Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 03, pp. 1–26. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu INTERNAL STABILIZATION OF INTERCONNECTED HEAT-WAVE EQUATIONS XIU-FANG YU, JUN-MIN WANG, HAN-WEN ZHANG Abstract. This article concerns the internal stabilization problem of 1-D in- terconnected heat-wave equations, where information exchange and the two actuators occur at the adjacent side of the two equations. By designing an inverse back-stepping transformation, the original system is converted into a dissipative target system. Moreover, we investigate the eigenvalues distribu- tion and the corresponding eigenfunctions of the closed-loop system by an asymptotic analysis method. This shows that the spectrum of the system can be divided into two families: one distributed along the a line parallel to the left side of the imaginary axis and symmetric to the real axis, and the other on the left half real axis. Then we work on the properties of the resolvent operator and we verify that the root subspace is complete. Finally, we prove that the closed-loop system is exponentially stable. 1. Introduction Parabolic-hyperbolic equations, as one of linear PDE-PDE systems or nonlinear, has been studied for several decades. It attracts many researchers because of the im- portant applications, such as fluid structure interaction models [3, 4, 21, 30], math- ematical biology [8], electromagnetic fields [6] and so on. For parabolic-hyperbolic systems, many works have been done: polynomial stability [21, 30], regularity anal- ysis [11], boundary controllability [2], global existence and asymptotic behavior of smooth solutions [31], and optimal linear-quadratic control [1]. Heat-wave coupled system is one of typical parabolic-hyperbolic systems. In [30], a 1-D heat-wave system coupled through an interface without any control input is proved to be polynomially stable by spectral analysis and it is also studied that null- controllability problem of the above system with the controller at one boundary. In [11], it is considered that regularity analysis for an abstract system of coupled heat-wave equations, and in [18], an optimal regularity result in Sobolev spaces is proved for the heat-wave system with mass on the interface. In [12], it is considered that spectrum and stability of a 1-D heat-wave coupled system, where dynamical boundary control is designed at the Neumann boundary of the wave equation. The stabilities of the heat-wave coupled systems in RN and R2 have presented in [21] and [5] respectively. More results of heat-wave coupled systems can also be refer to the references in [5, 11, 12, 18, 21, 30]. All of those works mentioned above 2020 Mathematics Subject Classification. 93C20, 93D15. Key words and phrases. Heat-wave equation; boundary control; spectrum; root subspace; exponential stability. ©2023. This work is licensed under a CC BY 4.0 license. Submitted September 2, 2022. Published January 6, 2023. 1 2 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 force the control at the boundary of the coupled heat-wave systems. Compared to those references and the results, in this paper, we put the controls at the internal connection point between the heat and wave equations, and both the heat and wave equations suffer the unstable terms of the system. For distributed parameter systems, the Riemannian geometrical method intro- duced in [28] is an efficient tool for the stabilization problem of the variable- coefficient systems. In [16], the Euler-Bernoulli plate with variable coefficients is stabilized by nonlinear internal feedback using geometry method. The backstep- ping method is another control strategy, which is applied to design controllers to stabilize systems in many aspects, such as to deal with PDEs with space-dependent diffusivity or time-dependent reactivity [24], a wave equation with an internal spa- tially varying antidamping term [23], coupled ODE-hyperbolic equations [25]. In [13], it is considered that the stabilization problem of a 1-d wave equation where the instability is at its free end and control is on the opposite end. By backstep- ping transformation, the controller and the observer are then designed. However, for some complex systems, one controller can not achieve desirable effect, which inspires us to add the number of controllers or other paths to improve system per- formance. In [9], two controllers are designed to stabilize the Orr-Sommerfeld equa- tion, the Squire equation and ODE cascaded system with matched disturbances, and in [27], the pointwise feedback stabilization problem of a 1-D Euler-Bernoulli beam equation with time delay outputs are considered, where two collocated sensors are presented at the arbitrary internal position. Unstable heat equation Anti-stable wave equation- � ? 