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1. | Department of Mathematics, United States Naval Academy, Annapolis, MD 21402, United States |
2. | Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588 |
References:
[1] |
Elvise Berchio, Alberto Farina, Alberto Ferrero and Filippo Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem,, J. Differential Equations, 252 (2012), 2596.
doi: 10.1016/j.jde.2011.09.028. |
[2] |
M. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée,, Comptes Rendus de l'Academie des Sciences Paris, 247 (1958), 431.
|
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. Journal, 63 (1991), 615.
doi: 10.1215/S0012-7094-91-06325-8. |
[4] |
C. de Silva, "Vibration and Shock Handbook,", Mechanical Engineering, (2005). Google Scholar |
[5] |
G. Fragnelli and D. Mugnai, Stability of solutions for nonlinear wave equations with a positive-negative damping,, Discrete and Continuous Dynamical Systems Series S, 4 (2011), 615.
doi: 10.3934/dcdss.2011.4.615. |
[6] |
G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations,, SIAM J. Control Optim., 47 (2008), 2520.
doi: 10.1137/070689735. |
[7] |
R. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183.
|
[8] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160.
doi: 10.1016/0022-1236(87)90044-9. |
[9] |
C. Gui, W. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbbR^n$,, Comm. Pure Appl. Math., 45 (1992), 1153.
doi: 10.1002/cpa.3160450906. |
[10] |
Y. Han and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves,, Bull. Sci. Math., 124 (2000), 415.
doi: 10.1016/S0007-4497(00)00141-X. |
[11] |
L. Hsiao and Tai-Ping Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping,, Comm. Math. Phys., 143 (1992), 599.
|
[12] |
S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl., 16 (1963), 305.
|
[13] |
P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation,, Nonlinearity, 22 (2009), 1653.
doi: 10.1088/0951-7715/22/7/009. |
[14] |
P. Karageorgis and W. Strauss, Instability of steady states for nonlinear wave and heat equations,, J. Differential Equations, 241 (2007), 184.
doi: 10.1016/j.jde.2007.06.006. |
[15] |
M. Kawashita, H. Nakazawa and H. Soga, Non decay of the total energy for the wave equation with the dissipative term of spatial anisotropy,, Nagoya Math. J., 174 (2004), 115.
|
[16] |
S. Konabe and T. Nikuni, Coarse-grained finite-temperature theory for the Bose condensate in optical lattices,, Journal of Low Temperature Physics, 150 (2008), 12. Google Scholar |
[17] |
A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, SIAM Review, 32 (1990), 537.
doi: 10.1137/1032120. |
[18] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $\mathbbR^n$,, J. Differential Equations, 95 (1992), 304.
doi: 10.1016/0022-0396(92)90034-K. |
[19] |
E. Lieb and M. Loss, "Analysis,", $2^{nd}$ edition, 14 (2001).
|
[20] |
A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (): 169.
|
[21] |
K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping,, J. Differential Equations, 131 (1996), 171.
doi: 10.1006/jdeq.1996.0159. |
[22] |
P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients,, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 609.
doi: 10.3934/dcdss.2009.2.609. |
[23] |
P. Radu, G. Todorova and B. Yordanov, Diffusion phenomenon in Hilbert spaces and applications,, J. Differential Equations, 250 (2011), 4200.
doi: 10.1016/j.jde.2011.01.024. |
[24] |
P. Reverberi, P. Bagnerini, L. Maga and A. G. Bruzzone, On the non-linear Maxwell-Cattaneo equation with non-constant diffusivity: Shock and discontinuity waves,, International Journal of heat and Mass Transfer, 51 (2008), 5327. Google Scholar |
[25] |
J. Shatah and W. Strauss, Spectral condition for instability,, in, 255 (2000), 189.
doi: 10.1090/conm/255/03982. |
[26] |
B. Simon, Schrödinger semigroups,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447.
doi: 10.1090/S0273-0979-1982-15041-8. |
[27] |
G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods,, Aerosp. Sci. Technolo., 1 (1997), 545. Google Scholar |
[28] |
P. Souplet and Q. Zhang, Stability for semilinear parabolic equations with decaying potentials in $\mathbbR^n$ and dynamical approach to the existence of ground states,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 683.
doi: 10.1016/S0294-1449(02)00098-7. |
[29] |
G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464.
doi: 10.1006/jdeq.2000.3933. |
[30] |
P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur,, Comptes Rendus Acad. Sci., 246 (1958), 3154.
|
[31] |
J. Wirth, Wave equations with time-dependent dissipation. II. Effective dissipation,, J. Differential Equations, 232 (2007), 74.
doi: 10.1016/j.jde.2006.06.004. |
[32] |
B. Yordanov and Q. Zhang, Finite-time blow up for wave equations with a potential,, SIAM J. Math. Anal., 36 (2005), 1426.
