Figure 1.
Domain
-
Previous Article
A generalized complex Ginzburg-Landau equation: Global existence and stability results
- CPAA Home
- This Issue
-
Next Article
Spectral properties of ordinary differential operators admitting special decompositions
May
2021, 20(5): 1987-2020.
doi: 10.3934/cpaa.2021055
Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation
| 1. | Department of Mathematics, State University of Paraíba, Campina Grande, PB 58429-500, Brazil |
| 2. | Department of Mathematics, State University of Maringá, Maringá, PR 87020-900, Brazil |
| 3. | Department of Mathematics, Federal University of Technology-Paraná, Cornélio Procópio, PR 86300-000, Brazil |
We are concerned with the transmission problem of nonlinear viscoelastic waves in a heterogeneous medium, establishing the well-posedness of solutions and the exponential stability of the related energy functional. We introduce an auxiliary problem to prove the exponential stability and the proof combines an observability inequality and microlocal analysis tools.
Keywords:
Nonlinear wave equation,
transmission problem,
viscoelastic effect,
viscoelastic wave equation,
exponential stability.
Mathematics Subject Classification:
Primary: 35L05; Secondary: 35L53, 35B40.
Citation:
Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure and Applied Analysis,
2021, 20
(5)
: 1987-2020.
doi: 10.3934/cpaa.2021055
![]()
References:
| [1] |
M. S. Alves, J. Muñoz Rivera, M. Sepúlveda and O. Vera Villagrán,
The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math., 74 (1992), 345-365.
doi: 10.1137/130923233. |
| [2] |
M. S. Alves, J. E. Muñoz Rivera, M. Sepúlveda, O. Vera Villagrán and M. Zegarra Garay,
The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), 483-497.
doi: 10.1002/mana.201200319. |
| [3] |
J. A. D. Appleby, M. Fabrizio, B. Lazzari and D. W. Reynolds,
On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.
|
| [4] |
L. Boltzmann,
Zur Theorie der elastischen Nachwirkung, Wien. Ber., 70 (1874), 275-306.
|
| [5] |
L. Boltzmann,
Zur Theorie der elastischen Nachwirkung, Wied. Ann., 5 (1878), 430-432.
|
| [6] |
N. Burq and P. Gérard, Contrôle Optimal des Équations Aux Dérivées Partielles, 2001. Available from: http://www.math.u-psud.fr/ burq/articles/coursX.pdf. |
| [7] |
F. Cardoso and G. Vodev, Boundary stabilization of transmission problems, J. Math. Phys., 51(2010), 023512.
doi: 10.1063/1.3277163. |
| [8] |
M. Cavalcanti, L. Fatori and Ma To Fu,
Attractors for wave equations with degenerate memory, J. Differ. Equ., 260 (2016), 56-83.
doi: 10.1016/j.jde.2015.08.050. |
| [9] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Silva and A. Y. S. Franco,
Exponential stability for the wave model with localized memory in a past history framework, J. Differ. Equ., 264 (2018), 6535-6584.
doi: 10.1016/j.jde.2018.01.044. |
| [10] |
M. M. Cavalcanti, E. R. S. Coelho and V. N. Domingos Cavalcanti,
Exponential stability for a transmission problem of a viscoelastic wave equation, Appl. Math. Optim., 81 (2020), 621-650.
doi: 10.1007/s00245-018-9514-9. |
| [11] |
M. Conti, E. M. Marchini and V. Pata,
A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216.
doi: 10.1016/j.na.2013.08.015. |
| [12] |
M. Conti, E. M. Marchini and V. Pata,
Global attractors for nonlinear viscoelastic equations with memory, Commun. Pure Appl. Anal., 15 (2016), 1893-1913.
doi: 10.3934/cpaa.2016021. |
| [13] |
M. Conti, E. M. Marchini and V. Pata,
Non classical diffusion with memory, Math. Meth. Appl. Sci., 38 (2015), 948-958.
doi: 10.1002/mma.3120. |
| [14] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
| [15] |
V. Danese, P. Geredeli and V. Pata,
Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2881-2904.
doi: 10.3934/dcds.2015.35.2881. |
| [16] |
T. Duyckaerts, X. Zhang and E. Zuazua,
On the optimality of the observability inequalities for parabolic and
hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1-41.
