Abstract
We consider a system of transmission of the wave equation with Neumann feedback control that contains a distributed delay term and that acts on the exterior boundary. We prove under some assumptions that the solutions decay exponentially in an appropriate energy space. To establish this result, we introduce a suitable energy function and use multipliers technique method and compactness-uniqueness argument.
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Abdallah, C., Dorato, P., Benitez-Read, J., Byrne, R.: Delayed positive feedback can stabilize oscillatory systems. in: ACC’ 1993, San Francisco, pp. 3106–3107 (1993)
Ammari, K., Gerbi, S.: Interior feedback stabilization of wave equations with dynamic boundary delay. Z. Anal. Anwend. 36(3), 297–327 (2017)
Batzel, J.J., Kappel, F., Timischl-Teschl, S.: A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans. J. Math. Biol. 50(3), 293–335 (2005)
Bayili, G., Aissa, A.B., Nicaise, S.: Same decay rate of second order evolution equations with or without delay. Syst. Control Lett. 141, 104700 (2020)
Benseghir, A.: Existence and exponential decay of solutions for transmission problems with delay. Electron. J. Differ. Eq. 214(212), 1–11 (2014)
Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control. Optim. 26(3), 697–713 (1988)
Datko, R., Lagnese, J., Polis, M.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control. Optim. 24(1), 152–156 (1986)
Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology: Physical Origins and Classical Methods. Springer-Verlag, Berlin, Heidelberg (1990)
Engel, K.-J., Nagel, R., Brendle, S.: One-Parameter Semigroups for Linear Evolution Equations, vol. 194. Springer-Verlag, New York (2000)
Ghecham, W., Rebiai, S.-E., Sidiali, F.Z.: Stabilization of coupled wave equations with boundary or internal distributed delay. Appl. Anal. 100(14), 3096–3115 (2021)
Hale, J., Verduyn Lunel, S.: Introduction to Functional Differential Equations. Springer-Verlag, New York (1993)
Lasiecka, I., Triggiani, R.: Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations. In: Lecture Notes in Pure and Applied Mathematics, vol. 188, pp. 215–243. Marcel Dekker, New York (1997)
Lasiecka, I., Triggiani, R.: Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25(2), 189–224 (1992)
Lions, J.-L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1: Contrôlabilité exacte. Collection de recherche en mathématiques appliquées, Masson, Paris (1988)
Lions, J.-L., Magenes, E.: Problemes aux Limites non Homogenes et Applications, vol. 1. Dunod, Paris (1968)
Liu, W.: Stabilization and controllability for the transmission wave equation. IEEE Trans. Autom. Control 46(12), 1900–1907 (2001)
Liu, W., Williams, G.: The exponential stability of the problem of transmission of the wave equation. Bull. Aust. Math. Soc. 57(2), 305–327 (1998)
Liu, G.: Well-posedness and exponential decay of solutions for a transmission problem with distributed delay. Electron. J. Differ. Eq. 174, 1–13 (2017)
Liu, W.: Exponential stability of the energy of the wave equation with variable coefficients and a boundary distributed delay. Z. Naturforschung A 69(10–11), 547–552 (2014)
Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control. Optim. 45(5), 1561–1585 (2006)
Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integr. Eq. 21(9–10), 935–958 (2008)
Ning, Z.-H., Yan, Q.-X.: Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback. J. Math. Anal. Appl. 367(1), 167–173 (2010)
Ning, Z.-H., Shen, C.-X., Zhao, X.: Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback. J. Math. Anal. Appl. 405(1), 148–155 (2013)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)
Rebiai, S.-E., Sidiali, F.Z.: Uniform exponential stability of the transmission wave equation with a delay term in the boundary feedback. IMA J. Math. Control. Inf. 33(1), 1–20 (2016)
Suh, I.H., Bien, Z.: Use of time delay action in the controller design. IEEE Trans. Automat. Control 25(3), 600–603 (1980)
Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM: Control, Optim. Calculus Variat. 12(4), 770–785 (2006)
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Appendix A Proof of the identity (3.10)
Appendix A Proof of the identity (3.10)
We multiply both sides of (1.1) by \(2\,h(x).\nabla y_{i}(x,t)+(divh(x)-\alpha )y_{i}(x,t)\) and integrate over \(\Omega _{i}\times (0,T), i=1,2;\) we have
We sum up (A1) for i, we obtain
Below, we compute the terms on the left-hand side of (A2).