6 qvt(1, t) rux(1, t) U1(t) U2(t) Figure 1. An interconnected heat-wave system In this article, we consider the stabilization of 1-D coupled heat-wave system where it is free at x = 0 and x = 2 while the information of heat and wave equation are exchanged with each other and two inputs flow into two equations respectively at the point x = 1 (see Figure 1). ut(x, t) = uxx(x, t) + cu(x, t), 0 < x < 1, t > 0, vtt(x, t) = vxx(x, t) + 2dvt(x, t), 1 < x < 2, t > 0, u(0, t) = v(2, t) = 0, t > 0, u(1, t) = qvt(1, t) + U1(t), vx(1, t) = rux(1, t) + U2(t), t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x), vt(x, 0) = v1(x), (1.1) where c > 0, d > 0 are constants, q, r 6= 0, u(x, t), w(x, t) are the state of the heat and wave equations respectively, U1(t) and U2(t) are two input controls forced on the heat and wave respectively, and u0(x), v0(x), v1(x) are the initial datum. In the middle point x = 1, there is the weak connection conditions between the heat and wave equations, u(1, t) = qvt(1, t), vx(1, t) = rux(1, t), EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 3 where the output qvt(1, t) of the wave is flowing into the heat equation, and the heat flux rux(1, t) is feeding into the wave equation. It is known that (1) when c is large enough, the heat equation has the finite unstable eigenvalues; and (2) owing to the term 2dvt(x, t), the wave equation has infinitely many unstable eigenvalues. So we need to design the controls U1(t) and U2(t) to make the closed-loop system exponentially stable. This article is organized as follows. By an invertible backstepping transforma- tion, two boundary feedback controllers are designed to stabilize the original system in Section 2. In Section 3, we show the target system is well-posed by the semigroup approach. After investigating the eigenvalues and the corresponding eigenfunctions, we prove the root subspace is complete in Section 4 and the closed-loop system is exponentially stable by the Riesz basis method in Section 5. 2. Transformations By defining w(x, t) = v(2 − x, t), 0 < x < 1, t > 0, system (1.1) is transformed into the following coupled PDEs in the domain {(x, t) : 0 < x < 1, t > 0}, ut(x, t) = uxx(x, t) + cu(x, t), wtt(x, t) = wxx(x, t) + 2dwt(x, t), u(0, t) = w(0, t) = 0, u(1, t) = qwt(1, t) + U1(t), wx(1, t) = −rux(1, t)− U2(t), u(x, 0) = u0(x), w(x, 0) = w0(x), wt(x, 0) = w1(x). (2.1) where constants c > 0, d > 0, q 6= 0, r 6= 0. Let H1 L(0, 1) = {f ∈ H1(0, 1)|f(0) = 0} with H1 norm, and we define the projection Π : L2(0, 1)×H1 L(0, 1)× L2(0, 1)→ L2(0, 1)×H1 L(0, 1)× L2(0, 1) (u(x), w(x), wt(x)) 7→ (y(x), z(x), zt(x)), which maps system (2.1) into the target system yt(x, t) = yxx(x, t), 0 < x < 1, t > 0, ztt(x, t) = zxx(x, t)− 2αzt(x, t)− α2z(x, t), 0 < x < 1, t > 0, y(0, t) = z(0, t) = 0, t > 0, y(1, t) = pzt(1, t), zx(1, t) = −pyx(1, t), t > 0, y(x, 0) = y0(x), z(x, 0) = z0(x), zt(x, 0) = z1(x), 0 < x < 1, (2.2) where α > 0, and p 6= 0 are two constants. Moreover, the projection satisfies y(x) = u(x)− ∫ x 0 k1(x, y)u(y)dy, (2.3) and z(x) = h(x)w(x)− ∫ x 0 k2(x, y)w(y)dy − ∫ x 0 k3(x, y)wt(y)dy, (2.4) zt(x) = h(x)wt(x) + k3y(x, x)w(x)− k3(x, x)wx(x) + k3(x, 0)wx(0) − ∫ x 0 k3yy(x, y)w(y)dy − ∫ x 0 [k2(x, y) + 2dk3(x, y)]wt(y)dy . (2.5) 4 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 The control laws are U1(t) = ∫ 1 0 k1(1, y)u(y, t)dy − p ∫ 1 0 k3yy(1, y)w(y, t)dy − p ∫ 1 0 [k2(1, y) + 2dk3(1, y)]wt(y, t)dy − pk3(1, 1)wx(1, t) + pk3(1, 0)wx(0, t) + pk3y(1, 1)w(1, t) + [ph(1)− q]wt(1, t), U2(t) = −r ∫ 1 0 k1x(1, y)u(y, t)dy − r p ∫ 1 0 k2x(1, y)w(y, t)dy − r p ∫ 1 0 k3x(1, y)wt(y, t)dy − rk1(1, 1)u(1, t)− r p k3(1, 1)wt(1, t) + r p [h′(1)− k2(1, 1)]w(1, t) + rh(1)− p p wx(1, t), where h(x) = cosh(d + α)x. The kernel functions k1(x, y), k2(x, y), and k3(x, y) satisfy k1xx(x, y)− k1yy(x, y) = ck1(x, y), d dx (k1(x, x)) = − c 2 , k1(x, 0) = 0, (2.6) and k2xx(x, y)− k2yy(x, y)− 2ak3yy(x, y) = α2k2(x, y), k3xx(x, y)− k3yy(x, y) = 2ak2(x, y) + (α2 + 4ad)k3(x, y), k2(x, x) = (a+ d) sinh ax+ (a2 2 − α2 2 − ad ) x cosh ax, k3(x, x) = − sinh ax, k2(x, 0) = 0, k3(x, 0) = 0, (2.7) where a = d+ α, the domain is the triangle Ω1 = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x}. (2.8) The computation process is shown in the appendix. Now we consider the existence of kernel functions (2.6) and (2.7). Using succes- sive approximations method, (2.6) has solution k1(x, y) = −cy I1( √ c(x2 − y2))√ c(x2 − y2) , x 6= y, (2.9) where I1(·) is the modified Bessel function of order one (see [14, Chapter 4]). The existence of (2.7) is included in the following theorem. Theorem 2.1. System (2.7) have a unique solution (k2(x, y), k3(x, y)) ∈ C2(Ω1). Proof. We introduce