Nonlinear Analysis 95 (2014) 743–755 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Existence of traveling waves in van der Waals fluids with viscosity and capillarity effects Mai Duc Thanh a,∗ , Nguyen Dinh Huy b , Nguyen Huu Hiep b , Dao Huy Cuong c a Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam b Faculty of Applied Science, University of Technology, 268 Ly Thuong Kiet, str District 10, Ho Chi Minh City, Viet Nam c Nguyen Huu Cau High School, 07 Nguyen Anh Thu, Trung Chanh, Hoc Mon, Ho Chi Minh City, Viet Nam article info Article history: Received August 2012 Accepted 16 October 2013 Communicated by Enzo Mitidieri Keywords: van der Waals fluid Traveling wave Shock wave Viscosity Capillarity Lyapunov stability abstract We establish the existence of traveling waves of non-isentropic van der Waals fluids with viscosity and capillarity effects The method developed the one for simpler models The nonconvex equation of state of the fluid causes much difficulty in evaluating the related quantities, and so the argument and the analysis are much more involved than the convex equation of state The point is to estimate the pressure along the Hugoniot curves such that a Lyapunov function can be defined in an appropriate way © 2013 Elsevier Ltd All rights reserved Introduction In this paper we study the existence of traveling waves in van der Waals fluids with the effects of viscosity and capillarity The viscous–capillary model is given by v t − ux = 0, λ ut + p x = ux − (µvx )xx , v x λ Et + (up)x = uux − (u(µvx )x )x + (µux vx )x , v x (1.1) for x ∈ R and t > As usual, the symbols ρ, v = 1/ρ, S , p, ε, T and u denote the density, specific volume, entropy, pressure, internal energy, temperature, and velocity, respectively, and E =ε+ u2 + µ vx2 (1.2) is the total energy The quantities λ and µ represent the viscosity and capillarity coefficients, respectively For simplicity, throughout we assume that λ and µ are positive constants Besides, a van der Waals fluid is characterized by the equation ∗ Corresponding author Tel.: +84 2211 6965; fax: +84 3724 4271 E-mail addresses: mdthanh@hcmiu.edu.vn, mdthanh1@gmail.com (M.D Thanh), dinhhuy56@hcmut.edu.vn (N.D Huy), nguyenhuuhiep47@yahoo.com (N.H Hiep), cuongnhc82@gmail.com (D.H Cuong) 0362-546X/$ – see front matter © 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.na.2013.10.017 744 M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 of state of the form p= RT − v−b a v2 , (1.3) where a > 0, b > and R > are constants Nonclassical (non-Lax) shocks have been known to appear in van der Waals fluids, and they exist at the expenses of Lax shocks by the nucleation criterion, which prefers non-Lax shocks over Lax shocks Moreover, Riemann solvers with kinetics use both kinds of Lax shocks and of nonclassical (non-Lax) shocks, see [1–3], for example Traveling waves associated with nonclassical shocks in the Euler equations for van der Waals fluids with viscosity and capillarity effects were obtained by Bedjaoui–LeFloch [4] The question is whether or not the Euler equations for van der Waals fluids with viscosity and capillarity effects can also possess traveling waves associated with Lax shocks? Our aim in this paper is to seek a positive answer to this question Therefore, our results indicate that the viscous–capillary models are