# American Institute of Mathematical Sciences

December  2016, 11(4): 655-692. doi: 10.3934/nhm.2016013

## Decay rates for $1-d$ heat-wave planar networks

 1 Department of Mathematics, Tianjin University, Tianjin 300072 2 Deusto Tech, University of Deusto, 48007 Bilbao, Basque Country, Spain

Received  December 2015 Revised  May 2016 Published  October 2016

The large time decay rates of a transmission problem coupling heat and wave equations on a planar network is discussed.
When all edges evolve according to the heat equation, the uniform exponential decay holds. By the contrary, we show the lack of uniform stability, based on a Geometric Optics high frequency asymptotic expansion, whenever the network involves at least one wave equation.
The (slow) decay rate of this system is further discussed for star-shaped networks. When only one wave equation is present in the network, by the frequency domain approach together with multipliers, we derive a sharp polynomial decay rate. When the network involves more than one wave equation, a weakened observability estimate is obtained, based on which, polynomial and logarithmic decay rates are deduced for smooth initial conditions under certain irrationality conditions on the lengths of the strings entering in the network. These decay rates are intrinsically determined by the wave equations entering in the system and are independent on the heat equations.
Citation: Zhong-Jie Han, Enrique Zuazua. Decay rates for $1-d$ heat-wave planar networks. Networks and Heterogeneous Media, 2016, 11 (4) : 655-692. doi: 10.3934/nhm.2016013
##### References:
 [1] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410. [2] K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2007), 327-343. doi: 10.1007/s10492-007-0018-1. [3] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM J. Control Optim., 39 (2000), 1160-1181. doi: 10.1137/S0363012998349315. [4] M. Alves, J. Muñoz Rivera, M. Sepúlveda and O. V. Villagrán, The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math., 74 (2014), 345-365. doi: 10.1137/130923233. [5] V. M. Babich, The higher-dimensional WKB method or ray method. Its analogues and generalizations, in Partial Differential Equations V, Encyclopedia of Mathematical Sciences, Springer-Verlag, Berlin/New York, 34 (1999), 91-131, 241-247. doi: 10.1007/978-3-642-58423-7_3. [6] J. von Below, A characteristic equation associated to an eigenvalue problem on $C^2$-networks, Linear Algebra Appl., 71 (1985), 309-325. doi: 10.1016/0024-3795(85)90258-7. [7] J. von Below, Classical solvability of linear parabolic equations on networks, J. Differ. Equations, 72 (1988), 316-337. doi: 10.1016/0022-0396(88)90158-1. [8] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [9] R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathématiques et Applications 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3. [10] C. Farhat, M. Lesoinne and P. LeTallec, Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity, Comp. Meth. Appl. Mech. Eng., 157 (1998), 95-114. doi: 10.1016/S0045-7825(97)00216-8. [11] Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Appl. Math., 110 (2010), 511-533. doi: 10.1007/s10440-009-9459-8. [12] Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks, Networks and Heterogeneous Media, 5 (2010), 315-334. doi: 10.3934/nhm.2010.5.315. [13] Z. J. Han and G. Q. Xu, Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs, Netw. Heterog. Media, 6 (2011), 297-327. doi: 10.3934/nhm.2011.6.297. [14] J. H. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159. doi: 10.1007/s00033-012-0274-0. [15] J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, in Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8. [16] Z. Liu and R. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4. [17] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, 1999. [18] R. von Loon, P. D. Anderson, J. de Hart and F. P. T. Baaijens, A combined fictitious domain/adaptive meshing method for fluid-structure interaction in heart valves, Int. J. Numer. Meth. Fluids, 46 (2004), 533-544. [19] Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42. [20] D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 337 (2008), 174-196. doi: 10.1016/j.jmaa.2007.03.080. [21] F. Ali Mehmeti, A characterization of a generalized $C^\infty$-notion on nets, Integr. Equat. Oper. Th, 9 (1986), 753-766. doi: 10.1007/BF01202515. [22] H. Morand and R. Ohayon, Fluid Structure Interaction: Applied Numerical Methods, Wiley, New York, 1995. [23] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425. [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [25] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [26] M. E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. [27] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Contr. Optim, 48 (2009), 2771-2797. doi: 10.1137/080733590. [28] G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled networks of strings, SIAM J. Control Optim., 47 (2008), 1762-1784. doi: 10.1137/060649367. [29] X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differ. Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004. [30] X. Zhang ang E. Zuazua, Control, observation and polynomial decay for a coupled heat-wave system, C. R. Acad. Sci. Paris, Ser. I, 336 (2003), 823-828. doi: 10.1016/S1631-073X(03)00204-8. [31] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. An., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x. [32] E. Zuazua, Null control of a 1-d model of mixed hyperbolic-parabolic type, in Optimal Control and Partial Differential Equations, J. L. Menaldi et al., eds., IOS Press, 2001, 198-210.

