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Non-existence of global solutions to nonlinear wave equations with positive initial energy
Department of Mathematics, Koç University, Rumelifeneri Yolu, Sariyer 34450, Istanbul, Turkey |
We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a finite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained.
References:
| [1] |
A. B. Aliyev and A. A. Kazimov,
Global non-existence of solutions with fixed positive initial energy of the Cauchy problem for a system of Klein-Gordon equations, Differ. Equ., 51 (2015), 1563-1568.
|
| [2] |
A. B. Alshin, M. O. Korpusov and A. G. Sveshnikov,
Blow up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Analysis and Applications, 15. Walter de Gruyter and Co., Berlin, 2011. xii+648 pp. |
| [3] |
P. Aviles and J. Sandefur,
Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58 (1985), 404-427.
|
| [4] |
B. A. Bilgin and V. K. Kalantarov,
Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations, J. Math. Anal. Appl., 403 (2013), 89-94.
|
| [5] |
E. H. de Brito,
Nonlinear initial-boundary value problems, Nonlinear Anal., 11 (1987), 125-137.
|
| [6] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko,
Strongly damped wave problems: bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 2310-2333.
|
| [7] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998. xiv+186 pp. |
| [8] |
T. Cazenave,
Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.
|
| [9] |
Sh. P. Chen and R. Triggiani,
Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.
|
| [10] |
F. Gazzola and M. Squassina,
Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006), 185-207.
|
| [11] |
H. A. Erbay, S. Erbay and A. Erkip,
Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations, Nonlinear Anal., 95 (2014), 313-322.
|
| [12] |
V. A. Galaktionov and S. I. Pohozaev,
Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Anal., 53 (2003), 453-466.
|
| [13] |
S. J. Jakubov,
Solvability of the Cauchy problem for abstract quasilinear hyperbolic equations of second order and their applications, Trans. Moscow Math. Soc., 23 (1970), 36-59.
|
| [14] |
V. K Kalantarov and O. A. Ladyzhenskaya,
The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math., 10 (1978), 53-70.
|
| [15] |
R. J. Knops, H. A. Levine and L. E. Payne,
Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal., 55 (1974), 52-72.
|
| [16] |
M. O. Korpusov,
Blow-up of the solution of a nonlinear system of equations with positive energy, Theoret. and Math. Phys., 171 (2012), 725-738.
|
| [17] |
M. O. Korpusov,
On the blow-up of solutions of a dissipative wave equation of Kirchhoff type with a source and positive energy, Sib. Math. J., 52 (2011), 471-483.
|
| [18] |
N. Kutev, N. Kolkovska and M. Dimova,
Nonexistence of global solutions to new ordinary differential inequality and applications to nonlinear dispersive equations, Math. Methods Appl. Sci., 39 (2016), 2287-2297.
|
| [19] |
I. Lasiecka and A. Stahel,
The wave equation with semilinear Neumann boundary conditions, Nonlinear Anal., 15 (1990), 39-58.
|
| [20] |
H. A. Levine,
Instability and nonexistence of global solutions to nonlinear wave equations of
the form $P{{u}_{tt}} = -Au+F(u)$, Trans. Am. Math. Soc., 192 (1974), 1-21.
|
| [21] |
H. A. Levine,
Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
|
| [22] |
H. A. Levine and L. E. Paine,
Nonexistence theorems for the heat equations with nonlinear boundary conditions and for porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334.
|
| [23] |
H. A. Levine, S. R. Park and J. Serrin,
Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228 (1998), 181-205.
|
| [24] |
H. A. Levine and R. A. Smith,
A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math., 374 (1987), 1-23.
|
| [25] |
H. A. Levine and G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy,
Proc. Amer. Math. Soc., 129 (2001), 793-805 |
| [26] |
S. A. Messaoudi and B. Said-Houari,
Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277-287.
|
| [27] |
L. T. Ngoc and N. T. Long,
Existence, blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions, Appl. Math., 61 (2016), 165-196.
|
| [28] |
S. R. Park,
Nonexistence of global solutions of some quasilinear initial-boundary value problems, J. Korean Math. Soc., 34 (1997), 623-632.
|
| [29] |
L. E. Payne and D. H. Sattinger,
Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.
|
| [30] |
E. Pişkin and N. Polat,
Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesq-type equation, Turkish J. Math., 38 (2014), 706-727.
|
| [31] |
P. Pucci and J. Serrin,
Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.
|
| [32] |
B. Straughan,
Further global nonexistence theorems for abstract nonlinear wave equations, Proc. Amer. Math. Soc., 48 (1975), 381-390.
|
| [33] |
B. Straughan,
Explosive Instabilities in Mechanics Springer, 1998. |
| [34] |
M. Tsutsumi,
On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193.
|
| [35] |
Y. Wang,
A Sufficient condition for finite time blow up of the nonlinear Klein -Gordon equations with arbitrary positive initial energy, Proc. Amer. Math.Soc., 136 (2008), 3477-3482.
|
| [36] |
R. Zeng, Ch. Mu and Sh. Zhou,
A blow-up result for Kirchhoff-type equations with high energy, Math. Methods Appl. Sci., 34 (2011), 479-486.