appropriate for applications involving both kinds of Lax shocks and nonclassical shock waves The existence of traveling waves has attracted many authors Recently, Thanh [5] studied the existence of traveling waves in the Euler equations for polytropic ideal fluids with viscosity and capillarity So, the result is applied for convex equations of state In this work, the viscosity and capillarity coefficients are slightly different from the ones in [5] More interestingly, the fluid is of van der Waals type (1.3) This is a typical nonconvex equation of state Consequently, the system of Euler equations may be of mixed type as an elliptic–hyperbolic model, and the characteristic fields are not entirely genuinely nonlinear The analysis and the argument will be much more involved for van der Waals fluids, since they possess complicated features not only in the characteristic fields, but also in the Rankine–Hugoniot relations, and the admissibility criteria for shock waves, etc However, we can establish the existence of traveling waves of (1.1) for van der Waals fluids when the viscosity and capillarity are suitably chosen The point is that we can estimate the pressure along the Hugoniot curves such that we can suitably define a Lyapunov function We will show that the viscous–capillary model (1.1) for van der Waals fluids can still yield nice properties of the corresponding Lyapunov function Accordingly, the level sets near the saddle point of the Lyapunov function will be proved to provide sharp estimates for the attraction domain of the asymptotically stable equilibrium point A stable trajectory from the saddle point will then be shown to enter the attraction domain of the asymptotically stable equilibrium point This saddle-to-stable connection gives us the traveling wave of (1.1) We observe that traveling waves corresponding to a given non-Lax shock for viscous–capillary models were considered by LeFloch and his collaborators and students, see [6–9,4,10,11] Traveling waves corresponding to a given Lax shock for viscous–capillary models were obtained by Thanh [12–15,5] Traveling waves were considered earlier for diffusive–dispersive scalar equations by Bona and Schonbek [16], Jacobs, McKinney, and Shearer [17] Traveling waves and admissibility criteria of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod [18,19] and Fan [20,21], Shearer and Yang [22] See also [23–25] for related works This paper is organized as follows In Section we provide basic concepts and properties of the Euler equations for van der Waals fluids In Section we study the system of ordinary differential equations which are derived from (1.1) for the traveling wave Its equilibria and the stability of equilibria using linearization will be presented In Section 4, we define a Lyapunov function, estimate the attraction domain, and establish the existence of the traveling wave Preliminaries 2.1 Hyperbolicity Several equations of state of a van der Waals fluid other than (1.3) are given by d T = exp (v − b)2/3 ε= 3R T− 2S 3R − , a 3(v − b) a = p+ − , v v v a where d > is a parameter, see [26] Substituting T from the last system into (1.3), one obtains an equation of state of the form p = p(v, S ), where p(v, S ) = Rd (v − b)5/3 exp 2S 3R − − a v2 See Fig Consider the fluid dynamics equations in the Lagrangian coordinates