the following variable transformation: ξ = x+ y, η = x− y, and denote k2(x, y) = G(ξ, η), k3(x, y) = H(ξ, η). (2.10) Then (2.7) can be rewritten as Gξη(ξ, η) = α2 4 G(ξ, η) + a 2 Hξξ(ξ, η)− aHξη(ξ, η) + a 2 Hηη(ξ, η), (2.11) Hξη(ξ, η) = a 2 G(ξ, η) + adH(ξ, η), (2.12) EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 5 G(ξ, 0) = g( ξ 2 ), H(ξ, 0) = − sinh a 2 ξ, (2.13) G(ξ, ξ) = 0, H(ξ, ξ) = 0, (2.14) where the domain is Ω2 = {(ξ, η)| 0 ≤ ξ + η ≤ 2, 0 ≤ η ≤ ξ}, (2.15) and g( ξ 2 ) = (a+ d) sinh a 2 ξ + (a2 4 − α2 4 − ad 2 ) ξ cosh a 2 ξ. (2.16) Integrating (2.11) and (2.12) with respect to η from 0 to η, and then with respect to ξ from η to ξ, we obtain G(ξ, η) = G(η, η) +G(ξ, 0)−G(η, 0) + α2 4 ∫ ξ η ∫ η 0 G(τ, s) ds dτ + a 2 ∫ ξ η ∫ η 0 Hξξ(τ, s) ds dτ − a ∫ ξ η ∫ η 0 Hξη(τ, s) ds dτ + a 2 ∫ ξ η ∫ η 0 Hηη(τ, s) ds dτ, H(ξ, η) = H(η, η) +H(ξ, 0)−H(η, 0) + a 2 ∫ ξ η ∫ η 0 G(τ, s) ds dτ + ad ∫ ξ η ∫ η 0 H(τ, s) ds dτ. By (2.13) and (2.14), we have G(ξ, η) = g( ξ 2 )− g( η 2 ) + α2 4 ∫ ξ η ∫ η 0 G(τ, s) ds dτ − a ∫ ξ η ∫ η 0 Hξη(τ, s) ds dτ + a 2 ∫ ξ η ∫ η 0 Hξξ(τ, s) ds dτ + a 2 ∫ ξ η ∫ η 0 Hηη(τ, s) ds dτ, (2.17) H(ξ, η) = sinh a 2 η − sinh a 2 ξ + a 2 ∫ ξ η ∫ η 0 G(τ, s) ds dτ + ad ∫ ξ η ∫ η 0 H(τ, s) ds dτ, (2.18) where g(·) = (a+ d) sinh a ·+ · ( a2 2 − α2 2 − ad ) cosh a·. Differentiating H(ξ, η) with respect to ξ and η, we have Hξ(ξ, η) = −a 2 cosh a 2 ξ + a 2 ∫ η 0 G(ξ, s)ds+ ad ∫ η 0 H(ξ, s)ds, Hη(ξ, η) = a 2 cosh a 2 η + a 2 ∫ ξ η G(τ, η)dτ − a 2 ∫ η 0 G(η, s)ds + ad ∫ ξ η H(τ, η)dτ − ad ∫ η 0 H(η, s)ds, Hξη(ξ, η) = a 2 G(ξ, η) + adH(ξ, η), 6 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 Hξξ(ξ, η) = −a 2 4 sinh a 2 ξ + a 2 ∫ η 0 Gξ(ξ, s)ds+ ad ∫ η 0 Hξ(ξ, s)ds, Hηη(ξ, η) = a2 4 sinh a 2 η + a 2 ∫ ξ η Gη(τ, η)dτ − a 2 ∫ η 0 Gη(η, s)ds + ad ∫ ξ η Hη(τ, η)dτ − ad ∫ η 0 Hη(η, s)ds. We set up the following recursion for n = 0, 1, 2, . . .: Hn+1(ξ, η) = a 2 ∫ ξ η ∫ η 0 Gn(τ, s) ds dτ + ad ∫ ξ η ∫ η 0 Hn(τ, s) ds dτ, Hn+1 ξ (ξ, η) = a 2 ∫ η 0 Gn(ξ, s)ds+ ad ∫ η 0 Hn(ξ, s)ds, Hn+1 η (ξ, η) = a 2 ∫ ξ η Gn(τ, η)dτ − a 2 ∫ η 0 Gn(η, s)ds+ ad ∫ ξ η Hn(τ, η)dτ − ad ∫ η 0 Hn(η, s)ds, Hn+1 ξη (ξ, η) = a 2 Gn(ξ, η) + adHn(ξ, η), Hn+1 ξξ (ξ, η) = a 2 ∫ η 0 Gnξ (ξ, s)ds+ ad ∫ η 0 Hn ξ (ξ, s)ds, Hn+1 ηη (ξ, η) = a 2 ∫ ξ η Gnη (τ, η)dτ − a 2 ∫ η 0 Gnη (η, s)ds+ ad ∫ ξ η Hn η (τ, η)dτ − ad ∫ η 0 Hn η (η, s)ds, Gn+1(ξ, η) = α2 4 ∫ ξ η ∫ η 0 Gn(τ, s) ds dτ + a 2 ∫ ξ η ∫ η 0 Hn ξξ(τ, s) ds dτ − a ∫ ξ η ∫ η 0 Hn ξη(τ, s) ds dτ + a 2 ∫ ξ η ∫ η 0 Hn ηη(τ, s) ds dτ, with initial values G0(ξ, η) = g( ξ 2 )− g( η 2 ), H0(ξ, η) = sinh a 2 η − sinh a 2 ξ, H0 ξ (ξ, η) = −a 2 cosh a 2 ξ, H0 η (ξ, η) = a 2 cosh a 2 η, H0 ξη(ξ, η) = 0, H0 ξξ(ξ, η) = −a 2 4 sinh a 2 ξ, H0 ηη(ξ, η) = a2 4 sinh a 2 η. We denote M = max { sup x∈Ω2 |g′(x)|, sup x∈Ω2 |a cosh a 2 x|, sup x∈Ω2 |a 2 4 sinh a 2 x| } , (2.19) and we have |G0(ξ, η)| = |g( ξ 2 )− g( η 2 )| ≤ |ξ − η| 2 sup x∈Ω2 |g′(x)| ≤ sup x∈Ω2 |g′(x)| ≤M, |H0(ξ, η)| = | sinh a 2 η − sinh a 2 ξ| ≤ 2 sup x∈Ω2 |a 2 cosh a 2 x| ≤M, EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 7 |H0 ξ (ξ, η)| = |a 2 cosh a 2 ξ| ≤ sup x∈Ω2 |a 2 cosh a 2 x| ≤M, |H0 η (ξ, η)| = |a 2 cosh a 2 η| ≤ sup x∈Ω2 |a 2 cosh a 2 x| ≤M, |H0 ξξ(ξ, η)| = |a 2 4 sinh a 2 ξ| ≤ sup x∈Ω2 |a 2 4 sinh a 2 x| ≤M, |H0 ηη(ξ, η)| = |a 2 4 sinh a 2 η| ≤ sup x∈Ω2 |a 2 4 sinh a 2 x| ≤M, |H0 ξη(ξ, η)| = 0 ≤M. Assume that the following expressions hold for some n ∈ N: |Gn(ξ, η)| ≤MKn (ξ + η)n n! , |Gnξ (ξ, η)| ≤MKn (ξ + η)n n! , |Gnη (ξ, η)| ≤MKn (ξ + η)n n! , |Hn(ξ, η)| ≤MKn (ξ + η)n n! , |Hn ξ (ξ, η)| ≤MKn (ξ + η)n n! , |Hn η (ξ, η)| ≤MKn (ξ + η)n n! , |Hn ξη(ξ, η)| ≤MKn (ξ + η)n−1 (n− 1)! , |Hn ξξ(ξ, η)| ≤MKn (ξ + η)n−1 (n− 1)! , |Hn ηη(ξ, η)| ≤MKn (ξ + η)n−1 (n− 1)! , (2.20) where M is given by (2.19) and K is a positive constant. Then we obtain |Hn+1(ξ, η)| ≤ |a| 2 ∫ ξ η ∫ η 0 |Gn(τ, s)| ds dτ + |a|d ∫ ξ η ∫ η 0 |Hn(τ, s)| ds dτ ≤ ( |a| 2 + |a|d )MKn n! ∫ ξ η ∫ η 0 (τ + s)n ds dτ ≤ ( |a| 2 + |a|d ) MKn (n+ 1)! |ξ − η|(ξ + η)n+1 ≤ ( |a|+ 2|a|d ) MKn (n+ 1)! (ξ + η)n+1, |Hn+1 ξ (ξ, η)| ≤ |a| 2 ∫ η 0 |Gn(ξ, s)|ds+ |a|d ∫ η 0 |Hn(ξ, s)|ds ≤ ( |a| 2 + |a|d )MKn n! ∫ η 0 (ξ + s)nds ≤ ( |a| 2 + |a|d ) MKn (n+ 1)! (ξ + η)n+1, |Hn+1 η (ξ, η)| ≤ |a| 2 ∫ ξ η |Gn(τ, η)|dτ + |a| 2 ∫ η 0 |Gn(η, s)|ds + |a|d ∫ ξ η |Hn(τ, η)|dτ + |a|d ∫ η 0 |Hn(η, s)|ds ≤ ( |a| 2 + |a|d )MKn n! ∫ ξ 0 (s+ η)nds ≤ ( |a| 2 + |a|d ) MKn (n+ 1)! (ξ + η)n+1, 8 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 |Hn+1 ξη (ξ, η)| ≤ |a| 2 |Gn(ξ, η)|+ |a|d|Hn(ξ, η)| ≤ ( |a| 2 + |a|d) MKn n! (ξ + η)n, |Hn+1 ξξ (ξ, η)| ≤ |a| 2 ∫ η 0 |Gnξ (ξ, s)|ds+ |a|d ∫ η 0 |Hn ξ (ξ, s)|ds ≤ ( |a| 2 + |a|d ) MKn (n+ 1)! (ξ + η)n+1 ≤ ( |a| 2 + |a|d )MKn n! (ξ + η)n, |Hn+1 ηη (ξ, η)| ≤ |a| 2 ∫ ξ 0 |Gnη (τ, η)|dτ + |a|d ∫ ξ 0 |Hn η (τ, η)|dτ ≤ ( |a| 2 + |a|d )MKn n! (ξ + η)n, and |Gn+1(ξ, η)| ≤ α2 4 ∫ ξ η ∫ η 0 |Gn(τ, s)| ds dτ + |a| 2 ∫ ξ η ∫ η 0 |Hn ξξ(τ, s)| ds dτ + |a| ∫ ξ η ∫ η 0 |Hn ξη(τ, s)| ds dτ + |a| 2 ∫ ξ η ∫ η 0 |Hn ηη(τ, s)| ds dτ ≤ α2 4 MKn n! ∫ ξ η ∫ η 0 (τ + s)n ds dτ + 2|a|MKn (n− 1)! ∫ ξ η ∫ η 0 (τ + s)n−1 ds dτ = α2 4 MKn (n+ 1)! ∫ ξ η [(τ + η)n+1 − τn+1]dτ + 2|a|MKn n! ∫ ξ η [(τ + η)n − τn]dτ ≤ α2 4 MKn (n+ 1)! ∫ ξ η (τ + η)n+1dτ + 2|a|MKn n! ∫ ξ η (τ + η)ndτ ≤ α2 4 MKn (n+ 1)! |ξ − η|(ξ + η)n+1 + 2|a|MKn (n+ 1)! (ξ + η)n+1 ≤ ( 2|a|+ α2 2 ) MKn (n+ 1)! (ξ + η)n+1. Let K = max { |a|+ 2|a|d, 2|a|+ α2 2 } . Then (2.20) is true for n+ 1. Hence, the two series ∞∑ n=0 Gn(ξ, η), ∞∑ n=0 Hn(ξ, η), are convergent absolutely in Ω2 given by (2.15), and then G(ξ, η), H(ξ, η) exist, which are given by G(ξ, η) = ∞∑ n=0 Gn(ξ, η), H(ξ, η) = ∞∑ n=0 Hn(ξ, η). (2.21) EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 9 Since G(ξ, η), H(ξ, η) satisfy (2.17) and (2.18), we know that G(ξ, η), H(ξ, η) are of C2 in Ω2. This together with (2.10) yields the desired result. � Now, we prove that the projection Π−1 exists, where Π−1 : L2(0, 1)×H1 L(0, 1)× L2(0, 1)→ L2(0, 1)×H1 L(0, 1)× L2(0, 1) (y(x), z(x), zt(x)) 7→ (u(x), w(x), wt(x)). Actually, we let a = d+ α, u(x, t) = y(x, t) + ∫ x 0 s1(x, y)y(y, t)dy, (2.22) w(x, t) = l(x)z(x, t) + ∫ x 0 s2(x, y)z(y, t)dy + ∫ x 0 s3(x, y)zt(y, t)dy, (2.23) where s1xx(x, y)− s1yy(x, y) = cs1(x, y), d dx (s1(x, x)) = − c 2 , s1(x, 0) = 0, s2xx(x, y)− s2yy(x, y) + 2as3yy(x, y) = −α2s2(x, y) + 2α2as3(x, y), s3xx(x, y)− s3yy(x, y) = (4aα− α2)s3(x, y)− 2as2(x, y), s2(x, x) = −a 2 + α2 + dα a sinh ax− d2 2 x cosh ax, s3(x, x) = − sinh ax, s2(x, 0) = 0, s3(x, 0) = 0, (2.24) which are obtained from Appendix 6. In a similar way as the proof of Theorem 2.1, we obtain s1(x, y), s2(x, y) and s3(x, y) all exist, which implies that projection Π is invertible. 3. Well-posedness of system We consider system (2.2) in the Hilbert space H = L2(0, 1)×H1 L(0, 1)× L2(0, 1), H1 L(0, 1) = {f ∈ H1(0, 1)|f(0) = 0}, with the inner product 〈(g1, φ1, θ1), (g2, φ2, θ2)〉 = ∫ 1 0 g1(x)g2(x)dx+ α2 ∫ 1 0 φ1(x)φ2(x)dx + ∫ 1 0 [ φ′1(x)φ′2(x) + θ1(x)θ2(x) ] dx. We define a linear operator A : D(A) ⊂ H → H as follows A(g, φ, θ) = (g′′, θ, φ′′ − 2αθ − α2φ), ∀(g, φ, θ) ∈ D(A), D(A) = { (g, φ, θ) ∈ H2(0, 1)× (H2(0, 1) ∩H1 L(0, 1))×H1 L(0, 1), g(0) = 0, g(1) = pθ(1), φ′(1) = −pg′(1) } , (3.1) and system (2.2) can be rewritten as d dt (y(t), z(t), zt(t)) = A (y(t), z(t), zt(t)) , (y(0), z(0), zt(0)) = (y0, z0, z1). (3.2) First, we have the following result. 10 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 Lemma 3.1. The operator A defined by (3.1) is dissipative and generates a C0- semigroup eAt of contractions in H. Hence, system (3.2) is well-posed in the sense that for initial datum (y(·, 0), z(·, 0), zt(·, 0)) ∈ H, system (3.2) admits a unique solution (y(·, t), z(·, t), zt(·, t)) ∈ C(0,∞;H) (y(·, t), z(·, t), zt(·, t)) = eAt(y(·, 0), z(·, 0), zt(·, 0)). (3.3) Proof. Let (r,m, n) ∈ H. Solving A(g, φ, θ) = (r,m, n), we have g′′(x) = r(x), θ(x) = m(x), φ′′(x)− 2αθ(x)− α2φ(x) = n(x), g(0) = 0, φ(0) = 0, θ(0) = 0, g(1) = pθ(1), φ′(1) = −pg′(1), (3.4) which yields g(x) = pm(1)x− x ∫ 1 0 ∫ τ 0 r(s) ds dτ + ∫ x 0 ∫ τ 0 r(s) ds dτ, m(0) = 0, θ(x) = m(x), φ(x) = C sinhαx+ ∫ x 0 α−1 sinhα(x− s)[2αm(s) + n(s)]ds, C = ∫ 1 0 ∫ τ 0 pr(s) α coshα ds dτ − p2m(1) α coshα − ∫ 1 0 pr(s) + [2αm(s) + n(s)] coshα(1− s) α coshα ds. (3.5) Hence, (3.4) has a unique solution, which implies that A−1 exists and is compact. Moreover, the spectrum of A, σ(A), consists of isolated eigenvalues of finite alge- braic multiplicity only. 〈A(g, φ, θ), (g, φ, θ)〉 = 〈(g′′, θ, φ′′ − 2αθ − α2φ), (g, φ, θ)〉 = ∫ 1 0 g′′(x)g(x)dx+ α2 ∫ 1 0 θ(x)φ(x)dx + ∫ 1 0 [ θ′(x)φ′(x) + [φ′′(x)− 2αθ(x)− α2φ(x)]θ(x) ] dx = g(x)g′(x) ∣∣1 0 − ∫ 1 0 |g′(x)|2dx+ ∫ 1 0 θ′(x)φ′(x)dx+ ∫ 1 0 φ′′(x)θ(x)dx − 2α ∫ 1 0 |θ(x)|2dx+ α2 ∫ 1 0 [θ(x)φ(x)− φ(x)θ(x)]dx = g(x)g′(x) ∣∣1 0 + φ′(x)θ(x) ∣∣1 0 − ∫ 1 0 |g′(x)|2dx− 2α ∫ 1 0 |θ(x)|2dx + ∫ 1 0 [θ′(x)φ′(x)− θ′(x)φ′(x)]dx+ α2 ∫ 1 0 [θ(x)φ(x)− φ(x)θ(x)]dx. Since α > 0, we obtain Re〈A(g, φ, θ), (g, φ, θ)〉 = − ∫ 1 0 [g′(x)]2dx− 2α ∫ 1 0 θ2(x)dx ≤ 0. (3.6) EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 11 Hence, A is dissipative and generates a C0-semigroup eAt of contractions in H by Lumer-Philips Theorem ([20]). Moreover, system (3.2) is well-posed satisfying (3.3). � Next, we consider the eigenvalues problem of operator A. Let A(g, φ, θ) = λ(g, φ, θ), (g, φ, θ) ∈ D(A). We have θ(x) = λφ(x) and g′′(x) = λg(x), φ′′(x)− 2αλφ(x)− α2φ(x) = λ2φ(x), g(0) = 0, φ(0) = 0, g(1) = pλφ(1), φ′(1) = −pg′(1), (3.7) which has general solution (g(x), φ(x)), g(x) = c1e √ λx + c2e − √ λx, φ(x) = c3e (λ+α)x + c4e −(λ+α)x, (3.8) where c1, c2, c3 and