Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem with heat radiation

Acta Mech 235, 6089–6101 (2024) https://doi.org/10.1007/s00707-024-04035-5 ORIGINAL PAPER José R. Fernández ·Ramón Quintanilla Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem with heat radiation Received: 6 February 2024 / Revised: 2 July 2024 / Accepted: 5 July 2024 / Published online: 30 July 2024 © The Author(s) 2024 Abstract In this work, we study some qualitative properties arising in the solution of a thermoelastic problem with heat radiation. The so-called Moore–Gibson–Thompson equation is used to model the heat conduction. By using the logarithmic convexity argument, the uniqueness and instability of solutions are proved without imposing any condition on the elasticity tensor. Then, the existence of solutions is obtained assuming that the elastic tensor is positive definite applying the theory of linear semigroups, and the exponential energy decay is shown in the one-dimensional case. Finally, we consider the one-dimensional quasi-static version and we assume that the elastic coefficient is negative. The existence and decay of solutions are proved, and a justification of the quasi-static approach is also provided. 1 Introduction The classical theory of thermoelasticity is based on the Fourier law; however, this law has received much criticism since, on the one hand, it is compatible with the instantaneous propagation of the thermal waves (which contradicts the causality principle) and, on the other hand, it does not describe, in a suitable way, the heat conduction phenomena at low temperatures [1,2]. For this reason, several alternative theories have been proposed to describe the heat conduction and the thermoelasticity. Perhaps the first one was formulated by Cattaneo and Maxwell [3], where the heat equation is a second order in time damped hyperbolic equation. This heat equation brings to two different thermoelastic theories [4,5]. At the 1990s, there were proposed two interesting theories. One was introduced by Green and Naghdi [6–10] and it was based on a rigorous axiomatic approach. In fact, there were three theories that the authors called types, I, II and III, respectively. In particular, theory I recovered the classical theory and theory III also violated the causality principle. On the other hand, Tzou [11] suggested the introduction of two delay parameters in the classical Fourier law. Unfortunately, this theory brings to ill posed problems [12], but it has been very successful to consider Taylor’s polynomials as an approximation. In 2007, Roy Choudhuri [13] suggested to combine the type III theory with the dual-phase-lag to obtain the so-called three-phase-lag theory. Few years later [14], it was proposed to combine the type III theory with the introduction of a relaxation parameter (as Cattaneo and Maxwell) to obtain the Moore–Gibson– Thompson (MGT) equation, which also brings to the MGT thermoelasticty. We recall that this basic law is written as qi+τ˙qi=(kijα,i),j+(k∗ ijθ,i),j,(1) J. R. Fernández (B ) Departamento de Matemática Aplicada I, Universidade de Vigo, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain E-mail: [email protected] R. Quintanilla Departamento de Matemáticas, E.S.E.I.A.A.T., Universitat Politécnica de Catalunya, Colom 11, 08222 Terrassa, Barcelona, Spain E-mail: [email protected] 6090 J. R. Fernández, R. Quintanilla where q=(qi)is the heat flux vector, αis the thermal displacement which satisfies ˙α=θ,θbeing the temperature, and τis a relaxation parameter.1 The thermoelasticity