-
Term \(2\int _{0}^{T}\int _{\Omega }\partial _{t}^{2}y(x,t)h(x).\nabla y(x,t)dxdt\) Integration by parts in t yields
$$\begin{aligned}&2\int _{0}^{T}\int _{\Omega }\partial _{t}^{2}y(x,t)h(x).\nabla y(x,t)dxdt=2 \left[ \int _{\Omega }\partial _{t}y(x,t)h(x).\nabla y(x,t)dx\right] _{0}^{T} \nonumber \\&-\quad 2\int _{0}^{T}\int _{\Omega }\partial _{t}y(x,t)h(x).\nabla \partial _{t}y(x,t)dxdt \nonumber \\&\quad = 2\left[ \int _{\Omega }\partial _{t}y(x,t)h(x).\nabla y(x,t)dx\right] _{0}^{T}\nonumber \\&\quad -\int _{0}^{T}\int _{\Omega }h(x).\nabla ((\partial _{t}y(x,t))^{2})dxdt. \end{aligned}$$(A3)Applying Green’s theorem to the second integral on the right-hand side of (A3), we obtain
$$\begin{aligned}&2\int _{0}^{T}\int _{\Omega }\partial _{t}^{2}y(x,t)h(x).\nabla y(x,t)dxdt\nonumber \\&\quad =2 \left[ \int _{\Omega }\partial _{t}y(x,t)h(x).\nabla y(x,t)dx\right] _{0}^{T} \nonumber \\&\quad \quad -\int _{0}^{T}\int _{\Gamma }(\partial _{t}y(x,t))^{2}h(x).\nu (x)d\Gamma dt \mathbf {+}\int _{0}^{T}\int _{\Omega }(\partial _{t}y(x,t))^{2}divh(x)dxdt. \end{aligned}$$(A4) -
Term \(\int _{0}^{T}\int _{\Omega }\partial _{t}^{2}y(x,t)(divh(x)-\alpha )y(x,t)dxdt\) Using again integration by parts in t, we obtain
$$\begin{aligned}&\int _{0}^{T}\int _{\Omega }\partial _{t}^{2}y(x,t)(divh(x)-\alpha )y(x,t)dxdt \nonumber \\&\quad = \left[ \int _{\Omega }\partial _{t}y(x,t)(divh(x)-\alpha )y(x,t)dx\right] _{0}^{T}- \int _{0}^{T}\int _{\Omega }(\partial _{t}y(x,t))^{2}div(h(x)-\alpha )dxdt. \end{aligned}$$(A5) -
Term \(2\int _{0}^{T}\int _{\Omega _{1}}a_{1}\Delta y_{1}(x,t))h(x).\nabla y_{1}(x,t)dxdt+2\int _{0}^{T}\int _{\Omega _{2}}a_{2}\Delta y_{2}(x,t))h(x).\nabla y_{2}(x,t)dxdt\) From Green’s theorem, we have,
$$\begin{aligned}&2a_{1}\int _{0}^{T}\int _{\Omega _{1}}\Delta y_{1}(x,t)h(x).\nabla y_{1}(x,t)dxdt\nonumber \\&\quad =2a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\frac{\partial y_{1}(x,t) }{\partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt \nonumber \\&\quad \quad + 2a_{1}\int _{0}^{T}\int _{\Gamma _{0}}\frac{\partial y_{1}(x,t)}{\partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt\nonumber \\&\quad \quad - 2a_{1}\int _{0}^{T}\int _{\Omega _{1}}\nabla y_{1}(x,t).\nabla (h(x).\nabla y_{1}(x,t))dxdt. \end{aligned}$$(A6)Applying the identity
$$\begin{aligned} \nabla w(x).\nabla (h(x).\nabla w(x))=J(x)\nabla w(x).\nabla w(x)+\frac{1}{2} h(x).\nabla \left( \left| \nabla w(x)\right| ^{2}\right) \end{aligned}$$to the last integral on the right hand side of (A6), we find