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##### References:
 [1] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410. [2] K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2007), 327-343. doi: 10.1007/s10492-007-0018-1. [3] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force, SIAM J. Control Optim., 39 (2000), 1160-1181. doi: 10.1137/S0363012998349315. [4] M. Alves, J. Muñoz Rivera, M. Sepúlveda and O. V. Villagrán, The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math., 74 (2014), 345-365. doi: 10.1137/130923233. [5] V. M. Babich, The higher-dimensional WKB method or ray method. Its analogues and generalizations, in Partial Differential Equations V, Encyclopedia of Mathematical Sciences, Springer-Verlag, Berlin/New York, 34 (1999), 91-131, 241-247. doi: 10.1007/978-3-642-58423-7_3. [6] J. von Below, A characteristic equation associated to an eigenvalue problem on $C^2$-networks, Linear Algebra Appl., 71 (1985), 309-325. doi: 10.1016/0024-3795(85)90258-7. [7] J. von Below, Classical solvability of linear parabolic equations on networks, J. Differ. Equations, 72 (1988), 316-337. doi: 10.1016/0022-0396(88)90158-1. [8] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [9] R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathématiques et Applications 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3. [10] C. Farhat, M. Lesoinne and P. LeTallec, Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity, Comp. Meth. Appl. Mech. Eng., 157 (1998), 95-114. doi: 10.1016/S0045-7825(97)00216-8. [11] Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Appl. Math., 110 (2010), 511-533. doi: 10.1007/s10440-009-9459-8. [12] Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks, Networks and Heterogeneous Media, 5 (2010), 315-334. doi: 10.3934/nhm.2010.5.315. [13] Z. J. Han and G. Q. Xu, Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs, Netw. Heterog. Media, 6 (2011), 297-327. doi: 10.3934/nhm.2011.6.297. [14] J. H. Hao and Z. Liu, Stability of an abstract system of coupled hyperbolic and parabolic equations, Z. Angew. Math. Phys., 64 (2013), 1145-1159. doi: 10.1007/s00033-012-0274-0. [15] J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, in Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8. [16] Z. Liu and R. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4. [17] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, 1999. [18] R. von Loon, P. D. Anderson, J. de Hart and F. P. T. Baaijens, A combined fictitious domain/adaptive meshing method for fluid-structure interaction in heart valves, Int. J. Numer. Meth. Fluids, 46 (2004), 533-544. [19] Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42. [20] D. Mercier and V. Régnier, Spectrum of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 337 (2008), 174-196. doi: 10.1016/j.jmaa.2007.03.080. [21] F. Ali Mehmeti, A characterization of a generalized $C^\infty$-notion on nets, Integr. Equat. Oper. Th, 9 (1986), 753-766. doi: 10.1007/BF01202515. [22] H. Morand and R. Ohayon, Fluid Structure Interaction: Applied Numerical Methods, Wiley, New York, 1995. [23] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425. [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [25] J. Rauch, X. Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [26] M. E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. [27] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-d networks, SIAM J. Contr. Optim, 48 (2009), 2771-2797. doi: 10.1137/080733590. [28] G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled networks of strings, SIAM J. Control Optim., 47 (2008), 1762-1784. doi: 10.1137/060649367. [29] X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differ. Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004. [30] X. Zhang ang E. Zuazua, Control, observation and polynomial decay for a coupled heat-wave system, C. R. Acad. Sci. Paris, Ser. I, 336 (2003), 823-828. doi: 10.1016/S1631-073X(03)00204-8. [31] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. An., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x. [32] E. Zuazua, Null control of a 1-d model of mixed hyperbolic-parabolic type, in Optimal Control and Partial Differential Equations, J. L. Menaldi et al., eds., IOS Press, 2001, 198-210.
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