theory based on the law (1) has received much attention in the last five years (see, among others, [16–20]). In this paper, we center our attention in this theory, but we assume that our solid is a thermal conductor that radiates the thermal energy to the surrounding in a linearized form of Stefan–Boltzman law (that is, proportional to the relative temperature). It is worth saying that, in this case, we obtain a heat equation which is similar to the dual-phase-lag equation, but the coupling with the mechanical part is different. Therefore, we want to show several qualitative properties of the solutions to the problem determined by this system. It is worth recalling that a recent study concerning radiating rods with MGT thermoelasticity was presented in [21]. It is well known that the axioms of thermomechanics do not bring to the positivity of the elasticity tensor. For this reason, it is important to identify the properties of the solutions in this case. In this way, we will use the logarithmic convexity argument to show the uniqueness and instability of solutions.2Later, we will assume the positivity of the elasticity tensor and we will provide the existence of solutions as well as the exponential decay in the one-dimensional case. Moreover, we will restrict again to this one-dimensional setting, in a isotropic and homogeneous problem, when the elasticity is negative. We know that we cannot expect the stability of solutions, but we will consider the quasi-static problem (that is, the deformations are so slow that the secondorder time derivatives can be neglected), and we will prove that, if the absolute value of the elasticity is not very large, we can also conclude the existence and the decay of the solutions. The paper is organized in the following way. In the next section, the problem and the required assumptions on the constitutive tensors will be described. Then, in Sect.3we will show that this multi-dimensional problem has a unique solution and that it will be unstable in certain cases. We note that any condition will be imposed on the elasticity tensor. The existence of solutions will be considered in Sect.4. By using the theory of linear semigroups and assuming that the elastic tensor is positive definite, we will show that our Cauchy problem has at least a solution. Later, we will restrict ourselves to the one-dimensional setting in Sect.5, and we will prove that the solutions to our problem decay in an exponential way when the elastic coefficient is positive. Finally, we will ask ourselves in Sect.6what happens in the case that the elastic coefficient is negative, but assuming also that the acceleration of the system can be neglected (that is, we will consider the quasi-static version of our problem). We will show that the problem has a unique solution and that it will be decay exponentially. A justification of this quasi-static approach will be also given. 2 Basic assumptions In this section, we describe the problem that we will study in this paper. We consider the problem determined by the MGT thermoelasticity with heat radiation, but we will work in the case that the radiation which excess temperature is relatively small. Therefore, the lost by radiation is approximately linear in the temperature and the corresponding system is ρ¨ui=Cijkluk,l−βij(˙ θ+τ¨ θ),j,(2) c(τ ... θ+¨ θ) =−βij ˙ui,j+k∗ ij ˙ θ,j+kijθ,j,i−μ∗˙ θ, (3) where uidenotes the function vi+τai(velocity plus τtimes the acceleration), θis the temperature, ρis the mass density, cis the mass capacity, μ∗is the radiation coefficient, Cijkl is the elasticity tensor, k∗ ij is the thermal conductivity and kij is the rate thermal conductivity. As usual, we have the following symmetries: Cijkl =Cklij,kij =kji,k∗ ij =k∗ ji.