$$\begin{aligned}&2a_{1}\int _{0}^{T}\int _{\Omega _{1}}\Delta y_{1}(x,t)h(x).\nabla y_{1}(x,t)dxdt\\&\quad =2a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\frac{\partial y_{1}(x,t) }{\partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt \\&\quad\quad +2a_{1}\int _{0}^{T}\int _{\Gamma _{0}}\frac{\partial y_{1}(x,t)}{\partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt\\&\quad\quad -2a_{1}\int _{0}^{T}\int _{\Omega _{1}}J(x)\nabla y_{1}(x,t).\nabla y_{1}(x,t))dxdt \\&\quad\quad- a_{1}\int _{0}^{T}\int _{\Omega _{1}}h(x).\nabla \left( \left| \nabla y_{1}(x,t)\right| ^{2}\right) . \end{aligned}$$Another use of Green’s theorem yields
$$\begin{aligned}&2a_{1}\int _{0}^{T}\int _{\Omega _{1}}\Delta y_{1}(x,t)h(x).\nabla y_{1}(x,t)dxdt\nonumber \\&\quad =2a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\frac{\partial y_{1}(x,t) }{\partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt \nonumber \\&\quad\quad +2a_{1}\int _{0}^{T}\int _{\Gamma _{0}}\frac{\partial y_{1}(x,t)}{\partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt\nonumber \\&\quad\quad -2a_{1}\int _{0}^{T}\int _{\Omega _{1}}J(x)\nabla y_{1}(x,t).\nabla y_{1}(x,t))dxdt \nonumber \\&\quad \quad-a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt\nonumber \\&\quad \quad-a_{1}\int _{0}^{T}\int _{\Gamma _{0}}\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt \nonumber \\&\quad \quad+ a_{1}\int _{0}^{T}\int _{\Omega _{1}}\left| \nabla y_{1}(x,t)\right| ^{2}divh(x)dxdt. \end{aligned}$$(A7)For the intgral term \(2\int _{0}^{T}\int _{\Omega _{2}}a_{2}\Delta y_{2}(x,t))h(x).\nabla y_{2}(x,t)dxdt\), we proceed as above to find
$$\begin{aligned}&2a_{2}\int _{0}^{T}\int _{\Omega _{2}}\Delta y_{2}(x,t)h(x).\nabla y_{2}(x,t)d\Omega dt\nonumber \\&\quad =2a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\frac{\partial y_{2}(x,t)}{\partial \nu }h(x).\nabla y_{2}(x,t)d\Gamma dt \nonumber \\&\quad -2a_{2}\int _{0}^{T}\int _{\Gamma _{0}}\frac{\partial y_{2}(x,t)}{\partial \nu }h(x).\nabla y_{2}(x,t)d\Gamma dt\nonumber \\&\quad -2a_{2}\int _{0}^{T}\int _{\Omega _{2}}J(x)\nabla y_{2}(x,t).\nabla y_{2}(x,t))d\Omega dt \nonumber \\&\quad- a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\left| \nabla y_{2}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt\nonumber \\&\quad +a_{2}\int _{0}^{T}\int _{\Gamma _{0}}\left| \nabla y_{2}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt \nonumber \\&\quad +a_{2}\int _{0}^{T}\int _{\Omega _{2}}\left| \nabla y_{2}(x,t)\right| ^{2}divh(x)d\Omega dt. \end{aligned}$$(A8)Summing up (A7) and (A8) yields