vt − ux = 0, ut + px = 0, Et + (up)x = 0, x ∈ R, t > (2.1) M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 745 Fig A typical isentrope p = p(v, S ) of van der Waals fluids in the (v, p)-plane Choosing U = (v, u, S ), we can re-write (2.1) as vt − ux = 0, ut + pv (v, S )vx + pS (v, S )Sx = 0, St = The Jacobian matrix of the last system is given by A= pv −1 pS 0 Providing that −5Rd 2S 2a pv (v, S ) = exp − + < 0, 3(v − b)8/3 3R v the Jacobian matrix A admits three distinct real eigenvalues λ1 = − −pv (v, S ) < λ2 = < λ3 = −pv (v, S ), so that the system (1.2) is strictly hyperbolic 2.2 Shock waves and admissibility criteria A shock wave of (2.1) is a weak solution U of the form U (x, t ) = U− , U+ , if x < st , if x > st , (2.2) where U− and U+ are constant states, U− ̸= U+ , called the left-hand and right-hand states, respectively, and the constant s is the shock speed These quantities must satisfy the Rankine–Hugoniot relations s(v+ − v− ) + (u+ − u− ) = 0, −s(u+ − u− ) + p+ − p− = 0, p+ + p− ε+ − ε− + (v+ − v− ) = (2.3) Often, one fixes a left-hand state U− and consider the Hugoniot set H (U− ) consisting of all the right-hand states U = U+ that can be connected to U− by a shock wave (2.2) As well-known, the Hugoniot set contains the Hugoniot curves Hi (U− ), i = 1, associated with the first and the third characteristic fields, respectively, for van der Waals fluids can be parameterized by the specific volume v Precisely, the relations (2.3) for U = U+ determine the Hugoniot curves −p− v + (2ε− + p− v− )v − av + 3ab , v (4v − (3b + v− )) −(p(U− ; v) − p− ) u = u(U− ; v) := u− ± (v − v− ) , v − v− p = p(U− ; v) := 746 M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 where the plus sign corresponds to the 1-Hugoniot curve H1 (U− ), and the minus sign corresponds to the 3-Hugoniot curve H3 (U− ) The shock speed s along the Hugoniot curves is given by s = s(U− , U ) = ± − p(U− ; v) − p− v − v− , (2.4) provided p(U− ; v) − p− v − v− ≤ In (2.4), the minus sign corresponds to 1-shocks, and the plus sign corresponds to 3-shocks Recall that for any given state U− , the set of all states U+ can be connected to U− by a shock wave satisfying the Rankine–Hugoniot relations, and is called the Hugoniot set issuing from U− , and is denoted by H (U− ) The i-shock wave between the left-hand state U− and the right-hand state U+ with shock speed s(U− , U+ ) is said to be a Lax shock if it satisfies the Lax shock inequalities, see [27], λi (U+ ) < s(U− , U+ ) < λi (U− ), i = 1, (2.5) For example, for a 3-shock√ between a left-hand state U− = (v− , u− , S− ) and a right-hand state U+ = (v+ , u+ , S+ ) with the shock speed s(U− , U+ ) = −(p+ − p− )/(v+ − v− ), the Lax shock inequalities (2.5) read −pv (v+ , S+ ) < − p+ − p− v+ − v− < −pv (v− , S− ), or, pv (v+ , S+ ) > p+ − p− v+ − v− = −s2 > pv (v− , S− ) As well-known, the characteristic fields of the system (2.1) for a van der Waals fluid are not genuinely nonlinear For this kind of systems, one often uses Liu’s entropy condition, see [28], which imposes along Hugoniot curves that s(U− , U ) ≥ s(U+ , U− ) for any U between U+ and U− Thus, Liu’s entropy condition means that any discontinuity connecting the left-hand state U− and the right-hand state U+ satisfies –for 1-shocks p(U− , v) − p− v − v− ≥ p+ − p− = −s2 , v+ − v− for any v