(4) If we neglect in Eq. (3) the mechanical part, then we obtain an equation of the type known as “dual-phase-lag.” We note that we do not know any approach to this kind of “dual-phase-lag thermoelasticity” based on the logarithmic convexity arguments; however, we will do it now because the coupling term in this system is 1A thermomechanical justification for this proposition can be found in the work of Giorgi et al. [15]. 2It is worth noting that we are able to use this argument with the proposed coupling, but it does not seem easy to extend it to the case of dual-phase-lag thermoelasticity. Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem 6091 different from the usual one in the case of the dual-phase-lag. Here, it is provided by the function ˙ θ+τ¨ θ.This fact simplifies the argument. The problem we consider in this note is therefore determined by the system (2)–(3) in a bounded domain B⊂RN(N=1,2,3) with a smooth boundary ∂B, the Dirichlet boundary conditions: ui(x,t)=θ(x,t)=0 for a.e. x∈∂B,t≥0,(5) and the initial conditions: ui(x,0)=u0 i(x), ˙ui(x,0)=v0 i(x)for a.e. x∈B, θ(x,0)=θ0(x), ˙ θ(x,0)=ξ0(x), ¨ θ(x,0)=η0(x)for a.e. x∈B. (6) In this note, apart of the symmetry conditions (4), we also assume that (i) ρ(x),c(x)and μ∗are strictly positive. (ii) There exists a positive constant k0such that kijξiξj≥k0ξiξi(7) for every vector (ξi). (iii) The bilinear form Kij =k∗ ij −τkij is positive definite. We note that the energy of the system E(t)satisfies the equality: E(t)= Bρ˙ui˙ui+Cijklui,juk,l+c(˙ θ+τ¨ θ)2+τ|˙ θ|2+τKij ˙ θ,i˙ θ,j +kij(θ,i+τ˙ θ,i)(θ,j+τ˙ θ,j)dv+2 t  0 D(s)ds =E(0), (8) where the dissipation D(t)is given by D(t)= Bμ∗|˙ θ|2+Kij ˙ θ,i˙ θ,jdv. 3 Uniqueness and instability We note that we do not impose any condition on the elasticity tensor. Even in this situation, we will be able to prove the uniqueness of the solutions. Furthermore, in the case that E(0)<0 we will prove that the solutions increase in an exponential way. In view of the definition of the energy E(t)in (8), we see that, in the case that the elasticity tensor can be negative definite, the initial energy can be also negative. We are going to use the logarithmic convexity argument, which is strongly based on a good choice of a function which gives a measure of the solutions. Therefore, the key point is the definition of this function, but before we need to make a few comments. First, we consider a time integration of Eq. (3) and we find that c(˙ θ+τ¨ θ) =−βijui,j+(k∗ ijθ,i+kijω,i),j+c(ϑ0+τη0)+βiju0 i,j −(k∗ ijθ0 ,i),j−μ∗θ+μ∗θ0,(9) where ˙ω=θ. Let α(x)be a function which is the solution to the equation (kijα,i),j=c(ϑ0+τη0)+βiju0 i,j−(k∗ ijθ0 ,i),j+μ∗θ0 assuming null Dirichlet boundary conditions. 6092 J. R. Fernández, R. Quintanilla If we denote ψ(x,t)=α(x)+ω(x,t), we can write Eq. (3) in the form c(˙ θ+τ¨ θ) =−βijui,j+(kijψ,i+k∗ ijθ,i),j−μ∗θ. (10) We now define the function Gm(t,t0)= Bρuiui+kij(ψ,i+τθ,i)(ψ,j+τθ,j)+τKijθ,iθ,j+τμ∗|θ|2dv + t  0  BKijθ,iθ,j+μ∗|θ|2dvds+m(t+t0)2. (11) Here, mand t0are two nonnegative real numbers to be selected later. We can see that ˙ Gm(t,t0)=2 Bρui˙ui+kij(ψ,i+τθ,i)(θ,j+τ˙ θ,j)+τKijθ,i˙ θ,j+τμ∗θ˙ θdv + BKijθ0 ,iθ0 ,j+μ∗|θ0|2dv+2 t  0  BKijθ,i˙ θ,j+μ∗θ˙ θdvds+2m(t+t0), ¨ Gm(t,t0)=2 Bρui¨ui+kij(ψ,i+τθ,i)( ˙ θ,j+τ¨ θ,j)+τKijθ,i¨ θ,j+τμ∗θ¨ θ +Kijθ,i˙ θ,j+μ∗θ˙ θdv+2 Bρ˙ui˙ui+kij(θ,i+τ˙ θ,i)(θ,j+τ˙ θ,j) +τKij ˙ θ,i˙ θ,j+τμ∗|˙ θ|2dv+2m. We note that  Bkij(ψ,i+τθ,i)( ˙ θ,j+τ¨ θ,j)+τKijθ,i¨ θ,j+Kijθ,i˙ θ,j+μ∗θ˙ θ+τμ∗θ¨ θ+ρui¨uidv =− BCijklui,juk,l+c(˙ θ+τ¨ θ)2dv. Thus, we see that ¨ Gm(t,t0)=4 Bρ˙ui˙ui+kij(θ,i+τ˙ θ,i)(θ,j+τ˙ θ,j)+τKij ˙ θ,i˙ θ,j+τμ∗|˙ θ|2dv +4 t  0  BKij ˙ θ,i˙ θ,j+μ∗|˙ θ|2dvds+2(m−E(0)). If we denote ν 2= BKijθ0 ,iθ0 ,j+μ∗|θ0|2dv, we find that ¨ Gm(t,t0)Gm(t,t0)−˙ Gm(t,t0)−ν 22≥2(m+E(0))Gm(t,t0). (12) Now, we are going to prove an uniqueness result. To this end, it is sufficient to show that the only solution for the null initial conditions is the null solution. We note that, in this case, E(0)=0andν=0. Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem 6093 If we take m=0andt0=0, we find that ¨ G0(t,0)G0(t,0)−˙ G0(t,0)2≥0,(13) which implies that d2 dt2ln G0(t,0)≥0. Thus, we obtain that (see [22,23]) G0(t,0)≤G0(0,0)1−t TG0(T,0)t T for every 0 ≤t≤T.SinceG0(0,0)=0, we conclude that G0(t,0)=0foreveryt∈[0,T]. Therefore, we see that ui=θ=0 and the uniqueness follows. In the general case, we select m=−E(0)and we can obtain that Gm(t,t0)≥Gm(0,t0)˙ Gm(0,t0) ˙ Gm(0,t0)−νexp ˙ Gm(0,t0)−ν Gm(0,t0)−νGm(0,t0) ˙ Gm(0,t0)−ν, whenever t0is large enough to guarantee that ˙ Gm(0,t0)>ν. This last inequality leads to the instability of the solutions, and so, the following theorem is obtained. Theorem 1 Under the assumptions (4)and (i)–(iii), we have that (i) the problem determined by system (2)–(3), boundary conditions (5)and initial conditions (6)admits at most a unique solution. (ii) when condition E(0)<0holds, the solution is exponentially unstable. (iii) when E(0)=0,but ˙ G0(0,0)>ν, the solution is also exponentially unstable. 4 Existence Even if we have been able to obtain the uniqueness of solutions to the problem determined by the system (2)–(3), together with boundary conditions (5) and initial conditions (6) under the assumptions (i)–(iii), we cannot guarantee yet the existence of solutions. However, assuming also that the elasticity tensor is positive definite, from the definition of the energy (8) we can conclude that the solutions (when they exist) will be stable. Therefore, under the previous assumptions (i)–(iii) we will also suppose that the following condition holds: (iv) There exists a positive constant Msuch that  B Cijklui,juk,ldv≥M B ui,jui,jdv, for every vector field u=(ui)which vanishes at the boundary of B. 4.1 The Cauchy problem In this subsection, we will write our problem as a Cauchy problem in an adequate Hilbert space. Therefore, let us define the variational space: H=W1,2 0(B)×L2(B)×W1,2 0(B)×W1,2 0(B)×L2(B), where W1,2 0(B)=[W1,2 0(B)]Nand L2(B)=[L2(B)]N, equipped with the scalar product (u,v,θ,η,ξ),(u∗,v∗,θ∗,η ∗,ξ∗)= Bρviv∗ i+Cijklui,ju∗ k,l+c(η +τξ)(η∗+τξ∗) +τμ∗ηη∗+τkijη,iη∗ ,j+kij(θ,i+τη,i)(θ∗ ,j+τη∗ ,j)dv, 6094 J. R. Fernández, R. Quintanilla where the bar over a variable denotes the complex conjugate. It is clear that the norm associated with this scalar product is equivalent to the usual norm in the Hilbert space H. Therefore, we can write now our problem as follows: dU dt=AU,U(0)=(u0,v0,θ0,η 0,ξ0), (14) where Ais a matrix operator defined as A=⎛ ⎜ ⎜ ⎜ ⎝ 0I000 C00 β1β2 00 0 I0 00 0 0 I 0β3K1K2K3 ⎞ ⎟ ⎟ ⎟ ⎠ . Here, we have used the following operators: Cu =1 ρ(Cijkluk,l),j, β1η=−1 ρ(βijη),j, β2ξ=−τ ρ(βijξ),j, K1θ=1 cτ(kijθ,j),i, K2η=1 cτ(k∗ ijη,j),i−μ∗ cτη, K3ξ=−1 τξ, β3v=−1 cτβijvi,j. 4.2 Theorem of existence In this subsection, we will prove the existence of solutions to the abstract problem (14). To do this, we will use the theory of semigroups and we will show that the operator Agenerates a semigroup