• Previous Article
    Opinion Dynamics on a General Compact Riemannian Manifold
  • NHM Home
  • This Issue
  • Next Article
    Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics
September  2017, 12(3): 461-488. doi: 10.3934/nhm.2017020

Decay rates for elastic-thermoelastic star-shaped networks

1. 

School of Mathematics, Tianjin University, 300354 Tianjin, China

2. 

DeustoTech -Fundación Deusto, Avda. Universidades, 24, 48007 Bilbao, Basque Country, Spain

3. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

4. 

Facultad Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007 Bilbao, Basque Country, Spain

Received  September 2016 Revised  July 2017 Published  September 2017

Fund Project: The first author was supported by the Natural Science Foundation of China grant NSFC-61573252 and China Scholarship Council. The second author was supported by the Advanced Grant DYCON of the European Research Council Executive Agency, ICON of the ANR-2016-ACHN-0014-01 (France), FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the MTM2014-52347 Grant of the MINECO and a Humboldt Research Award at the University of Erlangen-Nürnberg

This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths.

First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.

Citation: Zhong-Jie Han, Enrique Zuazua. Decay rates for elastic-thermoelastic star-shaped networks. Networks & Heterogeneous Media, 2017, 12 (3) : 461-488. doi: 10.3934/nhm.2017020
References:
[1]

M. AlvesJ. Muñoz RiveraM. 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. Google Scholar

[2]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410. Google Scholar

[3]

K. AmmariM. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical and Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7. Google Scholar

[4]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Applicable Analysis, 86 (2007), 1529-1548. doi: 10.1080/00036810701734113. Google Scholar

[5]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1. Google Scholar

[6]

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. Google Scholar

[7]

J. W. S. Cassals, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge, UK, 1957. Google Scholar

[8]

R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 621-626. doi: 10.1016/S0764-4442(01)01876-6. Google Scholar

[9]

R. Dáger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 1087-1092. doi: 10.1016/S0764-4442(01)01942-5. Google Scholar

[10]

R. Dáger and E. Zuazua, Spectral boundary controllability of networks of strings, C. R. Math. Acad. Sci. Paris, 334 (2002), 545-550. doi: 10.1016/S1631-073X(02)02314-2. Google Scholar

[11]

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. Google Scholar

[12]

H. D. Fernández-SareJ. E. Muñoz-Rivera and R. Racke, Stability for a transmission problem in thermoelasticity with second sound, Journal of Thermal Stresses, 31 (2008), 1170-1189. Google Scholar

[13]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. Google Scholar

[14]

Z. J. Han and G. Q. Xu, Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping, Math. Meth. Appl. Sci., 38 (2015), 94-112. doi: 10.1002/mma.3052. Google Scholar

[15]

Z. J. Han and G. Q. Xu, Spectrum and stability analysis for a transmission problem in thermoelasticity with a concentrated mass, Z. Angew. Math. Phys., 66 (2015), 1717-1736. doi: 10.1007/s00033-015-0504-3. Google Scholar

[16]

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. Google Scholar

[17]

Z. J. Han and G. Q. Xu, Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation, International Journal of Control, 84 (2011), 458-475. doi: 10.1080/00207179.2011.561441. Google Scholar

[18]

D. B. HenryO. Lopes and A. Perissinitto Jr., On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Analysis: Theory, Methods & Applications, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U. Google Scholar

[19]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eqs., 1 (1985), 43-56. Google Scholar

[20]

S. JaffardM. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation, Journal of Differential Equations, 145 (1998), 184-215. doi: 10.1006/jdeq.1997.3385. Google Scholar

[21]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[22]

G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Archive Rat. Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078. Google Scholar

[23]

G. Lebeau and E. Zuazua, Decay rates for the linear system of three-dimensional system of thermoelasticity, Archive Rat. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. Google Scholar

[24]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6. Google Scholar

[25]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, 1999. Google Scholar

[26]

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. Google Scholar

[27]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42. Google Scholar

[28]

A. MarzocchiJ. E. Munoz Rivera and M. G. Naso, Asymptotic behaviour and expinential stability for a transmission problem in thermoelasticity, Math. Meth. Appl. Sci., 25 (2002), 955-980. doi: 10.1002/mma.323. Google Scholar

[29]

A. MarzocchiJ. E. Muñoz Rivera and M. G. Naso, Transmission problem in thermoelasticity with symmetry, IMA J. Appl. Math., 68 (2003), 23-46. doi: 10.1093/imamat/68.1.23. Google Scholar

[30]

S. A. Messaoudi and B. Said-Houari, Energy decay in a transmission problem in thermoelasticity of type Ⅲ, IMA J. Appl. Math., 74 (2009), 344-360. doi: 10.1093/imamat/hxp020. Google Scholar

[31]