doi: 10.1016/j.anihpc.2006.07.005. |
| [17] |
M. Fabrizio, C. Giorgi and V. Pata,
A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232.
doi: 10.1007/s00205-010-0300-3. |
| [18] |
H. D. Fernandéz Sare and J. E. Muñoz Rivera,
Analyticity of transmission problem to thermoelastic plates, Quart. Appl. Math., 69 (2011), 1-13.
doi: 10.1090/S0033-569X-2010-01187-6. |
| [19] |
L. Gagnon, Sufficient Conditions for the Controllability of Wave Equations with a Transmission Condition at the Interface, preprint, arXiv: 1711.00448. |
| [20] |
P. Gérard,
Microlocal defect measures, Commun. Partial Differ. Equ., 16 (1991), 1761-1794.
doi: 10.1080/03605309108820822. |
| [21] |
C. Giorgi, J. E. Muñoz Rivera and V. Pata,
Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.
doi: 10.1006/jmaa.2001.7437. |
| [22] |
M. Grasselli and V. Pata, Uniform attractors of non autonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (Eds. A. Lorenzi, B. Ruf), Birkhauser Verlag Basel/Switzerland, 50 (2002), 155–178.
doi: 10.1007/978-3-0348-8221-7_9. |
| [23] |
A. Guesmia and S. A. Messaoudi,
A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.
doi: 10.1016/j.nonrwa.2011.08.004. |
| [24] |
Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. Titi and D. Toundykov,
Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differ. Equ., 257 (2014), 3778-3812.
doi: 10.1016/j.jde.2014.07.009. |
| [25] |
M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha,
On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499.
doi: 10.1088/0951-7715/27/3/467. |
| [26] |
J. E. Lagnese,
Boundary controllability in problems of transmission for a class of Second order hyperbolic systems, ESAIM: Control Optim. Calc. Var., 2 (1997), 343-357.
doi: 10.1051/cocv:1997112. |
| [27] |
J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Coll. RMA, vol.8, Masson, Paris, 1988. |
| [28] |
W. Liu,
Stabilization and controllability for the transmission wave equation, IEEE Tran. Auto. Control, 46 (2001), 1900-1907.
doi: 10.1109/9.975473. |
| [29] |
K. Liu and Z. Liu,
Exponential decay of energy of vibrating strings with local viscoelasticity, ZAMP, 53 (2002), 265-280.
doi: 10.1007/s00033-002-8155-6. |
| [30] |
W. Liu and G. Williams,
The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc., 57 (1998), 305-327.
doi: 10.1017/S0004972700031683. |
| [31] |
S. Nicaise,
Boundary exact controllability of interface problems with singularities I: addition of the coefficients of
singularities, SIAM J. Control Optim., 34 (1996), 1512-1532.
doi: 10.1137/S0363012995282103. |
| [32] |
S. Nicaise,
Boundary exact controllability of interface problems with singularities II: addition of internal controls, SIAM J. Control Optim., 35 (1997), 585-603.
doi: 10.1137/S0363012995292032. |
| [33] |
T. Özsari, V. K. Kalantarov and I. Lasiecka,
Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control, J. Differ. Equ., 251 (2011), 1841-1863.
doi: 10.1016/j.jde.2011.04.003. |
| [34] |
V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333–360., |
| [35] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [36] |
J. E. Muñoz Rivera and M. G. Naso,
About asymptotic behavior for a transmission problem in hyperbolic thermoelasticity, Acta Appl. Math., 99 (2007), 1-27.
doi: 10.1007/s10440-007-9152-8. |
| [37] |
J. E. Muñoz Rivera and H. P. Oquendo,
The transmission problem of viscoelastic waves, Acta Appl. Math., 62 (2000), 1-21.
doi: 10.1023/A:1006449032100. |
| [38] |
J. Simon,
Compact Sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
| [39] |
V. Volterra,
Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.
doi: 10.1007/BF02418820. |
| [40] |
V. Volterra, LeÇons Sur Les Fonctions De Lignes, Gauthier-Villars, Paris, 1913. |
show all references
References:
| [1] |
M. S. Alves, J. Muñoz Rivera, M. Sepúlveda and O. Vera Villagrán,
The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math., 74 (1992), 345-365.