c4 are constants to be fixed. According to boundary conditions of the last line in (3.7), we obtain c1 + c2 = 0, c3 + c4 = 0, e √ λc1 + e− √ λc2 − pλeλ+αc3 − pλe−(λ+α)c4 = 0, p √ λe √ λc1 − p √ λe− √ λc2 + (λ+ α)eλ+αc3 − (λ+ α)e−(λ+α)c4 = 0. (3.9) Therefore, (3.9) has a nontrivial solution (c1, c2, c3, c4) if and only if∣∣∣∣∣∣∣∣ 1 1 0 0 0 0 1 1 e √ λ e− √ λ −pλeλ+α −pλe−(λ+α) p √ λe √ λ −p √ λe− √ λ (λ+ α)eλ+α −(λ+ α)e−(λ+α) ∣∣∣∣∣∣∣∣ = 0, which yields (λ+ α)(e √ λ − e− √ λ)(eλ+α + e−(λ+α)) + p2λ 3 2 (e √ λ + e− √ λ)(eλ+α − e−(λ+α)) = 0. (3.10) Moreover, by some linear algebra calculations, operator A has two branches of eigenfunctions (g1, φ1, λφ1) and (g2, φ2, λφ2), where g1(x) = ∣∣∣∣∣∣∣∣ 1 1 0 0 0 0 1 1 e √ λx e− √ λx 0 0 p √ λe √ λ −p √ λe− √ λ (λ+ α)eλ+α −(λ+ α)e−(λ+α) ∣∣∣∣∣∣∣∣ = −4(λ+ α) cosh(λ+ α) sinh √ λx, (3.11) φ1(x) = ∣∣∣∣∣∣∣∣ 1 1 0 0 0 0 1 1 0 0 e(λ+α)x e−(λ+α)x p √ λe √ λ −p √ λe− √ λ (λ+ α)eλ+α −(λ+ α)e−(λ+α) ∣∣∣∣∣∣∣∣ = 4p √ λ cosh √ λ sinh(λ+ α)x, (3.12) g2(x) = ∣∣∣∣∣∣∣∣ 1 1 0 0 0 0 1 1 e √ λ e− √ λ −pλeλ+α −pλe−(λ+α) e √ λx e− √ λx 0 0 ∣∣∣∣∣∣∣∣ = −4pλ sinh(λ+ α) sinh √ λx, (3.13) 12 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 φ2(x) = ∣∣∣∣∣∣∣∣ 1 1 0 0 0 0 1 1 e √ λ e− √ λ −pλeλ+α −pλe−(λ+α) 0 0 e(λ+α)x e−(λ+α)x ∣∣∣∣∣∣∣∣ = −4 sinh √ λ sinh(λ+ α)x. (3.14) Hence, the eigenvalues and its corresponding eigenfunctions of A are obtained in the following Theorem. Theorem 3.2. Let A be defined by (3.1) and ∆(λ) = (λ+ α)(e √ λ − e− √ λ)(eλ+α + e−(λ+α)) + p2λ 3 2 (e √ λ + e− √ λ)(eλ+α − e−(λ+α)). (3.15) (1) We have σ(A) = σp(A) = {λ ∈ C| ∆(λ) = 0}. (3.16) Moreover, the real part of elements in σ(A) are negative, which means that there is no eigenvalues distributed on the imaginary axis. (2) The eigenvalues {λ1n, λ1n, λ2n} have the following asymptotic expansions λ1n = −α− k1 p2 √ α2 + n2π2 + ( nπ + k2 p2 √ α2 + n2π2 ) i+O(n−1), λ2n = − (2n+ 1)2 4 π2 + 2p−2 +O(n−1), (3.17) where n ∈ N is sufficiently large, λ1n and λ1n are conjugate and k1 = √ −α+ √ α2 + n2π2 2 , k2 = √ α+ √ α2 + n2π2 2 . (3.18) (3) The corresponding eigenfunctions {Φ1n(x),Φ1n(x),Φ2n(x)}, with Φ1n(x) = (g1n(x), φ1n(x), λ1nφ1n(x)) T , Φ1n(x) = ( g1n(x), φ1n(x), λ1nφ1n(x) )T , Φ2n(x) = (g2n(x), φ2n(x), λ2nφ2n(x)) T , (3.19) satisfy the asymptotic expressions g1n(x) φ′1n(x) λ1nφ1n(x)  =  O(e− √ n) e(−l1+l2i)x + e(l1−l2i)x +O(n−1) e(−l1+l2i)x − e(l1−l2i)x +O(n−1)  , (3.20)  g2n(x) φ′2n(x) λ2nφ2n(x)  = e 2n+1 2 πix − e− 2n+1 2 πix βn(x) −βn(x) +O(n−1), (3.21) with l1 = k1 p2 √ α2 + n2π2 , l2 = nπ + k2 p2 √ α2 + n2π2 , βn(x) = (−1)n+12ip−1e(α+λ̃2n)(1−x), λ̃2n = − (2n+ 1)2 4 π2 + 2p−2. (3.22) EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 13 Proof. (1) From Lemma 3.1, we know Re(λ) ≤ 0. Now we show the real part of eigenvalues is strictly less than zero, that is, there is no eigenvalues located on imaginary axis. Let λ = iµ2 with real number µ 6= 0 and Φ(x) = (g(x), φ(x), θ(x)) ∈ D(A) be its associated eigenfunction. Then it then follows from (3.6) that g′(x) = 0, θ(x) = 0. According to θ(x) = iµ2φ(x) and (3.7), we have g(x) = 0, φ(x) = 0 and Φ(x) = 0. The proof of part (1) is complete. (2) It follows from (1) that Re(λ) ≤ 0 and arg λ ∈ (π2 , 3π 2 ). Let λ = ρ2, and then we have arg ρ ∈ (π4 , 3π 4 ) and (3.10) becomes (eρ + e−ρ) [ eρ 2+α − e−(ρ2+α) ] + ρ2 + α p2ρ3 (eρ − e−ρ) [ eρ 2+α + e−(ρ2+α) ] = 0. (3.23) To solve (3.23) with respect to ρ, we divide (π/4, 3π/4) into three sectors S1, S2, S3, where S1 = { ρ ∈ C : arg ρ ∈ ( π 4 , 5π 16 )}, S2 = {ρ ∈ C : arg ρ ∈ ( 11π 16 , 3π 4 )}, S3 = {ρ ∈ C | arg ρ ∈ [ 5π 16 , 11π 16 ]}. (i) When arg ρ ∈ S1, we have arg ρ2 ∈ (π2 , 5π 8 ) and Re(−ρ) = −|ρ| cos arg ρ < |ρ| cos 5π 16 < 0, which leads to eρ →∞, as |ρ| → ∞, eρ 2+α − e−(ρ2+α) +O(ρ−1) = 0, (3.24) p−2ρ−3e−ρ∆(ρ2) = eρ 2+α − e−(ρ2+α) +O(ρ−1), (3.25) where ∆(ρ2) is given by (3.15). It can be verified that {ρ̃2 n = −α ± nπi, n ∈ N} is the solution of eρ 2+α − e−(ρ2+α) = 0. By Rouche’s theorem, when n ∈ N is sufficiently large, (3.24) has a solution λ1n = ρ2 n = −α+ nπi+O(n− 1 2 ). By Taylor expansion and substituting ρn = √ −α+ nπi + O(n−1) into (3.23), we have a more accurate result, O(n− 1 2 ) = −ρ 2 n + α p2ρ3 n · 1− e−2ρn 1 + e−2ρn +O(ρ−2 n ) = − 1 p2 √ −α+ nπi +O(n−1), λ1n = −α− k1 p2 √ α2 + n2π2 + ( nπ + k2 p2 √ α2 + n2π2 ) i+O(n−1), where k1, k2 given by (3.18) are obtained from x2 + αx− 1 4n 2π2 = 0. (ii) When arg ρ ∈ S2, we have arg ρ2 ∈ ( 11π 8 , 3π 2 ) and Re(ρ) = |ρ| cos arg ρ < |ρ| cos 11π 16 < 0. Hence, e−ρ →∞, as |ρ| → ∞. Like in case i), we have λ1n = −α− k1 p2 √ α2 + n2π2 − ( nπ + k2 p2 √ α2 + n2π2 ) i+O(n−1), (3.26) and (3.25) holds. 