|
show all references
References:
| [1] |
A. B. Aliyev and A. A. Kazimov,
Global non-existence of solutions with fixed positive initial energy of the Cauchy problem for a system of Klein-Gordon equations, Differ. Equ., 51 (2015), 1563-1568.
|
| [2] |
A. B. Alshin, M. O. Korpusov and A. G. Sveshnikov,
Blow up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Analysis and Applications, 15. Walter de Gruyter and Co., Berlin, 2011. xii+648 pp. |
| [3] |
P. Aviles and J. Sandefur,
Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58 (1985), 404-427.
|
| [4] |
B. A. Bilgin and V. K. Kalantarov,
Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations, J. Math. Anal. Appl., 403 (2013), 89-94.
|
| [5] |
E. H. de Brito,
Nonlinear initial-boundary value problems, Nonlinear Anal., 11 (1987), 125-137.
|
| [6] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko,
Strongly damped wave problems: bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 2310-2333.
|
| [7] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998. xiv+186 pp. |
| [8] |
T. Cazenave,
Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.
|
| [9] |
Sh. P. Chen and R. Triggiani,
Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.
|
| [10] |
F. Gazzola and M. Squassina,
Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006), 185-207.
|
| [11] |
H. A. Erbay, S. Erbay and A. Erkip,
Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations, Nonlinear Anal., 95 (2014), 313-322.
|
| [12] |
V. A. Galaktionov and S. I. Pohozaev,
Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Anal., 53 (2003), 453-466.
|
| [13] |
S. J. Jakubov,
Solvability of the Cauchy problem for abstract quasilinear hyperbolic equations of second order and their applications, Trans. Moscow Math. Soc., 23 (1970), 36-59.
|
| [14] |
V. K Kalantarov and O. A. Ladyzhenskaya,
The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math., 10 (1978), 53-70.
|
| [15] |
R. J. Knops, H. A. Levine and L. E. Payne,
Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal., 55 (1974), 52-72.
|
| [16] |
M. O. Korpusov,
Blow-up of the solution of a nonlinear system of equations with positive energy, Theoret. and Math. Phys., 171 (2012), 725-738.
|
| [17] |
M. O. Korpusov,
On the blow-up of solutions of a dissipative wave equation of Kirchhoff type with a source and positive energy, Sib. Math. J., 52 (2011), 471-483.
|
| [18] |
N. Kutev, N. Kolkovska and M. Dimova,
Nonexistence of global solutions to new ordinary differential inequality and applications to nonlinear dispersive equations, Math. Methods Appl. Sci., 39 (2016), 2287-2297.
|
| [19] |
I. Lasiecka and A. Stahel,
The wave equation with semilinear Neumann boundary conditions, Nonlinear Anal., 15 (1990), 39-58.
|
| [20] |
H. A. Levine,
Instability and nonexistence of global solutions to nonlinear wave equations of
the form $P{{u}_{tt}} = -Au+F(u)$, Trans. Am. Math. Soc., 192 (1974), 1-21.
|
| [21] |
H. A. Levine,
Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
|
| [22] |
H. A. Levine and L. E. Paine,
Nonexistence theorems for the heat equations with nonlinear boundary conditions and for porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334.
|
| [23] |
H. A. Levine, S. R. Park and J. Serrin,
Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228 (1998), 181-205.
|
| [24] |
H. A. Levine and R. A. Smith,
A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math., 374 (1987), 1-23.
|
| [25] |
H. A. Levine and G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy,
Proc. Amer. Math. Soc., 129 (2001), 793-805 |
| [26] |
S. A. Messaoudi and B. Said-Houari,
Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277-287.
|
| [27] |
L. T. Ngoc and N. T. Long,
Existence, blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions, Appl. Math., 61 (2016), 165-196.
|
| [28] |
S. R. Park,
Nonexistence of global solutions of some quasilinear initial-boundary value problems, J. Korean Math. Soc., 34 (1997), 623-632.
|
| [29] |
L. E. Payne and D. H. Sattinger,
Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.
|
| [30] |
E. Pişkin and N. Polat,
Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesq-type equation, Turkish J. Math., 38 (2014), 706-727.
|
| [31] |
P. Pucci and J. Serrin,
Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.
|
| [32] |
B. Straughan,
Further global nonexistence theorems for abstract nonlinear wave equations, Proc. Amer. Math. Soc., 48 (1975), 381-390.
|
| [33] |
B. Straughan,
Explosive Instabilities in Mechanics Springer, 1998. |
| [34] |
M. Tsutsumi,
On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173-193.
|
| [35] |
Y. Wang,
A Sufficient condition for finite time blow up of the nonlinear Klein -Gordon equations with arbitrary positive initial energy, Proc. Amer. Math.Soc., 136 (2008), 3477-3482.
|
| [36] |
R. Zeng, Ch. Mu and Sh. Zhou,
A blow-up result for Kirchhoff-type equations with high energy, Math. Methods Appl. Sci., 34 (2011), 479-486.
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