doi: 10.1137/130923233. |
| [2] |
M. S. Alves, J. E. Muñoz Rivera, M. Sepúlveda, O. Vera Villagrán and M. Zegarra Garay,
The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), 483-497.
doi: 10.1002/mana.201200319. |
| [3] |
J. A. D. Appleby, M. Fabrizio, B. Lazzari and D. W. Reynolds,
On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.
|
| [4] |
L. Boltzmann,
Zur Theorie der elastischen Nachwirkung, Wien. Ber., 70 (1874), 275-306.
|
| [5] |
L. Boltzmann,
Zur Theorie der elastischen Nachwirkung, Wied. Ann., 5 (1878), 430-432.
|
| [6] |
N. Burq and P. Gérard, Contrôle Optimal des Équations Aux Dérivées Partielles, 2001. Available from: http://www.math.u-psud.fr/ burq/articles/coursX.pdf. |
| [7] |
F. Cardoso and G. Vodev, Boundary stabilization of transmission problems, J. Math. Phys., 51(2010), 023512.
doi: 10.1063/1.3277163. |
| [8] |
M. Cavalcanti, L. Fatori and Ma To Fu,
Attractors for wave equations with degenerate memory, J. Differ. Equ., 260 (2016), 56-83.
doi: 10.1016/j.jde.2015.08.050. |
| [9] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Silva and A. Y. S. Franco,
Exponential stability for the wave model with localized memory in a past history framework, J. Differ. Equ., 264 (2018), 6535-6584.
doi: 10.1016/j.jde.2018.01.044. |
| [10] |
M. M. Cavalcanti, E. R. S. Coelho and V. N. Domingos Cavalcanti,
Exponential stability for a transmission problem of a viscoelastic wave equation, Appl. Math. Optim., 81 (2020), 621-650.
doi: 10.1007/s00245-018-9514-9. |
| [11] |
M. Conti, E. M. Marchini and V. Pata,
A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94 (2014), 206-216.
doi: 10.1016/j.na.2013.08.015. |
| [12] |
M. Conti, E. M. Marchini and V. Pata,
Global attractors for nonlinear viscoelastic equations with memory, Commun. Pure Appl. Anal., 15 (2016), 1893-1913.
doi: 10.3934/cpaa.2016021. |
| [13] |
M. Conti, E. M. Marchini and V. Pata,
Non classical diffusion with memory, Math. Meth. Appl. Sci., 38 (2015), 948-958.
doi: 10.1002/mma.3120. |
| [14] |
C. M. Dafermos,
Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.
doi: 10.1007/BF00251609. |
| [15] |
V. Danese, P. Geredeli and V. Pata,
Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2881-2904.
doi: 10.3934/dcds.2015.35.2881. |
| [16] |
T. Duyckaerts, X. Zhang and E. Zuazua,
On the optimality of the observability inequalities for parabolic and
hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1-41.
doi: 10.1016/j.anihpc.2006.07.005. |
| [17] |
M. Fabrizio, C. Giorgi and V. Pata,
A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232.
doi: 10.1007/s00205-010-0300-3. |
| [18] |
H. D. Fernandéz Sare and J. E. Muñoz Rivera,
Analyticity of transmission problem to thermoelastic plates, Quart. Appl. Math., 69 (2011), 1-13.
doi: 10.1090/S0033-569X-2010-01187-6. |
| [19] |
L. Gagnon, Sufficient Conditions for the Controllability of Wave Equations with a Transmission Condition at the Interface, preprint, arXiv: 1711.00448. |
| [20] |
P. Gérard,
Microlocal defect measures, Commun. Partial Differ. Equ., 16 (1991), 1761-1794.
doi: 10.1080/03605309108820822. |
| [21] |
C. Giorgi, J. E. Muñoz Rivera and V. Pata,
Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.
doi: 10.1006/jmaa.2001.7437. |
| [22] |
M. Grasselli and V. Pata, Uniform attractors of non autonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (Eds. A. Lorenzi, B. Ruf), Birkhauser Verlag Basel/Switzerland, 50 (2002), 155–178.
doi: 10.1007/978-3-0348-8221-7_9. |
| [23] |
A. Guesmia and S. A. Messaoudi,
A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.
doi: 10.1016/j.nonrwa.2011.08.004. |
| [24] |
Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. Titi and D. Toundykov,
Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differ. Equ., 257 (2014), 3778-3812.
