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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

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.
Citation: Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations and 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), 2596-2616. 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.

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