14 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 (iii) When arg ρ ∈ S3, we have arg ρ2 ∈ [ 5π 8 , 11π 8 ] and Re(ρ2) = |ρ2| cos arg ρ2 ≤ |ρ2| cos 5π 8 < 0, Hence, e−ρ 2 →∞, as |ρ| → ∞ and − p−2ρ−3eρ 2+α∆(ρ2) = eρ + e−ρ +O(ρ−1). (3.27) Equation (3.23) can be rewritten as eρ + e−ρ +O(ρ−1) = 0. (3.28) By applying Rouche’s theorem again, we have ρn = ρ̃+O(n−1), ρ̃ = 2n+ 1 2 πi. (3.29) Furthermore, ρ2 n = ρ̃2 +O(1). Substituting (3.29) into (3.23), we have O(n−1) = −ρ 2 n + α p2ρ3 n e2(ρ2n+α) + 1 e2(ρ2n+α) − 1 +O(ρ−2 n ) = p−2ρ̃−1 +O(ρ̃−2), λ2n = ρ2 n = ρ̃2 + 2p−2 +O(ρ̃−1) = − (2n+ 1)2 4 π2 + 2p−2 +O(n−1), for sufficiently large integer n. Collecting the above three cases above, the eigenvalues of A are given by (3.17). (3) Now we are in a position to solve the corresponding eigenfunctions. Let Φ1n(x) =  g1n(x) φ1n(x) λ1nφ1n(x)  = 1/ cosh √ λ1n 2pλ 3 2 1n  g1(x, λ1n) φ1(x, λ1n) λ1φ1(x, λ1n)  , Φ2n(x) =  g2n(x) φ2n(x) λ2nφ2n(x)  = 1/ sinh(λ2n + α) −2pλ2n  g2(x, λ2n) φ2(x, λ2n) λ2φ2(x, λ2n)  , where g1(x), φ1(x), g2(x), φ2(x) are given by (3.11), (3.12), (3.13) and (3.14). Then (g1n(x), φ′1n(x), λ1nφ1n(x)) T = 1/ cosh √ λ1n 2pλ 3 2 1n  −4(λ1n + α) cosh(λ1n + α) sinh √ λ1nx 4p √ λ1n(λ1n + α) cosh √ λ1n cosh(λ1n + α)x 4pλ 3 2 1n cosh √ λ1n sinh(λ1n + α)x  =  −2(λ1n+α) cosh(λ1n+α) sinh √ λ1nx pλ 3 2 1n cosh √ λ1n 2(λ1n+α) cosh(λ1n+α)x λ1n 2 sinh(λ1n + α)x.  , (3.30) EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 15 and (g2n(x), φ′2n(x), λ2nφ2n(x)) T = 1/ sinh(λ2n + α) −2pλ2n  −4pλ2n sinh(λ2n + α) sinh √ λ2nx −4(λ2n + α) sinh √ λ2n cosh(λ2n + α)x −4λ2n sinh √ λ2n sinh(λ2n + α)x  =  2 sinh √ λ2nx 2(λ2n+α) sinh √ λ2n cosh(λ2n+α)x pλ2n sinh(λ2n+α) 2 sinh √ λ2n sinh(λ2n+α)x p sinh(λ2n+α)  . (3.31) For (3.30), we have the following estimates for (3.17): −2(λ1n + α) cosh(λ1n + α) sinh √ λ1nx pλ 3 2 1n cosh √ λ1n = −2(λ1n + α)(1− e−2 √ λ1nx) pλ 3 2 1ne √ λ1n(1−x)(1 + e−2 √ λ1n) = O(e− √ n), 2(λ1n + α) cosh(λ1n + α)x λ1n = e(−l1+l2i)x + e(l1−l2i)x +O(n−1), 2 sinh(λ1n + α)x = e(−l1+l2i)x − e(l1−l2i)x +O(n−1), where l1, l2 are given by (3.22). Using (3.17) and (3.29), we estimate (3.31), where the first term satisfies 2 sinh √ λ2nx = e 2n+1 2 πix − e− 2n+1 2 πix +O(n−1), the second term satisfies 2(λ2n + α) sinh √ λ2n cosh(λ2n + α)x pλ2n sinh(λ2n + α) = 2 sinh(2n+1 2 πi) cosh(α− (2n+1)2 4 π2 + 2p−2)x p sinh(α− (2n+1)2 4 π2 + 2p−2) +O(n−1) = e2[α− (2n+1)2 4 π2+2p−2]x + 1 e2[α− (2n+1)2 4 π2+2p−2] − 1 (−1)n2ip−1e[α− (2n+1)2 4 π2+2p−2](1−x) +O(n−1) = (−1)n+12ip−1e[α− (2n+1)2 4 π2+2p−2](1−x) +O(n−1), and the third term satisfies 2 sinh √ λ2n sinh(λ2n + α)x p sinh(λ2n + α) = 2 sinh(2n+1 2 πi) sinh(α− (2n+1)2 4 π2 + 2p−2)x p sinh(α− (2n+1)2 4 π2 + 2p−2) +O(n−1) = e2[α− (2n+1)2 4 π2+2p−2]x − 1 e2[α− (2n+1)2 4 π2+2p−2] − 1 (−1)n2ip−1e[α− (2n+1)2 4 π2+2p−2](1−x) +O(n−1) = (−1)n2ip−1e[α− (2n+1)2 4 π2+2p−2](1−x) +O(n−1). Hence, the eigenfunctions of operator A satisfy (3.20) and (3.21). � Moreover, the eigenvalues of A satisfy the following. Theorem 3.3. Let A be defined by (3.1). Then all λ = ρ2 ∈ σ(A) with sufficiently large moduli are algebraically simple. 16 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 Proof. One can check that the multiplicity of each λ ∈ σ(A) with sufficiently large modulus, as a pole of R(λ,A), is less than or equal to the multiplicity of λ as a zero of the entire function ∆(ρ2) with respect to ρ. On the other hand, it can be verified that λ is geometrically simple. Moreover, all zeros of ∆(ρ2) = 0 in (3.23) with large moduli are simple, then the result follows from the general formula ma ≤ p · mg (see [17, p. 148]), where p denotes the order of the pole of the resolvent operator and ma, mg denote the algebraic and geometric multiplicities respectively. � Now, we investigate the eigenvalues and eigenfunctions of A∗, the adjoint of operator A, which is defined as A∗(g, φ, θ) = (g′′,−θ,−φ′′ − 2αθ + α2φ), ∀(g, φ, θ) ∈ D(A∗) D(A∗) = { (g, φ, θ) ∈ H2(0, 1)× (H2(0, 1) ∩H1 L(0, 1))×H1 L(0, 1), g(0) = 0, g(1) = pθ(1), φ′(1) = pg′(1) } . (3.32) Let A∗(g, φ, θ) = λ∗(g, φ, θ), (g, φ, θ) ∈ D(A∗) and we have θ(x) = −λ∗φ(x) and g′′(x) = λ∗g(x), −φ′′(x) + 2αλ∗φ(x) + α2φ(x) = −(λ∗)2φ(x), g(0) = 0, φ(0) = 0, g(1) = −pλ∗φ(1), φ′(1) = pg′(1). (3.33) A straightforward computation shows that λ∗ satisfies (3.10) and it follows that σ(A∗) = σ(A) = {λ1n, λ1n, λ2n} given by (3.17). Moreover, A∗ has the eigenfunc- tions Ψ1n(x) = (g1n(x), φ1n(x),−λ1nφ1n(x)) T , Ψ1n(x) = ( g1n(x), φ1n(x),−λ1nφ1n(x) )T , Ψ2n(x) = ( g2n(x), φ2n(x),−λ2nφ2n(x) )T , (3.34) satisfying (3.20) and (3.21). 