between v+ and v− ; ≤ p+ − p− = −s2 , v+ − v− for any v between v+ and v− –for 3-shocks p(U− ; v) − p− v − v− (2.6) 2.3 Equation of the entropy Let us now find the equation of conservation of energy in terms of the specific entropy S The left-hand side of the equation of conservation of energy of (1.1) can be re-written as Et + (up)x = εt + uut + upx + pux + µ vx2 = TSt − pvt + uut + upx + pux + t µ = TSt − p(vt − ux ) + u(ut + px ) + vx2 µ vx2 t t The 2nd term of the last equation vanishes due to the conservation of mass The 3rd term can be substituted by the equation of conservation of momentum in (1.1) Thus, it follows from the last system that Et + (up)x = TSt + u = TSt + u λ ux v λ ux v − (µvx )xx + x µ vx2 t − (µvx )xx + µvx vxt x (2.7) M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 747 The right-hand side of the equation of conservation of energy of (1.1) is given by λ uux v − (u(µvx )x )x = u x λ ux v + x λ u − ux (µvx )x − u(µvx )xx + (µux vx )x v x (2.8) Equating both sides of (1.1), using (2.7), (2.8) and vt = ux , and simplifying the terms, we get TSt = (µux vx ) − [µvx vxt + ux (µvx )x ] + λ u v x Since vxt = vtx = uxx , it is derived from the last equation that TSt = (µux vx ) − [µvx uxx + ux (µvx )x ] + λ u , v x or, equivalently, TSt = λ u v x The above argument shows that the system (1.1) is equivalent to the following system v t − ux = 0, λ ut + p x = ux − (µvx )xx , v x λ TSt = u2x v (2.9) Traveling waves and equilibria 3.1 Traveling waves A traveling waves of (1.1) connecting the left-hand state U− and the right-hand state U+ is a smooth solution of (1.1) depending on the variable U = U (y) = (v(y), u(y), S (y)), y = x − st , where s is a constant, and satisfying the boundary conditions lim U (y) = U± y→±∞ lim y→±∞ d dy U (y) = lim y→±∞ d2 dy2 (3.1) U (y) = Substituting U = U (y), y = (x − st ), into (2.9), we get −s v ′ − u ′ = 0, λ ′ ′ ′ ′ −su + p = u − (µv ′ )′′ , (3.2) v λ −sTS ′ = (u′ )2 , v where (.)′ = d(.)/dy Eliminating u′ in the 2nd and the 3rd equations of (3.2) by substituting u′ from the first equation of (3.2), we obtain λ ′ v s v + p = −s v λ −TS ′ = s (v ′ )2 v ′ ′ ′ − (µv ′ )′′ , (3.3) Integrate the first equation of (3.3) on the interval (−∞, y) and use the boundary conditions (3.1) to get λ v s2 (v − v− ) + p(v, S ) − p(v− , S− ) = −s v ′ − (µv ′ )′ (3.4) Re-arranging terms of the last equation to get λ (µv ′ )′ = −s v ′ − (p(v, S ) − p(v− , S− )) − s2 (v − v− ) v (3.5) 748 M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 We can also simplify the 2nd equation in (3.3) Indeed, multiplying both sides of Eq (3.4) by sv ′ one gets λ v s2 (v − v− )v ′ + s(p − p− )v ′ = −s (v ′ )2 − (µv ′ )′ v ′ (3.6) Add (3.6) to the 2nd equation in (3.3) side-by-side to get s2 (v − v− )v ′ + s(p − p− )v ′ − TS ′ = (µv ′ )′ v ′ Simplifying terms in the last equations yields s2 (v − v− )v ′ + (p − p− )v ′ − TS ′ = − 2µv ′ v ′′ , or s2 (v − v− )v ′ − p− v ′ − (−pv ′ + TS ′ ) = − 1 ′ µ(v ′ )2 Since ε(v, S )′ = −pv ′ + TS ′ , the last equation gives 1 ′ µ(v ′ )2 = ε(v, S )′ + p− v ′ − s2 (v − v− )v ′ Integrating the last equation in (−∞, y), we have µ(v ′ )2 = (ε(v, S ) − ε− ) + p− (v − v− ) − s2 (v − v− )2 (3.7) It follows from (3.5) and (3.7) that by setting w = µv ′ , we can re-write the system (3.3) as v′ = w , µ w′ = −s λ w − p + p− − s2 (v − v− ), µv (3.8) w2 s2 = (ε − ε− ) + p− (v − v− ) − (v − v− )2 , 2µ where p = p(v, S ), ε = ε(v, S ) The third equation in (3.8) determines the entropy as a function S = S (v, w) so that by substituting S = S (v, w) into p = p(v, S ) in the second equation of (3.8), we obtain a simpler system However, we can this in a direct way by substituting a 3(v − b) a ε = p+ − , v v into the third equation of (3.8) to get w2 a 3(v − b) a s2 = p+ − − ε− − p− (v − v− ) + (v − v− )2 2µ v v This yields p = p(v, S (v, w)) = w2 + p(U− ; v), 3µ(v − b) (3.9) where p˜ (U− ; v) := s2 a + ε− − p− (v − v− ) + (v − v− )2 − 3(v − b) v v a (3.10) Substitute p from (3.9) into the second equation of (3.8), we obtain the following × ordinary differential equations of first order v′ = w , µ w′ = −s λ w2 w− − p˜ (U− ; v) − p− + s2 (v − v− ) , µv 3µ(v − b) where p(U− ; v) is given by (3.10) (3.11) M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 749 Lemma 3.1 The system (3.8) is equivalent to the following system v′ = w , µ w ′ = −s w 2µ λ w − p + p+ − s2 (v − v+ ), µv = (ε − ε+ ) + p+ (v − v+ ) − s 2 (3.12) (v − v+ )2 Consequently, the system (3.11) is equivalent to the system v′ = w , µ w ′ = −s λ w2 w− − p˜ (U+ ; v) − p+ + s2 (v − v+ ) , µv 3µ(v − b) where p˜ (U+ ; v) := 3(v − b) a v + ε+ − p+ (v − v+ ) + s2 (v − v+ ) − a v2 In addition, it holds that p˜ (U− ; v) = p˜ (U+ ; v) Proof Equating the right-hand sides of the second equations of (3.8) and (3.12) to get −p + p− − s2 (v − v− ) = −p + p+ − s2 (v − v+ ) which gives s2 = − p+ − p− v+ − v− This is exactly the definition of the shock speed Equating the right-hand sides of the third equations of (3.8) and (3.12) to get (ε − ε− ) + p− (v − v− ) − s2 (v − v− )2 = (ε − ε+ ) + p+ (v − v+ ) − s2 (v − v+ )2 which gives (ε+ − ε− ) + p− (v − v− ) − p+ (v − v+ ) + s2 (v − v+ )2 − (v − v− )2 = Simplifying the last equation gives us ε+ − ε− + p+ + p− (v+ − v− ) = which is exactly the third Rankine–Hugoniot relation in (2.3) The remaining conclusion immediately follows 3.2 Equilibria and their stability Lemma 3.2 Given a 3-shock between the left-hand state (v− , u− , S− ) and the right-hand state (v+ , u+ , S+ ) with shock speed s satisfying the Lax shock inequalities (2.5) and the Liu entropy condition (2.6) Then, (v± , 0) are equilibria of the system of differential equations (3.11) and following conclusions hold (a) The equilibrium point (v− , 0) is a saddle point (b) The point (v+ , 0) is an asymptotically stable equilibrium point Proof Setting the right-hand side of (3.11), we get w = 0, p˜ (U− ; v) − p− + s2 (v − v− ) = Clearly, p(U− , v− ) = p− 750 M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 Moreover, it follows from (2.4) that p˜ (U− ; v+ ) = p− − s2 (v+ − v− ) = p+ Thus, the two points (v± , 0) are equilibria of the system (3.11) To investigate the stability of the equilibria (v± , 0), we use the linearization method Precisely, we will evaluate the Jacobian matrix of the system (3.11) and its eigenvalues at these equilibria The Jacobian matrix of the system (3.11) at these equilibria is given by B± = −(˜p′ (U− , v± ) + s2 ) − sλ , µv± which has the characteristic equation as ξ2 + sλ µv± ξ+ p˜ ′ (U− , v± ) + s2 µ = Since sλ/µv± > 0, the last equation admits two roots which have opposite signs or have two (complex) roots of negative real parts depending