of contractions in the Hilbert space H. We note that the domain of the operator is a dense subspace in H. At the same time, we see that ReAU,U=− Bμ∗|η|2+kijη,iη,jdv≤0, where the last inequality follows from the assumptions (i)–(iv). Then, in order to prove the existence of solutions it is enough to show that zero belongs to the resolvent of our operator (see [24]). We consider F=(f1,f2,f3,f4,f5)∈H. Thus, we need to show the existence of an element U= (u,v,θ,η,ξ)∈Dom(A)such that AU=F. That is, we must solve the following linear system: v=f1,η=f3,ξ=f4, Cu +β1η+β2ξ=f2, β3v+K1θ+K2η+K3ξ=f5. Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem 6095 We obtain in a straightforward way v,ηand ξ. Then, we need to solve the remaining system: (Cijkluk,l),j=ρf2i+(βij(f3+τf4)),j, (kijθ,j),i=cτf5+βij f1i,j+μ∗f3+(k∗ ij f3,i),j+τf4.(15) It is clear that we can obtain uand θsolving the system (15), and so, we can conclude that the operator A generates a contractive semigroup in view of the Lumer–Phillips corollary applied to the Hille–Yosida theorem. Therefore, we can write the following existence result. Theorem 2 Let the assumptions (i)–(iv) hold. Then, the operator Adefined previously is the infinitesimal generator of a C0contractive semigroup on H. Thus, for any initial data U(0)=(u0,v0,θ0,η 0,ξ0)∈H there exists a solution to Cauchy problem (14)with the following regularity: U∈C([0,∞);Dom(A)) ∩C1([0,∞);H). Remark 1 Since the solutions to our problem are determined by the existence of a semigroup of contractions, we can also conclude the continuous dependence with respect to the initial data and the supply terms (if they were imposed). That is, we have shown that our problem is well posed in the sense of Hadamard. 5 Exponential decay In this section, we will try to show the exponential decay of the solutions to our problem, but we will restrict to the one-dimensional setting. Thus, it is convenient to recall that, in this case, the system (2)–(3) becomes: ρ¨u=μuxx −β(˙ θx+τ¨ θx), c(τ ... θ+¨ θ) =−β˙ux+k∗˙ θxx +kθxx −μ∗˙ θ. We will need to assume that the coefficients ρ,μ,c,τ,k,k∗,μ∗and k∗−τkare positive and that β= 0. The one-dimensional domain is given by B=(0,1). We note that we have assumed that the length of the beam is equal to one for the sake of simplicity in the calculations. Therefore, we can write the abstract Cauchy problem (14) in the following form: A⎛ ⎜ ⎜ ⎜ ⎝ u v θ η ξ ⎞ ⎟ ⎟ ⎟ ⎠ =⎛ ⎜ ⎜ ⎜ ⎝ v ρ−1(μuxx −β(ηx+τξx)) η ξ (cτ)−1(−βvx+k∗ηxx +kθxx −μ∗η−cξ) ⎞ ⎟ ⎟ ⎟ ⎠ . It is convenient to note that the domain of this operator is made by the elements of the Hilbert space such that v∈W1,2 0(0,1),μuxx −βτξx∈L2(0,1)and k∗ηxx +kθxx ∈L2(0,1). Since our objective is to show the exponential decay, it will be enough to prove that the imaginary axis is contained at the resolvent of the operator Aand that the asymptotic condition lim sup |λ|→∞ (iλI−A)−1<∞(16) holds (see [24]). First, we will show that the imaginary axis is contained at the resolvent. We will proceed by contradiction, and so, let us assume that this condition is not satisfied. Since iλdoes not belong to the resolvent, there will exist a sequence of real numbers λn→λ= 0 and a sequence of unit norm vectors (un,v n,θ n,η n,ξ n)in the domain of the operator such that iλnun−vn→0inW1,2(0,1), iλnρvn−(μunxx −β(ηnx +τξnx)) →0inL2(0,1), iλnθn−ηn→0inW1,2(0,1), iλnηn−ξn→0inW1,2(0,1), iλncτξn−(−βvnx +k∗ηnxx +kθnxx −μ∗ηn−cξn)→0inL2(0,1). 