May 2018, 17(3): 987-999. doi: 10.3934/cpaa.2018048

Non-existence of global solutions to nonlinear wave equations with positive initial energy

Department of Mathematics, Koç University, Rumelifeneri Yolu, Sariyer 34450, Istanbul, Turkey

* Corresponding author: Varga K. Kalantarov

Received  February 2017 Revised  February 2017 Published  January 2018

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.

Citation: Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048
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. CarvalhoJ. 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. ErbayS. 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. KnopsH. 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. KutevN. 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. LevineS. 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. ZengCh. 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. CarvalhoJ. 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. ErbayS. 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. KnopsH. 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. KutevN. 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. LevineS. 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. ZengCh. 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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