on the sign of p′ (U− , v± ) + s2 < It follows from the Liu entropy condition that s2 ≤ − p(U− , v) − p− v − v− This implies that p˜ (U− ; v) = ≤ = = = a s2 a + ε− − p− (v − v− ) + (v − v− ) − 2 v a p(U− ; v) − p− a + ε− − p− (v − v− ) − (v − v− ) − 3(v − b) v v a p(U− ; v) + p− a + ε− − (v − v− ) − 3(v − b) v v a a + ε(U− ; v) − 3(v − b) v v p(U− ; v), v ∈ (v− , v+ ) 3(v − b) v (3.13) Observe that along the Hugoniot curves parameterized by the specific volume it holds that p′ (U− ; v− ) = pv (v− , S− ), and that p′ (U− ; v+ ) = p′ (U+ ; v+ ) = pv (v+ , S+ ) Moreover, since p(U− , v± ) = p± , it follows from (3.13) that p˜ ′ (U− ; v− ) ≤ p′ (U− ; v− ) = pv (v− , S− ), p˜ ′ (U− ; v+ ) ≥ p′ (U− ; v+ ) = pv (v+ , S+ ) By the Lax shock inequalities, we deduce that p˜ ′ (U− ; v− ) + s2 ≤ pv (v− , S− ) + s2 < 0, and that p˜ ′ (U− ; v+ ) + s2 ≥ pv (v+ , S+ ) + s2 > 0, (3.14) which terminates the proof Existence of traveling waves Given a shock wave of the form (2.2) of the hyperbolic system (2.1) connecting a given left-hand state U− = (v− , u− , S− ) to some right-hand state U+ = (v+ , u+ , S+ ) and propagating with the speed s = s(U− , U+ ), i = 1, We will assume that the shock satisfying the Lax shock inequalities (2.5) and the Liu entropy condition (2.6) For definitiveness, we consider a 3-shock between U− and U+ , where s > and v− < v+ Note that similar arguments can be made for other cases M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 751 The system (3.11) can be re-written as v′ = w , µ w ′ = −s (4.1) λ w2 w− − q(U− ; v), µv 3µ(v − b) where p˜ (U− ; v) = a s2 + ε− − p− (v − v− ) + (v − v− ) 3(v − b) v q(U− ; v) = p˜ (U− ; v) − p− + s2 (v − v− ) − a v2 , (4.2) 4.1 Lyapunov function Let us define a Lyapunov function candidate L(v, w) = µ v v+ q(U− ; z )dz + w2 , (4.3) where q(U− , v) is defined by (4.2) Obviously, L(v+ , 0) = It follows from (3.13) that q(U− ; v) ≤ p(U− ; v) − p− + s2 (v − v− ) = (v − v− ) p(U− ; v) − p− +s v − v− ≤ 0, v ∈ (v− , v+ ) Moreover, by the Lax shock inequalities, the last inequality is strict, at least for v near v− Thus, L(v, w) = µ v v+ q(U− ; z )dz + w2 > w2 ≥ 0, v ∈ (v− , v+ ) As seen in the proof of Lemma 3.2, p˜ (U− , v± ) = p± , which yields q(U− ; v± ) = Moreover, from (3.14) we have q′ (U− ; v+ ) = p˜ ′ (U− , v+ ) + s2 > Thus q(U− ; v) > for at least v+ < v < v˜ , for some v˜ > v+ Set ν = sup{v > v+ |q(U− ; v) > 0} (4.4) Assume in the following that ν v− q(U− ; v)dv > (4.5) It is easy to see that the value ν satisfies L(ν, 0) = µ ν v+ q(U− ; z )dz > max v− ≤v≤v+ L(v, 0) = L(v− , 0) > The above argument shows that the Lyapunov function candidate L is positive definite for v ∈ (v− , ν) We will show that it is in fact a Lyapunov function on the set b D := (v, w)|v− < v < ν, w > 3sλ −1 v (4.6) 752 M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 Fig The domain D of the Lyapunov function defined by (4.6) is above the curve C and on the right of the line v = b in the (v, w)-plane Lemma 4.1 Under the condition (4.5), the function L defined by (4.3) is a Lyapunov function in the set D given by (4.6) Precisely, the following conclusions hold L(v+ , 0) = 0, L(v, w) > 0, for v ∈ (v− , ν), v ̸= v+ , ˙L(v, w) < for w ̸= 0, w > 3sλ b − , v L˙ (w, z ) = on {w = 0}, (4.7) where L˙ denotes the derivative of L along trajectories of (4.1) Proof