$$\begin{aligned}&2\int _{0}^{T}\int _{\Omega }div(a(x)\nabla y(x,t))h(x).\nabla y(x,t) dxdt\nonumber \\&\quad =2a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\frac{\partial y_{1}(x,t)}{ \partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt\nonumber \\&\quad + 2a_{1}\int _{0}^{T}\int _{\Gamma _{0}}\frac{\partial y_{1}(x,t)}{\partial \nu }h(x).\nabla y_{1}(x,t)d\Gamma dt\nonumber \\&\quad - 2a_{1}\int _{0}^{T}\int _{\Omega _{1}}J(x)\nabla y_{1}(x,t).\nabla y_{1}(x,t))dxdt \nonumber \\&\quad - a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt\nonumber \\&\quad -a_{1}\int _{0}^{T}\int _{\Gamma _{0}}\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt \nonumber \\&\quad + a_{1}\int _{0}^{T}\int _{\Omega _{1}}\left| \nabla y_{1}(x,t)\right| ^{2}divh(x)dxdt\nonumber \\&\quad +2a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\frac{\partial y_{2}(x,t) }{\partial \nu }h(x).\nabla y_{2}(x,t)d\Gamma dt \nonumber \\&\quad - 2a_{2}\int _{0}^{T}\int _{\Gamma _{0}}\frac{\partial y_{2}(x,t)}{\partial \nu }h(x).\nabla y_{2}(x,t)d\Gamma dt \nonumber \\&\quad - 2a_{2}\int _{0}^{T}\int _{\Omega _{2}}J(x)\nabla y_{2}(x,t).\nabla y_{2}(x,t))dxdt \nonumber \\&\quad -a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\left| \nabla y_{2}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt\nonumber \\&\quad +a_{2}\int _{0}^{T}\int _{\Gamma _{0}}\left| \nabla y_{2}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt+ \nonumber \\&\quad a_{2}\int _{0}^{T}\int _{\Omega _{2}}\left| \nabla y_{2}(x,t)\right| ^{2}divh(x)dxdt. \end{aligned}$$(A9)We conclude from the boundary conditions (1.3) and (1.5) that
$$\begin{aligned} \nabla y_{1}(x,t)=\frac{\partial y_{1}(x,t)}{\partial \nu }\nu (x)\text { on }\Gamma _{1}\times (0,T), \end{aligned}$$(A10)and
$$\begin{aligned} \nabla (y_{2}(x,t)-y_{1}(x,t))=\frac{\partial (y_{2}(x,t)-y_{1}(x,t))}{ \partial \nu }\nu (x),\text { on }\Gamma _{0}\times (0,T). \end{aligned}$$Then
$$\begin{aligned}&\left| \nabla y_{2}(x,t)\right| ^{2} =\left| \nabla y_{1}(x,t)\right| ^{2}+2\left( \frac{\partial y_{2}(x,t)}{\partial \nu }-\frac{ \partial y_{1}(x,t)}{\partial \nu }\right) \frac{\partial y_{1}(x,t)}{\partial \nu } \nonumber \\&\quad +\left( \frac{\partial y_{2}(x,t)}{\partial \nu }-\frac{\partial y_{1}(x,t)}{ \partial \nu }\right) ^{2} =\left| \nabla y_{1}(x,t)\right| ^{2}+ \left( \frac{\partial y_{2}(x,t)}{ \partial \nu }\right) ^{2}\nonumber \\&\quad - \left( \frac{\partial y_{1}(x,t)}{\partial \nu }\right) ^{2},\text { on }\Gamma _{0}\times (0,T), \end{aligned}$$so on \(\Gamma _{0}\times (0,T),\)
