulation cannot occur. If there is one or three positive zeros of the polynomial $P(T)$, then this is so-called monostable case where the point $T=0$ is unstable with respect to the corresponding kinetic system (without diffusion). In this case, blood coagulation begins under any small perturbation of the trivial equilibrium which is not possible biologically. The only realistic from the physiological point of view situation is realized with two positive equilibria corresponding to the bistable case. The trivial equilibrium $T=0$ is stable, and blood coagulation begins if the initial perturbation exceeds some threshold level.

The initial production of blood factors after injury occurs due to the interaction of blood plasma with the damaged vessel wall. This initial quantity of blood factors should be sufficiently large in order to overcome the threshold level and to initiate blood coagulation. Therefore, it is important to determine these critical conditions. We show in this work that the threshold level of the initial quantity of blood factors is given by the pulse solution of the reaction-diffusion system. This solution exists if and only if the speed of the travelling wave is positive. Hence, we come to the following conditions of blood coagulation: the wave speed is positive and the initial condition should be greater than the pulse solution.

An approximate analytical method to determine the wave speed is suggested in [1]. The system of equation is reduced to a single equation by some asymptotic procedure. It is the same equation for the thrombin concentration

[TABLE]

as the equation considered above to construct the homotopy in the Leray-Schauder method. There is a simple analytical condition providing the positiveness of the wave speed for this equation: the integral $\int_{0}^{T^{-}}P(T)dT$ should be positive. If the wave speed is negative, then blood coagulation does not occur. If the wave speed is positive but it is less than some given physiological values, then blood coagulation is insufficient leading to possible bleeding disorders such as hemophilia. If the speed is too large, then excessive blood coagulation can lead to thrombosis. The analytical approach suggested in [1] gives a good approximation of the wave speed obtained for the system of equations and in the experiments.

The second criterium of blood coagulation concerns the initial condition. If the wave speed is positive, then, as we prove in this work, there exists a pulse solution. The initial condition should be greater than this pulse solution. For the scalar equation the solution will then locally converge to the second stable equilibrium. It is also proved that it converges to the travelling wave solution. For the system of equations, local convergence to the second stable equilibrium can also be proved using the method of upper and lower solutions. Convergence to the travelling wave is not proved but it can be expected since it takes place for the scalar equation approximating the system of equations. If the initial condition is less than the pulse solution, then the solution will decay converging to zero. Blood coagulation does not occur in this case. Mathematically, this convergence can be proved both for the scalar equation and for the system of equations by the same method of upper and lower solutions.

The results of this work essentially use the monotonicity property of the reaction-diffusion system. This property implies the applicability of the maximum principle and of some other mathematical methods. The model considered here is a simplification of more complete models of blood coagulation. Though they may not satisfy the monotonicity property, we can expect that similar qualitative properties of solutions will also be valid for these more complete models.

Finally, let us note that the approach developed in this work can be applied for some models of blood coagulation in flow. This question will be investigated in the subsequent works.

Acknowledgements.

The work was partially supported by the “RUDN University Program 5-100”.

Appendix A Expression of the function $\boldsymbol{\phi}$

The function $\boldsymbol{\phi}(T)=(\phi_{1}(T),...,\phi_{7}(T))$ such that $\mathbf{F}(\boldsymbol{\phi}(T),T)=0$ is given by the following expressions:

[TABLE]

For each $i=1,...,7$, the function $\phi_{i}$ satisfies $F_{i}(\phi_{i}(T),T)=0$ for $T\geq 0$. Then $\boldsymbol{\phi}$ is obtain by successively determining: $\phi_{3}$, $\phi_{4}$, $\phi_{7}$, $\phi_{5}$, $\phi_{2}$, $\phi_{6}$ and $\phi_{1}$. This order of computing will often be used in this work. It can be noticed that the functions $\phi_{i}$ are rational fractions with no positive poles.

The following properties of the functions $\phi_{i}$ are straightforward:

[TABLE]

Furthermore we note that for $i=3,...,7$ and $T<T^{0}$, $\phi_{i}(T)<\rho_{i}$.

Appendix B Description of $P$

In the Appendix A we saw that the $\phi_{i}$ are rational fractions, hence it is also the case for $P$ since $P$ satisfies (1.5). We have:

[TABLE]

where $P_{1}$ and $P_{2}$ are third degree polynomials with positive coefficients, hence $P_{2}(T)>0$ for $T\geq 0$. Then the numerator of $P$ denoted $R$ is:

[TABLE]

The computation of the coefficients of $R$ shows that it is a fourth degree polynomial: $R(T)=aT^{4}+bT^{3}+cT^{2}+dT$. Since the coefficient of $P_{1}$ and $P_{2}$ are positive, we can note that the coefficient $a$ is negative. Only the explicit form of the coefficient $d$ is used:

[TABLE]

Appendix C Jacobian matrix $(\mathbf{F}^{\tau})^{\prime}$

The matrix has the form:

[TABLE]

where $H^{\tau}=P^{\prime}(T)$,

[TABLE]

[TABLE]

[TABLE]