Let us consider the derivative of L along the trajectories of (4.1): L˙ (v, w) = ∇ L(v, w) · ⟨v ′ , w ′ ⟩ w λ w2 = ⟨µ p˜ (U− ; v) − p− + s2 (v − v− ) , w⟩ · , −s w − − p˜ (U− ; v) − p− + s2 (v − v− ) µ µv 3µ(v − b) λ w = −s w − µv 3µ(v − b) w −w sλ + = µ v 3(v − b) The last equality establishes the third line of (4.7) and implies that L˙ (v, w) < 0, w ̸= whenever sλ v w > 0, 3(v − b) + which holds if w > −3sλ(1 − b/v) = 3sλ b v −1 This completes the proof of Lemma 4.1 The set D defined by (4.6) is the region above the curve C : w = 3sλ vb − and on the right of the line v = b in the (v, w)-plane, see Fig 4.2 Estimating attraction domain First, we will define the elementary sets that include the estimates of the attraction domain as follows Set θ := 9b(1 − b/v− ) (v+ − v− )v+ (4.8) Assume that µ < θ λ2 (4.9) M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 753 Fig The estimation of the attraction domain Under the condition (4.9), let us choose a positive number M such that √ √ µs < M < θ sλ (4.10) Now, we fix an arbitrary value < ε < (v+ − v− )/2, and set Gε = (v, w) ∈ R2 |(v − v+ )2 + w ≤ |v+ − (v− + ε)|2 , v ≤ v+ M |v+ − ν|2 2 (v, w) ∈ R |(v − v+ ) + w ≤ |v+ − ν| , v ≥ v+ , (4.11) (M |v+ − (v− + ε)|)2 (see Fig 3) The sets Gε are located above the curve C in the (v, w)-plane, and so are included in the domain of the Lyapunov 2 function Indeed, one can check that the function (3sλ)2 (b/v − 1)2 − (v+ − v− )2 > 0, M2 Actually, the function h is then strictly increasing, and so h(v) := (v − v+ )2 + h(v) > h(b) = (v+ − b)2 − (v+ − v− )2 > 0, v + > v > b v > b This means that the closure of the sets Gε is located above the curve C in the (v, w)-plane, and so the conclusion follows Lemma 4.2 Let ∂ Gε denote the boundary of the set Gε defined by (4.11) It holds that L(v, w) > L(v− + ε, 0), for all (v, w) ∈ ∂ Gε \ {(v− + ε, 0)} Proof On the part ∂ Gε , v ≤ v+ , one has w = M (|v+ − (v− + ε)|2 − (v − v+ )2 ) Therefore, L(v, w)|(v,w)∈∂ Gε ,v≤v+ = v v+ g (z )dz + := ϕ(v), M2 (|v+ − (v− + ε)|2 − (v − v+ )2 ) v ∈ [v− + ε, v+ ] In view of Lemma 3.1, we have q(U− ; v) = q(U+ ; v) = p˜ (U+ ; v) − p+ + s2 (v − v+ ) ≤ p(U+ ; v) − p+ + s2 (v − v+ ) Therefore, dϕ(v) dv = µq(U+ ; v) − M (v − v+ ) p+ − p(U+ ; v) 2 = (v+ − v) M − µ +s v+ − v ≥ (v+ − v)(M − µs2 ) > 0, v < v+ (4.12) 754 M.D Thanh et al / Nonlinear Analysis 95 (2014) 743–755 So, ϕ(v) > ϕ(v− + ε), v ∈ (v− + ε, v+ ] This means that L(v, w) > L(v− + ε, 0), for all (v, w) ∈ ∂ Gε \ {(v− + ε, 0)}, v ≤ v+ Arguing similarly, we obtain L(v, w) > L(ν, 0), for all (v, w) ∈ ∂ Gε \ {(ν, 0)}, v ≥ v+ Therefore, we get (4.12) The proof of Lemma 4.2 is complete Using Lemmas 4.1 and 4.2, we can now establish the following basic result in estimating the attraction domain of the asymptotically stable equilibrium point Lemma 4.3 Fix an arbitrary < ε < (w+ − w− )/2, and let Gε be defined by (4.11) Then, for any positive number < δ < L(v− + ε, 0), the set Ωδ := {(v, w) ∈ Gε |L(v, w) ≤ δ} (4.13) is a compact set, lies entirely inside Gε , positively invariant with respect to (4.1), and has the point (v+ , 0) as an interior point The set Ωδ is an attraction set of the asymptotically stable equilibrium point (v+ , 0) We omit the proof, since it is similar to the one of Lemma 3.3, [5] Using the set Ωδ defined by (4.13), we set the union Ω= ΩL(v− +2ε,0) (4.14) 0