J. E. Muñoz Rivera and H. Portillo Oquendo, The transmission problem for thermoelastic beams, Journal of Thermal Stresses, 24 (2001), 1137-1158. doi: 10.1080/014957301753251665. Google Scholar

[32]

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. Google Scholar

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[34]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. Google Scholar

[35]

R. Racke, Thermoelasticity with second sound-Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. Google Scholar

[36]

W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201. doi: 10.1007/BF02392334. Google Scholar

[37]

F. Shel, Exponential stability of a network of elastic and thermoelastic materials, Math. Meth. Appl. Sci., 36 (2013), 869-879. doi: 10.1002/mma.2644. Google Scholar

[38]

L. N. Trefethen, Spectral Methods in Matlab, PA: SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. Google Scholar

[39]

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. Google Scholar

[40]

G. Q. XuD. 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. Google Scholar

[41]

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. Google Scholar

[42]

E. Zuazua, Controllability of the linear system of thermoelasticity, Journal de Mathématiques Pures et Appliquées, 74 (1995), 291-315. Google Scholar

show all references

References:
[1]

M. AlvesJ. Muñoz RiveraM. 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. Google Scholar

[2]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differential and Integral Equations, 17 (2004), 1395-1410. Google Scholar

[3]

K. AmmariM. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical and Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7. Google Scholar

[4]

K. Ammari, Asymptotic behaviour of some elastic planar networks of Bernoulli-Euler beams, Applicable Analysis, 86 (2007), 1529-1548. doi: 10.1080/00036810701734113. Google Scholar

[5]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1. Google Scholar

[6]

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. Google Scholar

[7]

J. W. S. Cassals, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge, UK, 1957. Google Scholar

[8]

R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 621-626. doi: 10.1016/S0764-4442(01)01876-6. Google Scholar

[9]

R. Dáger and E. Zuazua, Controllability of tree-shaped networks of vibrating strings, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), 1087-1092. doi: 10.1016/S0764-4442(01)01942-5. Google Scholar

[10]

R. Dáger and E. Zuazua, Spectral boundary controllability of networks of strings, C. R. Math. Acad. Sci. Paris, 334 (2002), 545-550. doi: 10.1016/S1631-073X(02)02314-2. Google Scholar

[11]

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. Google Scholar

[12]

H. D. Fernández-SareJ. E. Muñoz-Rivera and R. Racke, Stability for a transmission problem in thermoelasticity with second sound, Journal of Thermal Stresses, 31 (2008), 1170-1189. Google Scholar

[13]

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. Google Scholar

[14]

Z. J. Han and G. Q. Xu, Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping, Math. Meth. Appl. Sci., 38 (2015), 94-112. doi: 10.1002/mma.3052. Google Scholar

[15]

Z. J. Han and G. Q. Xu, Spectrum and stability analysis for a transmission problem in thermoelasticity with a concentrated mass, Z. Angew. Math. Phys., 66 (2015), 1717-1736. doi: 10.1007/s00033-015-0504-3. Google Scholar

[16]

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. Google Scholar

[17]

Z. J. Han and G. Q. Xu, Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation, International Journal of Control, 84 (2011), 458-475. doi: 10.1080/00207179.2011.561441. Google Scholar

[18]

D. B. HenryO. Lopes and A. Perissinitto Jr., On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Analysis: Theory, Methods & Applications, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U. Google Scholar

[19]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eqs., 1 (1985), 43-56. Google Scholar

[20]

S. JaffardM. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation, Journal of Differential Equations, 145 (1998), 184-215. doi: 10.1006/jdeq.1997.3385. Google Scholar

[21]

J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[22]

G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Archive Rat. Mech. Anal., 141 (1998), 297-329. doi: 10.1007/s002050050078. Google Scholar

[23]

G. Lebeau and E. Zuazua, Decay rates for the linear system of three-dimensional system of thermoelasticity, Archive Rat. Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. Google Scholar

[24]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-008-6122-6. Google Scholar

[25]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, vol. 398, Chapman and Hall/CRC, Boca Raton, 1999. Google Scholar

[26]

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. Google Scholar

[27]

Yu. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42. Google Scholar

[28]

A. MarzocchiJ. E. Munoz Rivera and M. G. Naso, Asymptotic behaviour and expinential stability for a transmission problem in thermoelasticity, Math. Meth. Appl. Sci., 25 (2002), 955-980. doi: 10.1002/mma.323. Google Scholar

[29]

A. MarzocchiJ. E. Muñoz Rivera and M. G. Naso, Transmission problem in thermoelasticity with symmetry, IMA J. Appl. Math., 68 (2003), 23-46. doi: 10.1093/imamat/68.1.23. Google Scholar

[30]

S. A. Messaoudi and B. Said-Houari, Energy decay in a transmission problem in thermoelasticity of type Ⅲ, IMA J. Appl. Math., 74 (2009), 344-360. doi: 10.1093/imamat/hxp020. Google Scholar