4. Completeness of root subspaces In this section, we show the root subspace of system (3.2) is complete. First, we give the following lemma. Lemma 4.1 ([10]). Let D(λ) = 1 + ∑n i=1Qi(λ)eαiλ, where Qi are polynomials of λ, αi, i = 1, 2, . . . , n are some complex numbers and n ∈ N+. Then for all λ outside those circles of radius ε > 0 that centered at the roots of D(·), we have |D(λ)| ≥ C(ε) > 0 for some constant C(ε) that depends only on ε. Theorem 4.2. Let A be defined by (3.1) and for λ ∈ ρ(A), let R(λ,A) = (λI−A)−1 be the resolvent operator of A. Then there exists a constant M > 0 independent of λ such that R(λ,A) ≤M(1 + |λ|), (4.1) for all λ = ρ2 with ρ ∈ C lying outside all circles of radius ε > 0 that are centered at the zeros of ∆(ρ2). Proof. For any (g1, φ1, θ1) ∈ H, if (λI −A)(g, φ, θ) = (g1, φ1, θ1), (4.2) EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 17 and R(λ,A) = (λI −A)−1 is the resolvent operator of A, then (g, φ, θ) = R(λ,A)(g1, φ1, θ1). (4.3) Let λ 6= 0 and λ = ρ2 ∈ ρ(A), the resolvent set of A, which leads to ∆(λ) 6= 0. Solving (4.2), we obtain θ(x) = λφ(x)− φ1(x) and g′′(x)− λg(x) = −g1(x), φ′′(x)− (λ+ α)2φ(x) = −λφ1(x)− θ1(x)− 2αφ1(x), g(0) = 0, φ(0) = 0, g(1)− pλφ(1) = −pφ1(1), φ′(1) + pg′(1) = 0. (4.4) Now, we set the following functions, for 0 ≤ x ≤ 1, Q1(x, ξ) = 1 4 sign(x− ξ)ρ−1 [ eρ(x−ξ) − e−ρ(x−ξ) ] , Q2(x, ξ) = 1 4 sign(x− ξ)(ρ2 + α)−1 [ e(ρ2+α)(x−ξ) − e−(ρ2+α)(x−ξ)], F0(x) = − ∫ 1 0 Q1(x, ξ)g1(ξ)dξ, H0(x) = − ∫ 1 0 Q2(x, ξ) [ (ρ2 + 2α)φ1(ξ) + θ1(ξ) ] dξ. (4.5) According to the method of Green function [26, 10], for any (g1, φ1, θ1) ∈ H, the solution (g, φ, θ) of (4.3) has the following expressions g(x) = F (x, ρ) ∆(ρ2) , φ(x) = H(x, ρ) ∆(ρ2) , θ(x) = ρ2φ(x)− φ1(x), (4.6) where F (x, ρ) = ∣∣∣∣∣∣∣∣∣∣ eρx e−ρx 0 0 F0(x) 1 1 0 0 U1 0 0 1 1 U2 eρ e−ρ −pρ2eρ 2+α −pρ2e−(ρ2+α) U3 + pφ1(1) pρeρ −pρe−ρ (ρ2 + α)eρ 2+α −(ρ2 + α)e−(ρ2+α) U4 ∣∣∣∣∣∣∣∣∣∣ , H(x, ρ) = ∣∣∣∣∣∣∣∣∣∣ 0 0 e(ρ2+α)x e−(ρ2+α)x H0(x) 1 1 0 0 U1 0 0 1 1 U2 eρ e−ρ −pρ2eρ 2+α −pρ2e−(ρ2+α) U3 + pφ1(1) pρeρ −pρe−ρ (ρ2 + α)eρ 2+α −(ρ2 + α)e−(ρ2+α) U4 ∣∣∣∣∣∣∣∣∣∣ , with U1 = 1 4ρ ∫ 1 0 ( e−ρξ − eρξ ) g1(ξ)dξ, U2 = 1 4(ρ2 + α) ∫ 1 0 [ e−(ρ2+α)ξ − e(ρ2+α)ξ ][ (ρ2 + 2α)φ1(ξ) + θ1(ξ) ] dξ, U3 = pρ2 4(ρ2 + α) ∫ 1 0 [ e(ρ2+α)(1−ξ) − e−(ρ2+α)(1−ξ)][(ρ2 + 2α)φ1(ξ) + θ1(ξ) ] dξ − 1 4ρ ∫ 1 0 [ eρ(1−ξ) − e−ρ(1−ξ) ] g1(ξ)dξ, 18 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 U4 = −1 4 ∫ 1 0 [ e(ρ2+α)(1−ξ) + e−(ρ2+α)(1−ξ)][(ρ2 + 2α)φ1(ξ) + θ1(ξ) ] dξ − p 4 ∫ 1 0 [ eρ(1−ξ) + e−ρ(1−ξ) ] g1(ξ)dξ. Considering ρ ∈ S̄ = {ρ ∈ C | arg ρ ∈ (π/4, π/2]}, we have Re(−ρ) ≤ 0 or Re(ρ2) ≤ 0, (4.7) and ∆(ρ2) = ρ(ρ2 + α)eρ−ρ 2−α∆1(ρ), (4.8) where ∆(ρ2) is given by (3.15) with λ = ρ2, and ∆1(ρ) = p2ρ2 ρ2 + α (1 + e−2ρ) [ e2(ρ2+α) − 1 ] + ρ−1(1− e−2ρ) [ e2(ρ2+α) + 1 ] . (4.9) More accurately, ∆(ρ2) are expressed by (3.25), (3.27) in S1, S2, S3, respectively. Applying the following transformation for determinants F (x, ρ), H(x, ρ), that is, multiplying the first column by 1 4ρ ∫ 1 0 e−ρξg1(ξ)dξ, the second column by 1 4ρ ∫ 1 0 eρξg1(ξ)dξ, the third column by − 1 4(ρ2 + α) ∫ 1 0 e−(ρ2+α)ξ [ (ρ2 + 2α)φ1(ξ) + θ1(ξ) ] dξ, the fourth column by − 1 4(ρ2 + α) ∫ 1 0 e(ρ2+α)ξ [ (ρ2 + 2α)φ1(ξ) + θ1(ξ) ] dξ, and adding this expression to the last column, we have F (x, ρ) = (ρ2 + α)eρ−ρ 2−αF1(x, ρ), H(x, ρ) = ρeρ−ρ 2−αH1(x, ρ), Hx(x, ρ) = ρ(ρ2 + α)eρ−ρ 2−αH1x(x, ρ), (4.10) where F1(x, ρ) = ∣∣∣∣∣∣∣∣∣∣∣∣ e−ρ(1−x) e−ρx 0 0 ρ−1F̃0(x) e−ρ 1 0 0 ρ−1Ũ1 0 0 1 eρ 2+α Ũ2 ρ2+α 1 e−ρ −pρ2eρ 2+α −pρ2 Ũ3 + pφ1(1) pρ ρ2+α −pρ ρ2+αe −ρ eρ 2+α −1 Ũ4 ρ2+α , ∣∣∣∣∣∣∣∣∣∣∣∣ , H1(x, ρ) = ∣∣∣∣∣∣∣∣∣∣∣∣ 0 0 e(ρ2+α)x e(ρ2+α)(1−x) H̃0(x) ρ2+α e−ρ 1 0 0 ρ−1Ũ1 0 0 1 eρ 2+α Ũ2 ρ2+α ρ−1 ρ−1e−ρ −pρeρ2+α −pρ Ũ3+pφ1(1) ρ pρ −pρe−ρ (ρ2 + α)eρ 2+α −(ρ2 + α) Ũ4 ∣∣∣∣∣∣∣∣∣∣∣∣ , EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 19 H1x(x, ρ) = ∣∣∣∣∣∣∣∣∣∣∣∣ 0 0 e(ρ2+α)x −e(ρ2+α)(1−x) H̃′0(x) ρ2+α e−ρ 1 0 0 ρ−1Ũ1 0 0 1 eρ 2+α Ũ2 ρ2+α ρ−1 ρ−1e−ρ −pρeρ2+α −pρ Ũ3+pφ1(1) ρ pρ −pρe−ρ (ρ2 + α)eρ 2+α −(ρ2 + α) Ũ4 ∣∣∣∣∣∣∣∣∣∣∣∣ , and F̃0(x) = 1 2 ∫ x 0 e−ρ(x−ξ)g1(ξ)dξ + 1 2 ∫ 1 x e−ρ(ξ−x)g1(ξ)dξ, H̃0(x) = −1 2 ∫ x 0 e(ρ2+α)(x−ξ)[(ρ2 + 2α)φ1(ξ) + θ1(ξ)]dξ − 1 2 ∫ 1 x e(ρ2+α)(ξ−x)[(ρ2 + 2α)φ1(ξ) + θ1(ξ)]dξ, H̃ ′0(x) = 1 2 ∫ 1 x e(ρ2+α)(ξ−x)[(ρ2 + 2α)φ1(ξ) + θ1(ξ)]dξ − 1 2 ∫ x 0 e(ρ2+α)(x−ξ)[(ρ2 + 2α)φ1(ξ) + θ1(ξ)]dξ, Ũ1 = 1 2 ∫ 1 0 e−ρξg1(ξ)dξ, Ũ2 = −1 2 ∫ 1 0 e(ρ2+α)ξ[(ρ2 + 2α)φ1(ξ) + θ1(ξ)]dξ, Ũ3 = 1 2ρ ∫ 1 0 e−ρ(1−ξ)g1(ξ)dξ + pρ2 2(ρ2 + α) ∫ 1 0 e(ρ2+α)(1−ξ)[(ρ2 + 2α)φ1(ξ) + θ1(ξ)]dξ, Ũ4 = −p 2 ∫ 1 0 e−ρ(1−ξ)g1(ξ)dξ − 1 2 ∫ 1 0 e(ρ2+α)(1−ξ)[(ρ2 + 2α)φ1(ξ) + θ1(ξ)]dξ. According to (4.7), F1(x, ρ), H1(x, ρ), H1x(x, ρ) are bounded after a direct calcu- lation of matrix determinant. From (4.8) and (4.10), (4.6) becomes g(x) = F1(x, ρ) ρ∆1(ρ) , φ(x) = H1(x, ρ) (ρ2 + α)∆1(ρ) , φ′(x) = H1x(x, ρ) ∆1(ρ) , θ(x) = ρ2φ(x)− φ1(x) = ρ2H1(x, ρ) (ρ2 + α)∆1(ρ) − φ1(x). It follows from Lemma 4.1 that there exists a constant M1 > 0, such that |g(x)| ≤ M1 |ρ| {∫ 1 0 [|g1(ξ)|+ (ρ2 + 2α)|φ1(ξ)|+ |θ1(ξ)|]dξ + |pφ1(1)| } , |φ′(x)| ≤M1 {∫ 1 0 [|g1(ξ)|+ (ρ2 + 2α)|φ1(ξ)|+ |θ1(ξ)|]dξ + |pφ1(1)| } , 20 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 |θ(x)| ≤M1 {∫ 1 0 [|g1(ξ)|+ (ρ2 + 2α)|φ1(ξ)|+ |θ1(ξ)|]dξ + |pφ1(1)| } + |φ1(x)|. It is known that |f(x)| ≤ ‖f ′‖L1 ≤ ‖f ′‖L2 and ‖f‖L1 ≤ ‖f‖L2 hold for any function f(x) ∈ H1 L(0, 1). Hence, |λ|−1|g(x)| ≤ M1 |ρ| [ |λ|−1‖g1(ξ)‖L2 + (1 + 2α|λ|−1)‖φ1(ξ)‖L2 + |λ|−1‖θ1(ξ)‖L2 + |λ|−1‖pφ′1‖L2 ] , |λ|−1|φ′(x)| ≤M1 [ |λ|−1‖g1(ξ)‖L2 + (1 + 2α|λ|−1)‖φ1(ξ)‖L2 + |λ|−1‖θ1(ξ)‖L2 + |λ|−1‖pφ′1‖L2 ] , |λ|−1|θ(x)| ≤M1 [ |λ|−1‖g1(ξ)‖L2 + (1 + 2α|λ|−1)‖φ1(ξ)‖L2 + |λ|−1‖θ1(ξ)‖L2 + |λ|−1‖pφ′1‖L2 ] + |λ|−1‖φ′1‖L2 , which leads to ‖(g, φ, θ)‖ ≤M2|λ‖|(g1, φ1, θ1)‖, |λ| > K > 1, ‖(g, φ, θ)‖ ≤M3‖(g1, φ1, θ1)‖, M3 > M2, |λ| ≤ K, where M2, M3, K are positive constants independent of λ. Therefore, ‖(g, φ, θ)‖ ≤M(1 + |λ|)‖(g1, φ1, θ1)‖, for some constant M > 0 independent of λ and all λ = ρ2 with ρ ∈ S̄ lying outside all circles of radius ε > 0 that are centered at the zeros of ∆(ρ2). Finally, we have similar results for ρ ∈ Ŝ = {ρ ∈ C : arg ρ ∈ [π/2, 3π/4]} ([19, pp. 56-60]) which yields (4.1). The proof is complete. � Now, we show the root subspace of system (3.2) is complete. Theorem 4.3. Let A be defined by (3.1). Then both the root subspace of A and A∗ are complete in H, that is, Sp(A∗) = Sp(A) = H. Proof. We only show the completeness for the root subspace of A since the proof for A∗ is almost the same. From [7, Lemma 5 p.2355] the following orthogonal decomposition holds: H = σ∞(A∗)⊕ Sp(A), where σ∞(A∗) consists of those Y ∈ H so that R(λ,A∗)Y is an analytic function of λ anywhere in the whole complex plane. Hence, Sp(A) = H if and only if σ∞(A∗) = {0}. Let Y ∈ σ∞(A∗). According to ‖R(λ,A∗)‖ = ‖R(λ,A)‖ and Theorem 4.2, we obtain ‖R(λ,A∗)Y ‖ ≤M(1 + |λ|)Y, ∀λ ∈ C, for some constant M > 0 by the maximum modulus principle. From [15, Theorem 1 p. 3], R(λ,A∗)Y is a polynomial with degree of λ at most 1, that is, R(λ,A∗)Y = Y0 + λY1 for some Y0, Y1 ∈ H. Thus Y = (λ−A∗)(Y0 + λY1). Since A∗ is a closed operator, Y1 ∈ D(A∗) and so does Y0. Therefore, λ2Y1 + λ(Y0 −A∗Y1)−A∗Y0 = Y, ∀λ ∈ C. Comparing the coefficient of λ2, λ and λ0 in both sides of the above equation, we obtain Y1 = Y0 = Y = 0. The proof is complete. � EJDE-2023/03 INTERCONNECTED HEAT-WAVE EQUATIONS 21 5. Riesz basis property and exponential stability In this section, we show the Riesz basis generation and the exponential stability of system (3.2). For this purpose, we recall the following lemmas: Lemma 5.1. Let {ei}∞i=1 and {e∗i }∞i=1 be two approximately normalized and biorthog- onal sequences. Then {ei}∞i=1 and {e∗i }∞i=1 are Riesz basis for a Hilbert space H if and only if [29, p. 27] (a) both {ei}∞i=1 and {e∗i } ∞ i=1 are complete in H; (b) both {ei}∞i=1 and {e∗i } ∞ i=1 are Bessel sequences in H, that is, for any f ∈ H, two sequences {〈f, ei〉}∞i=1 and {〈f, e∗i 〉}∞i=1 belong to `2. Lemma 5.2 ([22, Lemma 3.2]). Let {µn} be a sequence which has asymptotic expressions µn = α(n+ iβ lnn) +O(1), α 6= 0, n = 1, 2, 3, . . . , where β is a real number. If µn satisfies supn Re(µn) <∞, the sequence {eµnx}∞n=1 is a Bessel sequence in L2(0, 1). Lemma 5.3. The sequences {e(−l1+l2i)x}∞n=0, {e(l1−l2i)x}∞n=0, {e 2n+1 2 πix}∞n=0, {e− 2n+1 2 πix}∞n=0, {e(α− (2n+1)2 4 π2+2p−2)(1−x)}∞n=0, are Bessel sequences in L2(0, 1), where l1, l2 are given by (3.22). Proof. (i) Let α1 = πi, β1 = 0, α2 = −πi, β2 = 0, we have {e(−l1+l2i)x}∞n=0, {e(l1−l2i)x}∞n=0, are Bessel sequences from Lemma 5.2. (ii) Let α3 = πi, β3 = 0, α4 = −πi, β4 = 0, we have {e 2n+1 2 πix}∞n=0, {e− 2n+1 2 πix}∞n=0, are two Bessel sequences. With α5 = −π2, β5 = 0, we obtain {e(α− (2n+1)2 4 π2+2p−2)(1−x)}∞n=0, is also a Bessel sequence. The proof is complete. � Now we can establish the Riesz basis property of system (3.2). Theorem 5.4. Let A be defined by (3.1). Then the generalized eigenfunctions of A form a Riesz basis for H. Proof. Let σ(A) = {λ1n, λ1n, λ2n}∞n=1 be the eigenvalues of A. By Theorem 3.2 and Theorem 3.3, we have that each eigenvalue of A with sufficient large modulus is simple, and hence there exists an integer N > 0 such that all λ1n, λ1n, λ2n with n ≥ N , are algebraically simple. If the algebraic multiplicities of λ1n, λ2n for n ≤ N are m1n,m2n respectively, we have the generalized eigenfunctions Φ1n,1,Φ2n,1 satisfy (A− λ1n)m1nΦ1n,1 = 0, (A− λ1n)m1n−1Φ1n,1 6= 0, (A− λ2n)m2nΦ2n,1 = 0, (A− λ2n)m2n−1Φ2n,1 6= 0, 22 X.-F. YU, J.-M. WANG, H.-W. ZHANG EJDE-2023/03 and Φ1n,j = (A− λ1n)j−1Φ1n,1, j = 2, 3, . . . ,m1n, Φ2n,p = (A− λ2n)p−1Φ2n,1, p = 2, 3, . . . ,m2n, where Φ1n,m1n and Φ2n,m2n are the eigenfunctions of A with respect to λ1n and λ2n respectively. Assume Φ1n and Φ2n are the normalized eigenfunctions of A corresponding to λ1n and λ2n with n ≥ N respectively. Then A and A∗ have linearly independent bi-orthogonal generalized eigenfunctions{ {Φ1n,j ,Φ1n,j}m1n j=1 , {Φ2n,p}m2n p=1 } n