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A doubly nonlinear CahnHilliard system with nonlinear viscosity
Nonexistence 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 nonexistence of solutions with fixed positive initial energy of the Cauchy problem for a system of KleinGordon equations, Differ. Equ., 51 (2015), 15631568. 
[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), 404427. 
[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), 8994. 
[5] 
E. H. de Brito, Nonlinear initialboundary value problems, Nonlinear Anal., 11 (1987), 125137. 
[6] 
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Strongly damped wave problems: bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 23102333. 
[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 KleinGordon equations, J. Funct. Anal., 60 (1985), 3655. 
[9] 
Sh. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 1555. 
[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), 185207. 
[11] 
H. A. Erbay, S. Erbay and A. Erkip, Thresholds for global existence and blowup in a general class of doubly dispersive nonlocal wave equations, Nonlinear Anal., 95 (2014), 313322. 
[12] 
V. A. Galaktionov and S. I. Pohozaev, Blowup and critical exponents for nonlinear hyperbolic equations, Nonlinear Anal., 53 (2003), 453466. 
[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), 3659. 
[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), 5370. 
[15] 
R. J. Knops, H. A. Levine and L. E. Payne, Nonexistence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal., 55 (1974), 5272. 
[16] 
M. O. Korpusov, Blowup of the solution of a nonlinear system of equations with positive energy, Theoret. and Math. Phys., 171 (2012), 725738. 
[17] 
M. O. Korpusov, On the blowup of solutions of a dissipative wave equation of Kirchhoff type with a source and positive energy, Sib. Math. J., 52 (2011), 471483. 
[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), 22872297. 
[19] 
I. Lasiecka and A. Stahel, The wave equation with semilinear Neumann boundary conditions, Nonlinear Anal., 15 (1990), 3958. 
[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), 121. 
[21] 
H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138146. 
[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), 319334. 
[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), 181205. 
[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), 123. 
[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), 793805 
[26] 
S. A. Messaoudi and B. SaidHouari, Global nonexistence of positive initialenergy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277287. 
[27] 
L. T. Ngoc and N. T. Long, Existence, blowup and exponential decay for a nonlinear Love equation associated with Dirichlet conditions, Appl. Math., 61 (2016), 165196. 
[28] 
S. R. Park, Nonexistence of global solutions of some quasilinear initialboundary value problems, J. Korean Math. Soc., 34 (1997), 623632. 
[29] 
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273303. 
[30] 
E. Pişkin and N. Polat, Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesqtype equation, Turkish J. Math., 38 (2014), 706727. 
[31] 
P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203214. 
[32] 
B. Straughan, Further global nonexistence theorems for abstract nonlinear wave equations, Proc. Amer. Math. Soc., 48 (1975), 381390. 
[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), 173193. 
[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), 34773482. 
[36] 
R. Zeng, Ch. Mu and Sh. Zhou, A blowup result for Kirchhofftype equations with high energy, Math. Methods Appl. Sci., 34 (2011), 479486. 
show all references
References:
[1] 
A. B. Aliyev and A. A. Kazimov, Global nonexistence of solutions with fixed positive initial energy of the Cauchy problem for a system of KleinGordon equations, Differ. Equ., 51 (2015), 15631568. 
[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), 404427. 
[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), 8994. 
[5] 
E. H. de Brito, Nonlinear initialboundary value problems, Nonlinear Anal., 11 (1987), 125137. 
[6] 
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Strongly damped wave problems: bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 23102333. 
[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 KleinGordon equations, J. Funct. Anal., 60 (1985), 3655. 
[9] 
Sh. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 1555. 
[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), 185207. 
[11] 
H. A. Erbay, S. Erbay and A. Erkip, Thresholds for global existence and blowup in a general class of doubly dispersive nonlocal wave equations, Nonlinear Anal., 95 (2014), 313322. 
[12] 
V. A. Galaktionov and S. I. Pohozaev, Blowup and critical exponents for nonlinear hyperbolic equations, Nonlinear Anal., 53 (2003), 453466. 
[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), 3659. 
[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), 5370. 
[15] 
R. J. Knops, H. A. Levine and L. E. Payne, Nonexistence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal., 55 (1974), 5272. 
[16] 
M. O. Korpusov, Blowup of the solution of a nonlinear system of equations with positive energy, Theoret. and Math. Phys., 171 (2012), 725738. 
[17] 
M. O. Korpusov, On the blowup of solutions of a dissipative wave equation of Kirchhoff type with a source and positive energy, Sib. Math. J., 52 (2011), 471483. 
[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), 22872297. 
[19] 
I. Lasiecka and A. Stahel, The wave equation with semilinear Neumann boundary conditions, Nonlinear Anal., 15 (1990), 3958. 
[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), 121. 
[21] 
H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138146. 
[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), 319334. 
[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), 181205. 
[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), 123. 
[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), 793805 
[26] 
S. A. Messaoudi and B. SaidHouari, Global nonexistence of positive initialenergy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277287. 
[27] 
L. T. Ngoc and N. T. Long, Existence, blowup and exponential decay for a nonlinear Love equation associated with Dirichlet conditions, Appl. Math., 61 (2016), 165196. 
[28] 
S. R. Park, Nonexistence of global solutions of some quasilinear initialboundary value problems, J. Korean Math. Soc., 34 (1997), 623632. 
[29] 
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273303. 
[30] 
E. Pişkin and N. Polat, Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesqtype equation, Turkish J. Math., 38 (2014), 706727. 
[31] 
P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203214. 
[32] 
B. Straughan, Further global nonexistence theorems for abstract nonlinear wave equations, Proc. Amer. Math. Soc., 48 (1975), 381390. 
[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), 173193. 
[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), 34773482. 
[36] 
R. Zeng, Ch. Mu and Sh. Zhou, A blowup result for Kirchhofftype equations with high energy, Math. Methods Appl. Sci., 34 (2011), 479486. 
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