doi: 10.1137/S0036141004440198. |
show all references
References:
[1] |
Elvise Berchio, Alberto Farina, Alberto Ferrero and Filippo Gazzola, Existence and stability of entire solutions to a semilinear fourth order elliptic problem,, J. Differential Equations, 252 (2012), 2596.
doi: 10.1016/j.jde.2011.09.028. |
[2] |
M. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée,, Comptes Rendus de l'Academie des Sciences Paris, 247 (1958), 431.
|
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. Journal, 63 (1991), 615.
doi: 10.1215/S0012-7094-91-06325-8. |
[4] |
C. de Silva, "Vibration and Shock Handbook,", Mechanical Engineering, (2005). Google Scholar |
[5] |
G. Fragnelli and D. Mugnai, Stability of solutions for nonlinear wave equations with a positive-negative damping,, Discrete and Continuous Dynamical Systems Series S, 4 (2011), 615.
doi: 10.3934/dcdss.2011.4.615. |
[6] |
G. Fragnelli and D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations,, SIAM J. Control Optim., 47 (2008), 2520.
doi: 10.1137/070689735. |
[7] |
R. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183.
|
[8] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160.
doi: 10.1016/0022-1236(87)90044-9. |
[9] |
C. Gui, W. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbbR^n$,, Comm. Pure Appl. Math., 45 (1992), 1153.
doi: 10.1002/cpa.3160450906. |
[10] |
Y. Han and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves,, Bull. Sci. Math., 124 (2000), 415.
doi: 10.1016/S0007-4497(00)00141-X. |
[11] |
L. Hsiao and Tai-Ping Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping,, Comm. Math. Phys., 143 (1992), 599.
|
[12] |
S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl., 16 (1963), 305.
|
[13] |
P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation,, Nonlinearity, 22 (2009), 1653.
doi: 10.1088/0951-7715/22/7/009. |
[14] |
P. Karageorgis and W. Strauss, Instability of steady states for nonlinear wave and heat equations,, J. Differential Equations, 241 (2007), 184.
doi: 10.1016/j.jde.2007.06.006. |
[15] |
M. Kawashita, H. Nakazawa and H. Soga, Non decay of the total energy for the wave equation with the dissipative term of spatial anisotropy,, Nagoya Math. J., 174 (2004), 115.
|
[16] |
S. Konabe and T. Nikuni, Coarse-grained finite-temperature theory for the Bose condensate in optical lattices,, Journal of Low Temperature Physics, 150 (2008), 12. Google Scholar |
[17] |
A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis,, SIAM Review, 32 (1990), 537.
doi: 10.1137/1032120. |
[18] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $\mathbbR^n$,, J. Differential Equations, 95 (1992), 304.
doi: 10.1016/0022-0396(92)90034-K. |
[19] |
E. Lieb and M. Loss, "Analysis,", $2^{nd}$ edition, 14 (2001).
|
[20] |
A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (): 169.
|
[21] |
K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping,, J. Differential Equations, 131 (1996), 171.
doi: 10.1006/jdeq.1996.0159. |
[22] |
P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients,, Discrete and Continuous Dynamical Systems Series S, 2 (2009), 609.
doi: 10.3934/dcdss.2009.2.609. |
[23] |
P. Radu, G. Todorova and B. Yordanov, Diffusion phenomenon in Hilbert spaces and applications,, J. Differential Equations, 250 (2011), 4200.
doi: 10.1016/j.jde.2011.01.024. |
[24] |
P. Reverberi, P. Bagnerini, L. Maga and A. G. Bruzzone, On the non-linear Maxwell-Cattaneo equation with non-constant diffusivity: Shock and discontinuity waves,, International Journal of heat and Mass Transfer, 51 (2008), 5327. Google Scholar |
[25] |
J. Shatah and W. Strauss, Spectral condition for instability,, in, 255 (2000), 189.
doi: 10.1090/conm/255/03982. |
[26] |
B. Simon, Schrödinger semigroups,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447.
doi: 10.1090/S0273-0979-1982-15041-8. |
[27] |
G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods,, Aerosp. Sci. Technolo., 1 (1997), 545. Google Scholar |
[28] |
P. Souplet and Q. Zhang, Stability for semilinear parabolic equations with decaying potentials in $\mathbbR^n$ and dynamical approach to the existence of ground states,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 683.
doi: 10.1016/S0294-1449(02)00098-7. |
[29] |
G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464.
doi: 10.1006/jdeq.2000.3933. |
[30] |
P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur,, Comptes Rendus Acad. Sci., 246 (1958), 3154.
|
[31] |
J. Wirth, Wave equations with time-dependent dissipation. II. Effective dissipation,, J. Differential Equations, 232 (2007), 74.
doi: 10.1016/j.jde.2006.06.004. |
[32] |
B. Yordanov and Q. Zhang, Finite-time blow up for wave equations with a potential,, SIAM J. Math. Anal., 36 (2005), 1426.
doi: 10.1137/S0036141004440198. |
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