doi: 10.1016/j.jde.2014.07.009. |
| [25] |
M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha,
On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, 27 (2014), 467-499.
doi: 10.1088/0951-7715/27/3/467. |
| [26] |
J. E. Lagnese,
Boundary controllability in problems of transmission for a class of Second order hyperbolic systems, ESAIM: Control Optim. Calc. Var., 2 (1997), 343-357.
doi: 10.1051/cocv:1997112. |
| [27] |
J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Coll. RMA, vol.8, Masson, Paris, 1988. |
| [28] |
W. Liu,
Stabilization and controllability for the transmission wave equation, IEEE Tran. Auto. Control, 46 (2001), 1900-1907.
doi: 10.1109/9.975473. |
| [29] |
K. Liu and Z. Liu,
Exponential decay of energy of vibrating strings with local viscoelasticity, ZAMP, 53 (2002), 265-280.
doi: 10.1007/s00033-002-8155-6. |
| [30] |
W. Liu and G. Williams,
The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc., 57 (1998), 305-327.
doi: 10.1017/S0004972700031683. |
| [31] |
S. Nicaise,
Boundary exact controllability of interface problems with singularities I: addition of the coefficients of
singularities, SIAM J. Control Optim., 34 (1996), 1512-1532.
doi: 10.1137/S0363012995282103. |
| [32] |
S. Nicaise,
Boundary exact controllability of interface problems with singularities II: addition of internal controls, SIAM J. Control Optim., 35 (1997), 585-603.
doi: 10.1137/S0363012995292032. |
| [33] |
T. Özsari, V. K. Kalantarov and I. Lasiecka,
Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control, J. Differ. Equ., 251 (2011), 1841-1863.
doi: 10.1016/j.jde.2011.04.003. |
| [34] |
V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333–360., |
| [35] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [36] |
J. E. Muñoz Rivera and M. G. Naso,
About asymptotic behavior for a transmission problem in hyperbolic thermoelasticity, Acta Appl. Math., 99 (2007), 1-27.
doi: 10.1007/s10440-007-9152-8. |
| [37] |
J. E. Muñoz Rivera and H. P. Oquendo,
The transmission problem of viscoelastic waves, Acta Appl. Math., 62 (2000), 1-21.
doi: 10.1023/A:1006449032100. |
| [38] |
J. Simon,
Compact Sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
| [39] |
V. Volterra,
Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.
doi: 10.1007/BF02418820. |
| [40] |
V. Volterra, LeÇons Sur Les Fonctions De Lignes, Gauthier-Villars, Paris, 1913. |
| [1] |
Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543 |
| [2] |
Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1221-1232. doi: 10.3934/dcdss.2021093 |
| [3] |
Zhiling Guo, Shugen Chai. Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022001 |
| [4] |
Belkacem Said-Houari, Flávio A. Falcão Nascimento. Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure and Applied Analysis, 2013, 12 (1) : 375-403. doi: 10.3934/cpaa.2013.12.375 |
| [5] |
Donghao Li, Hongwei Zhang, Shuo Liu, Qingiyng Hu. Blow-up of solutions to a viscoelastic wave equation with nonlocal damping. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022009 |
| [6] |
Menglan Liao. The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity. Evolution Equations and Control Theory, 2022, 11 (3) : 781-792. doi: 10.3934/eect.2021025 |
| [7] |
Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541 |
| [8] |
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 |
| [9] |
Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987 |
| [10] |
Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 |
| [11] |
Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 |
| [12] |
Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations and Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645 |
| [13] |
Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001 |
| [14] |
Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391 |
| [15] |
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 |
| [16] |
Jorge A. Esquivel-Avila. Nonexistence of global solutions for a class of viscoelastic wave equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4213-4230. doi: 10.3934/dcdss.2021134 |
| [17] |
Muhammad I. Mustafa. Viscoelastic plate equation with boundary feedback. Evolution Equations and Control Theory, 2017, 6 (2) : 261-276. doi: 10.3934/eect.2017014 |
| [18] |
Mohammad Akil, Haidar Badawi, Ali Wehbe. Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2991-3028. doi: 10.3934/cpaa.2021092 |
| [19] |
Teng Wang, Yi Wang. Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation. Kinetic and Related Models, 2019, 12 (3) : 637-679. doi: 10.3934/krm.2019025 |
| [20] |
Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]