6096 J. R. Fernández, R. Quintanilla If we take into account the dissipation, we can see that ηn→0inW1,2(0,1), and therefore, we find that θn→0 in W1,2(0,1). We can also conclude that λ−1 nξn→0inW1,2(0,1). Keeping in mind the last convergence, multiplying it by λ−1 nξn,wehaveξn→0inL2(0,1). Now, we will obtain that unalso tends to zero in W1,2(0,1).Wecanseethat λ−1 n(−βvnx +k∗ηnxx +kθnxx)→0inL2(0,1), and therefore, we conclude that −iβunx +λ−1 n(k∗ηnxx +kθnxx)→0inL2(0,1). (17) If we multiply it by unx,wefindthat −iβunx2+λ−1 nk∗ηnxx +kθnxx,unx→0, but we also have λ−1 nk∗ηnxx +kθnxx,unx=−λ−1 nk∗ηnx +kθnx,unxx+(k∗ηnx +kθnx)unx 1 0.(18) Since λ−1 nunxx is bounded, we can see that the first term tends to zero. On the other hand, we find that λ−1/2 nk∗ηnx +kθnxL∞(0,1)≤M1λ−1/2 nk∗ηnx +kθnx1/2k∗ηnxx +kθnxx1/2+M2λ−1/2 nk∗ηnx +kθnx, where M1and M2are two positive constants. Clearly, the second term tends to zero meanwhile the first one also does it because convergence (17) implies that λ−1 nk∗ηnxx +kθnxxis bounded. A rather similar inequality for λ−1/2 nunxL∞(0,1)allows to ensure that this term is also bounded. Therefore, the second term of (18) also tends to zero and we can conclude that un→0inW1,2(0,1). If we multiply the above second convergence by un, we can also show that vn→0inL2(0,1),andso,we arrive to a contradiction because we had assumed that the vectors had unit norm. Hence, we have shown that the imaginary axis is contained at the resolvent. In order to show the asymptotic condition, we can follow a similar argument in the case λn→∞.However, our previous proof can be adapted straightforwardly also to this case, because we have only needed to assume that λndoes not tend to zero. Therefore, we have proved the following. Theorem 3 Let us assume that the constitutive coefficients ρ,μ,c,τ,k,k ∗,μ∗and k∗−τk are positive and that β= 0. Then, the semigroup associated with the operator Ais exponentially stable; that is, there exist two positive constants M,ωsuch that U(t)≤Me−ωtU(0) for every U(0)∈Dom(A). 6Caseμ<0 In this final section, we will study the problem obtained when the elastic coefficient is negative (μ<0). Thanks to the arguments of the logarithmic convexity, it is obvious to see that the solutions can explode. Therefore, in order to study the corresponding problem we will focus on the solutions which are sufficiently small. That is to say, we will assume that the movement is so small than the second time derivative ¨uis almost negligible. Therefore, our system can be written as 0=μuxx −β(˙ θx+τ¨ θx), c(τ ... θ+¨ θ) =−β˙ux+k∗˙ θxx +kθxx −μ∗˙ θ. (19) If we integrate the first equation with respect to x,wefindthat μux−μux(0)=β(˙ θ+τ¨ θ). Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem 6097 If we integrate it again, we see that μu(x)−μxux(0)=β x  0 (˙ θ+τ¨ θ)(s)ds. Then, we obtain that −μux(0)=β1 0 (˙ θ+τ¨ θ)(s)ds, and so, we have ˙ux=1 μμ˙ux(0)+β(¨ θ+τ... θ)=1 μ⎡ ⎣−β 1  0 (¨ θ+τ... θ)(s)ds+β(¨ θ+τ... θ)⎤ ⎦. Therefore, our equation can be written as c(τ ... θ+¨ θ)+β2 μ(τ ... θ+τ¨ θ)−β2 μ 1  0 (τ ... θ+¨ θ)ds=k∗˙ θxx +kθxx −μ∗˙ θ. That is, it follows that c+β2 μ(τ ... θ+¨ θ)−β2 μ 1  0 (τ ... θ+¨ θ)ds=k∗˙ θxx +kθxx −μ∗˙ θ. (20) From now on, we will assume that c+β2 μ>0. Then, we can consider Eq. (20) in the Hilbert space: H=W1,2 0(0,1)×W1,2 0(0,1)×L2(0,1), equipped with the norm (θ,η,ξ)2= 1  0 c+β2 μ(τξ +η)2dx−β2 μ⎛ ⎝ 1  0 (τξ +η) dx⎞ ⎠ 2 + 1  0 k(θx+τηx)2dx +τK 1  0 |ηx|2dx+τμ∗ 1  0 η2dx. (21) 6.1 Existence and decay In this section, we will show the existence of solutions to the problem described above. We can define the operator Agiven by A⎛ ⎝ θ η ξ ⎞ ⎠=⎛ ⎝ η ξ M ⎞ ⎠, where M=L−1[k∗ηxx +kθxx −μ∗η]+τ−1ξ, being Lξ=τc+β2 μξ−τβ2 μ1 0 ξds.