$$\begin{aligned}&2a_{1}\frac{\partial y_{1}(x,t)}{\partial \nu }h(x).\nabla y_{1}(x,t)\nonumber \\&\quad -2a_{2}\frac{\partial y_{2}(x,t)}{\partial \nu }h(x).\nabla y_{2}(x,t)- a_{1}\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)\nonumber \\&\quad + a_{2}\left| \nabla y_{2}(x,t)\right| ^{2}h(x).\nu (x)\nonumber \\&\quad =2a_{1}\frac{ \partial y_{1}(x,t)}{\partial \nu }h(x).\nabla y_{1}(x,t)-2a_{2}\frac{ {\partial }y_{2}(x,t)}{\partial \nu }\bigg (\nabla y_{1}(x,t)\nonumber \\&\quad + \left( \frac{ \partial y_{2}(x,t)}{\partial \nu }-\frac{{\partial }y_{1}(x,t)}{ {\partial }\nu }\right) \nu (x)\bigg ).h(x)\nonumber \\&\quad - a_{1}\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)+a_{2}\bigg (\left| \nabla y_{1}(x,t)\right| ^{2}\nonumber \\&\quad + \left( \frac{\partial y_{2}(x,t)}{{\partial }\nu }\right) ^{2}-\left( \frac{{\partial }y_{1}(x,t) }{{\partial }\nu }\right) \bigg )h(x).\nu (x)\nonumber \\&\quad =\mathbf {-}2a_{1}\left( \frac{a_{1}}{a_{2}}-1\right) \left( \frac{\partial y_{1}(x,t)}{ \partial \nu }\right) ^{2}h(x).\nu (x)\nonumber \\&\quad\quad + (a_{2}-a_{1})\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)+\bigg (\frac{a_{1}^{2}}{a_{2}}\nonumber \\&\quad -a_{2}\bigg )\left( \frac{ \partial y_{1}(x,t)}{\partial \nu }\right) ^{2}h(x).\nu (x) \nonumber \\&\quad = (a_{2}-a_{1})\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)\nonumber \\&\quad - \frac{(a_{2}-a_{1})^{2}}{a_{2}}\left( \frac{\partial y_{1}(x,t)}{\partial \nu } \right) ^{2}h(x).\nu (x). \end{aligned}$$(A11)Insertion of (A10) and (A11) into (A9) results in
$$\begin{aligned}&2\int _{0}^{T}\int _{\Omega }div(a(x)\nabla y(x,t))h(x).\nabla y(x,t)dxdt\nonumber \\&\quad =a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\left( \frac{\partial y_{1}(x,t)}{ \partial \nu }\right) ^{2}h(x).\nu (x)d\Gamma dt \nonumber \\&\quad- (a_{1}-a_{2})\int _{0}^{T}\int _{\Gamma _{0}}\left| \nabla y_{1}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma \nonumber \\&\quad- \frac{(a_{2}-a_{1})^{2}}{a_{2} }\int _{0}^{T}\int _{\Gamma _{0}}\left( \frac{\partial y_{1}(x,t)}{\partial \nu } \right) ^{2}h(x).\nu (x)d\Gamma dt \nonumber \\&\quad+ 2a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\frac{\partial y_{2}(x,t)}{\partial \nu }h(x).\nabla y_{2}(x,t)d\Gamma dt\nonumber \\&\quad -a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\left| \nabla y_{2}(x,t)\right| ^{2}h(x).\nu (x)d\Gamma dt\nonumber \\&\quad- 2\int _{0}^{T}\int _{\Omega }a(x)J(x)\nabla y(x,t).\nabla y(x,t))dxdt\nonumber \\&\quad +\int _{0}^{T}\int _{\Omega }a(x)\left| \nabla y(x,t)\right| ^{2}divh(x)dxdt. \end{aligned}$$(A12) -
