### Electronic Research Archive

2020, Issue 2: 671-689. doi: 10.3934/era.2020035
Special Issues

# Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy

• Received: 01 December 2019 Revised: 01 March 2020
• Primary: 35L70, 35B44; Secondary: 35B40

• Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

Citation: Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy[J]. Electronic Research Archive, 2020, 28(2): 671-689. doi: 10.3934/era.2020035

### Related Papers:

• Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

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