-
Previous Article
Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III
- EECT Home
- This Issue
-
Next Article
Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system
Nonlinear instability of solutions in parabolic and hyperbolic diffusion
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), 2596-2616.
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), 431-433. |
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. Journal, 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[4] |
C. de Silva, "Vibration and Shock Handbook," Mechanical Engineering, CRC Press, 2005. |
[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), 615-622.
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), 2520-2539.
doi: 10.1137/070689735. |
[7] |
R. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203. |
[8] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.
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 $\mathbb{R}^{N}$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.
doi: 10.1002/cpa.3160450906. |
[10] |
Y. Han and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math., 124 (2000), 415-433.
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), 599-605. |
[12] |
S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl., 16 (1963), 305-330. |
[13] |
P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661.
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), 184-205.
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), 115-126. |
[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), 12-46. |
[17] |
A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Review, 32 (1990), 537-578.
doi: 10.1137/1032120. |
[18] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $\mathbb{R}^{N}$, J. Differential Equations, 95 (1992), 304-330.
doi: 10.1016/0022-0396(92)90034-K. |
[19] |
E. Lieb and M. Loss, "Analysis," $2^{nd}$ edition, Grad. Stud. Math., 14, Amer. Math. Soc., Providence, RI, 2001. |
[20] |
A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189. |
[21] |
K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188.
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), 609-629.
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), 4200-4218.
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), 5327-5332. |
[25] |
J. Shatah and W. Strauss, Spectral condition for instability, in "Nonlinear PDE's, Dynamics and Continuum Physics" (South Hadley, MA, 1998), Contemp. Math., 255, Amer. Math. Soc., Providence, RI, (2000), 189-198.
doi: 10.1090/conm/255/03982. |
[26] |
B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526.
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), 545-555. |
[28] |
P. Souplet and Q. Zhang, Stability for semilinear parabolic equations with decaying potentials in $\mathbb{R}^{N}$ and dynamical approach to the existence of ground states, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 683-703.
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), 464-489.
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), 3154-3155. |
[31] |
J. Wirth, Wave equations with time-dependent dissipation. II. Effective dissipation, J. Differential Equations, 232 (2007), 74-103.
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), 1426-1433.
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), 2596-2616.
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), 431-433. |
[3] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. Journal, 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[4] |
C. de Silva, "Vibration and Shock Handbook," Mechanical Engineering, CRC Press, 2005. |
[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), 615-622.
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), 2520-2539.
doi: 10.1137/070689735. |
[7] |
R. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203. |
[8] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.
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 $\mathbb{R}^{N}$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.
doi: 10.1002/cpa.3160450906. |
[10] |
Y. Han and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math., 124 (2000), 415-433.
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), 599-605. |
[12] |
S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl., 16 (1963), 305-330. |
[13] |
P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661.
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), 184-205.
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), 115-126. |
[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), 12-46. |
[17] |
A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Review, 32 (1990), 537-578.
doi: 10.1137/1032120. |
[18] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $\mathbb{R}^{N}$, J. Differential Equations, 95 (1992), 304-330.
doi: 10.1016/0022-0396(92)90034-K. |
[19] |
E. Lieb and M. Loss, "Analysis," $2^{nd}$ edition, Grad. Stud. Math., 14, Amer. Math. Soc., Providence, RI, 2001. |
[20] |
A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189. |
[21] |
K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188.
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), 609-629.
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), 4200-4218.
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), 5327-5332. |
[25] |
J. Shatah and W. Strauss, Spectral condition for instability, in "Nonlinear PDE's, Dynamics and Continuum Physics" (South Hadley, MA, 1998), Contemp. Math., 255, Amer. Math. Soc., Providence, RI, (2000), 189-198.
doi: 10.1090/conm/255/03982. |
[26] |
B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526.
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), 545-555. |
[28] |
P. Souplet and Q. Zhang, Stability for semilinear parabolic equations with decaying potentials in $\mathbb{R}^{N}$ and dynamical approach to the existence of ground states, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 683-703.
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), 464-489.
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), 3154-3155. |
[31] |
J. Wirth, Wave equations with time-dependent dissipation. II. Effective dissipation, J. Differential Equations, 232 (2007), 74-103.
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), 1426-1433.
doi: 10.1137/S0036141004440198. |
[1] |
J. Húska, Peter Poláčik, M.V. Safonov. Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations. Conference Publications, 2005, 2005 (Special) : 427-435. doi: 10.3934/proc.2005.2005.427 |
[2] |
Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917 |
[3] |
Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure and Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004 |
[4] |
Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2161-2177. doi: 10.3934/cpaa.2016032 |
[5] |
Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055 |
[6] |
Bartosz Bieganowski, Jaros law Mederski. Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (1) : 143-161. doi: 10.3934/cpaa.2018009 |
[7] |
Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure and Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883 |
[8] |
Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure and Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383 |
[9] |
Diego Berti, Andrea Corli, Luisa Malaguti. Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 6023-6046. doi: 10.3934/dcds.2021105 |
[10] |
M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057 |
[11] |
Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013 |
[12] |
Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256 |
[13] |
Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151 |
[14] |
Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737 |
[15] |
Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499 |
[16] |
Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389 |
[17] |
Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454 |
[18] |
Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697 |
[19] |
Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137 |
[20] |
Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]