[31]

J. E. Muñoz Rivera and H. Portillo Oquendo, The transmission problem for thermoelastic beams, Journal of Thermal Stresses, 24 (2001), 1137-1158. doi: 10.1080/014957301753251665. Google Scholar

[32]

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. Google Scholar

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[34]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. Google Scholar

[35]

R. Racke, Thermoelasticity with second sound-Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. Google Scholar

[36]

W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201. doi: 10.1007/BF02392334. Google Scholar

[37]

F. Shel, Exponential stability of a network of elastic and thermoelastic materials, Math. Meth. Appl. Sci., 36 (2013), 869-879. doi: 10.1002/mma.2644. Google Scholar

[38]

L. N. Trefethen, Spectral Methods in Matlab, PA: SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. Google Scholar

[39]

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. Google Scholar

[40]

G. Q. XuD. 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. Google Scholar

[41]

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. Google Scholar

[42]

E. Zuazua, Controllability of the linear system of thermoelasticity, Journal de Mathématiques Pures et Appliquées, 74 (1995), 291-315. Google Scholar

Figure 1.  Transmission problem in 1-d elasticity-thermoelasticity
Figure 2.  Star-shaped thermoelastic-elastic network
Figure A-1.  $u_1(x,t)$
Figure A-2.  $u_2(x,t)$
Figure A-3.  $u_3(x,t)$
Figure A-4.  $\theta_1(x,t)$
Figure A-5.  $\theta_2(x,t)$
Figure A-6.  $\theta_3(x,t)$
Figure B-1.  $u_1(x,t)$
Figure B-2.  $u_2(x,t)$
Figure B-3.  $u_3(x,t)$
Figure C-1.  $u_1(x,t)$
Figure C-2.  $u_2(x,t)$
Figure C-3.  $u_3(x,t)$
Figure C-4.  $\theta_1(x,t)$
Figure C-5.  $\theta_2(x,t)$
Figure D-1.  $u_1(x,t)$
Figure D-2.  $u_2(x,t)$
Figure D-3.  $u_3(x,t)$
Figure D-4.  $\theta_1(x,t)$
Figure E-1.  $u_1(x,t)$
Figure E-2.  $u_2(x,t)$
Figure E-3.  $u_3(x,t)$
Figure E-4.  $\theta_1(x,t)$
Figure F-1.  Logarithmic scale of energy for Case A, B, C
Figure F-2.  Energy for Case D, E
[1]

F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier. Dispersive waves with multiple tunnel effect on a star-shaped network. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 783-791. doi: 10.3934/dcdss.2013.6.783

[2]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[3]

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 & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[4]

Farah Abou Shakra. Asymptotics of wave models for non star-shaped geometries. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 347-362. doi: 10.3934/dcdss.2014.7.347

[5]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001

[6]

Zhong-Jie Han, Gen-Qi Xu. Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 57-77. doi: 10.3934/dcdsb.2012.17.57

[7]

Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100

[8]

Takahiro Hashimoto. Nonexistence of global solutions of nonlinear Schrodinger equations in non star-shaped domains. Conference Publications, 2007, 2007 (Special) : 487-494. doi: 10.3934/proc.2007.2007.487

[9]

Helmut Harbrecht, Thorsten Hohage. A Newton method for reconstructing non star-shaped domains in electrical impedance tomography. Inverse Problems & Imaging, 2009, 3 (2) : 353-371. doi: 10.3934/ipi.2009.3.353

[10]

Byung-Soo Lee. Strong convergence theorems with three-step iteration in star-shaped metric spaces. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 371-379. doi: 10.3934/naco.2011.1.371

[11]

Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589

[12]

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

[13]

Abdallah Ben Abdallah, Farhat Shel. Exponential stability of a general network of 1-d thermoelastic rods. Mathematical Control & Related Fields, 2012, 2 (1) : 1-16. doi: 10.3934/mcrf.2012.2.1

[14]

Zhuangyi Liu, Ramón Quintanilla. Time decay in dual-phase-lag thermoelasticity: Critical case. Communications on Pure & Applied Analysis, 2018, 17 (1) : 177-190. doi: 10.3934/cpaa.2018011

[15]

Raúl Ferreira, Julio D. Rossi. Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1469-1478. doi: 10.3934/dcds.2015.35.1469

[16]

Thinh Tien Nguyen. Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1651-1684. doi: 10.3934/dcds.2019073

[17]

Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 445-464. doi: 10.3934/dcds.2008.22.445

[18]

Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237

[19]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423

[20]

M. Carme Leseduarte, Antonio Magaña, Ramón Quintanilla. On the time decay of solutions in porous-thermo-elasticity of type II. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 375-391. doi: 10.3934/dcdsb.2010.13.375

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (50)
  • HTML views (17)
  • Cited by (0)

Other articles
by authors

[Back to Top]