May  2021, 20(5): 1821-1831. doi: 10.3934/cpaa.2021042

A note on the energy transfer in coupled differential systems

Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

* Corresponding author

Received  January 2021 Revised  January 2021 Published  May 2021 Early access  March 2021

We study the energy transfer in the linear system
$ \begin{cases} \ddot u+u+\dot u = b\dot v\\ \ddot v+v-\epsilon \dot v = -b\dot u \end{cases} $
made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter
$ b $
.
Citation: Monica Conti, Lorenzo Liverani, Vittorino Pata. A note on the energy transfer in coupled differential systems. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1821-1831. doi: 10.3934/cpaa.2021042
References:
[1]

F. Alabau-BoussouiraZ. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.  doi: 10.1051/cocv/2016011.

[2]

M. S. AlvesC. BuriolM. V. FerreiraJ. E. Muñoz RiveraM. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019.

[3]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differ. Equ., 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.

[4]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28. 

[5]

G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292.  doi: 10.1007/PL00005977.

[6]

M. ContiV. Pata and R. Quintanilla, Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptot. Anal., 120 (2020), 1-21.  doi: 10.3233/asy-191576.

[7]

F. Dell'Oro and V. Pata, On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differ. Equ., 257 (2014), 523-548.  doi: 10.1016/j.jde.2014.04.009.

[8]

R. Denk and F. Kammerlander, Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.  doi: 10.1093/imamat/hxy002.

[9]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2.

[10]

G. R. GoldsteinJ. A. Goldstein and G. Perla Menzala, On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199.  doi: 10.1090/S0033-569X-2012-01282-3.

[11]

J. 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.

[12] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974. 
[13]

J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047.

[14]

Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC, Boca Raton, 1999. doi: 0-8493-0615-9.

[15]

J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058.

[16]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.

[18]

M. L. SantosD. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012.

[19]

R. Triggiani, Heat-viscoelastic plate interaction: analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.  doi: 10.3934/eect.2018008.

show all references

References:
[1]

F. Alabau-BoussouiraZ. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.  doi: 10.1051/cocv/2016011.

[2]

M. S. AlvesC. BuriolM. V. FerreiraJ. E. Muñoz RiveraM. Sepúlveda and O. Vera, Asymptotic behaviour for the vibrations modeled by the standard linear solid model with a thermal effect, J. Math. Anal. Appl., 399 (2013), 472-479.  doi: 10.1016/j.jmaa.2012.10.019.

[3]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differ. Equ., 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.

[4]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1-28. 

[5]

G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum, 57 (1998), 278-292.  doi: 10.1007/PL00005977.

[6]

M. ContiV. Pata and R. Quintanilla, Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature, Asymptot. Anal., 120 (2020), 1-21.  doi: 10.3233/asy-191576.

[7]

F. Dell'Oro and V. Pata, On the stability of Timoshenko systems with Gurtin-Pipkin thermal law, J. Differ. Equ., 257 (2014), 523-548.  doi: 10.1016/j.jde.2014.04.009.

[8]

R. Denk and F. Kammerlander, Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.  doi: 10.1093/imamat/hxy002.

[9]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Rational Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2.

[10]

G. R. GoldsteinJ. A. Goldstein and G. Perla Menzala, On the overdamping phenomenon: a general result and applications, Quart. Appl. Math., 71 (2013), 183-199.  doi: 10.1090/S0033-569X-2012-01282-3.

[11]

J. 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.

[12] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974. 
[13]

J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.  doi: 10.1137/0523047.

[14]

Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC, Boca Raton, 1999. doi: 0-8493-0615-9.

[15]

J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.  doi: 10.1137/S0036142993255058.

[16]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

L. Perko, Differential equations and dynamical systems, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.

[18]

M. L. SantosD. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differ. Equ., 253 (2012), 2715-2733.  doi: 10.1016/j.jde.2012.07.012.

[19]

R. Triggiani, Heat-viscoelastic plate interaction: analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.  doi: 10.3934/eect.2018008.

Figure 1.  Plot of $ {{\mathtt{E}}} $ for $ \epsilon = 1 $ and $ b = 0.99 $ (black), $ b = 1 $ (blue) and $ b = 1.01 $ (red)
Figure 2.  Parametric plot of $ t\mapsto (u(t), \dot u(t)) $ for $ \epsilon = 1 $ and $ b = \sqrt{\frac{23}{13}+\frac{13}{23}-1} $
Figure 3.  $ {{\mathtt{E}}} $ for $ \epsilon = 1 $ and $ \boldsymbol{z}_0 = (1, 0, 0, 0) $ with different values of $ b $
Figure 4.  $ {{\mathtt{E}}} $ for $ \epsilon = 1 $ and $ \boldsymbol{z}_0 = (1, 0.5, 0, 0) $ with different values of $ b $
Figure 5.  Numerical $ u $ (blue) vs asymptotic $ u $ (red) for $ \epsilon = 1 $ with different values of $ b $ (and different time-scales)
Figure 6.  Numerical $ v $ (blue) vs asymptotic $ v $ (red) for $ \epsilon = 1 $ with different values of $ b $ (and different time-scales)
Figure 7.  Plot of $ {{\mathtt{E}}} $ for $ \epsilon = 0.5 $ and $ b = \sqrt{0.5}-0.1 $ (black), $ b = \sqrt{0.5} $ (blue) and $ b = \sqrt{0.5}+0.1 $ (red)
Figure 8.  Parametric plot of $ t\mapsto (u(t), \dot u(t)) $ for $ \epsilon = \frac12 $ and $ b = 1 $
Figure 9.  Parametric plot of $ t\mapsto (u(t), \dot u(t)) $ for $ \epsilon = \frac12 $ and $ b = 2 $
[1]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[2]

Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121

[3]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

[4]

Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations and Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019

[5]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273

[6]

Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020

[7]

Xiaoli Zhu, Fuyi Li, Ting Rong. Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2465-2485. doi: 10.3934/cpaa.2015.14.2465

[8]

Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599

[9]

Jian-Wen Sun, Seonghak Kim. Exponential decay for quasilinear parabolic equations in any dimension. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021280

[10]

Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations and Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008

[11]

Peng Sun. Exponential decay of Lebesgue numbers. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773

[12]

Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082

[13]

Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609

[14]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065

[15]

Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic and Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001

[16]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure and Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521

[17]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935

[18]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851

[19]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017

[20]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (233)
  • HTML views (130)
  • Cited by (0)

Other articles
by authors

[Back to Top]