Term \(a_{1}\int _{0}^{T}\int _{\Omega _{1}}\Delta y_{1}(x,t)(divh(x)-\alpha )y_{1}(x,t)dxdt+a_{2}\int _{0}^{T}\int _{\Omega _{2}}\Delta y_{2}(x,t)(divh(x)-\alpha )y_{2}(x,t)dxdt\) It follows from Green’s theorem that
$$\begin{aligned}&a_{1}\int _{0}^{T}\int _{\Omega _{1}}\Delta y_{1}(x,t)(divh(x)-\alpha )y_{1}(x,t)dxdt \\&+\quad a_{2}\int _{0}^{T}\int _{\Omega _{2}}\Delta y_{2}(x,t)(divh(x)-\alpha )y_{2}(x,t)dxdt \\&=\quad a_{1}\int _{0}^{T}\int _{\Gamma _{1}}\frac{\partial y_{1}(x,t)}{\partial \nu }(divh(x)-\alpha )y_{1}(x,t)d\Gamma dt \\&+\quad \quad a_{1}\int _{0}^{T}\int _{\Gamma _{0}} \frac{\partial y_{1}(x,t)}{\partial \nu }(divh(x)-\alpha )y_{1}(x,t)d\Gamma dt\\&-\quad \quad a_{1}\int _{0}^{T}\int _{\Omega _{1}}\left| \nabla y_{1}(x,t)\right| ^{2}(divh(x)-\alpha )dxdt \\&-\quad \quad a_{1}\int _{0}^{T}\int _{\Omega _{1}}y_{1}(x,t)\nabla y_{1}(x,t).\nabla (divh(x)-\alpha )dxdt \\&+\quad \quad a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\frac{\partial y_{2}(x,t)}{\partial \nu }(divh(x)-\alpha )y_{2}(x,t)d\Gamma dt \\&-\quad \quad a_{2}\int _{0}^{T}\int _{\Gamma _{0}} \frac{\partial y_{2}(x,t)}{\partial \nu }(divh(x)-\alpha )y_{2}(x,t)d\Gamma dt \\&-\quad \quad a_{2}\int _{0}^{T}\int _{\Omega _{2}}\left| \nabla y_{2}(x,t)\right| ^{2}(divh(x)-\alpha )dxdt\\&-\quad \quad a_{2}\int _{0}^{T}\int _{\Omega _{2}}y_{2}(x,t)\nabla y_{2}(x,t).\nabla (divh(x)-\alpha )dxdt. \end{aligned}$$Thus from (1.3), (1.5) and (1.6), we conclude that
$$\begin{aligned}&a_{1}\int _{0}^{T}\int _{\Omega _{1}}\Delta y_{1}(x,t)(divh(x)-\alpha )y_{1}(x,t)dxdt\nonumber \\&+\quad a_{2}\int _{0}^{T}\int _{\Omega _{2}}\Delta y_{2}(x,t)(divh(x)-\alpha )y_{2}(x,t)dxdt \nonumber \\&=\quad a_{2}\int _{0}^{T}\int _{\Gamma _{2}}\frac{\partial y_{2}(x,t)}{\partial \nu }(divh(x)-\alpha )y_{2}(x,t)d\Gamma dt \nonumber \\&- \quad \quad \int _{0}^{T}\int _{\Omega }a(x)\left| \nabla y(x,t)\right| ^{2}(divh(x)-\alpha )dxdt \nonumber \\&-\quad \quad \int _{0}^{T}\int _{\Omega }a(x)y(x,t)\nabla y(x,t).\nabla (divh(x)-\alpha )dxdt. \end{aligned}$$(A13)The desired identity follows now from (A2), (A4), (A5), (A12) and (A13).
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Moumen, L., Rebiai, SE. Exponential stability of the transmission wave equation with a distributed delay term in the boundary damping. Rend. Circ. Mat. Palermo, II. Ser 72, 3459–3486 (2023). https://doi.org/10.1007/s12215-022-00834-8
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DOI: https://doi.org/10.1007/s12215-022-00834-8