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June  2013, 2(2): 403-422. doi: 10.3934/eect.2013.2.403

Nonlinear instability of solutions in parabolic and hyperbolic diffusion

Stephen Pankavich 1, and  Petronela Radu 2,

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

Received  November 2012 Revised  December 2012 Published  March 2013

We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immediately applicable, including evolution equations with supercritical and exponential nonlinearities.
Keywords: steady states., variable coefficients, instability, Evolution equations, sign-changing damping.
Mathematics Subject Classification: Primary: 35B35, 35B05, 35B30; Secondary: 35L70, 35K5.
Citation: Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations & Control Theory, 2013, 2 (2) : 403-422. doi: 10.3934/eect.2013.2.403
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl., 16 (1963), 305. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

E. Lieb and M. Loss, "Analysis,", $2^{nd}$ edition, 14 (2001). Google Scholar

[20]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (): 169. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

B. Simon, Schrödinger semigroups,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447. doi: 10.1090/S0273-0979-1982-15041-8. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur,, Comptes Rendus Acad. Sci., 246 (1958), 3154. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl., 16 (1963), 305. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

E. Lieb and M. Loss, "Analysis,", $2^{nd}$ edition, 14 (2001). Google Scholar

[20]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (): 169. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

B. Simon, Schrödinger semigroups,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447. doi: 10.1090/S0273-0979-1982-15041-8. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

P. Vernotte, Les paradoxes de la théorie continue de l'équation de la chaleur,, Comptes Rendus Acad. Sci., 246 (1958), 3154. Google Scholar

[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. Google Scholar

[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. Google Scholar

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