Abstract
This paper proves the global existence of solution for a class of nonlinear wave equations with nonlinear combined power-type nonlinearities of different signs for the initial data at sup-critical energy level.
Highlights
The authors in [ ] first considered problem ( )-( ) and obtained the global existence and blow up of solutions for the sub-critical case E( ) < d, where E( ) is the initial energy and d is the depth of the potential well or the mountain pass level, which will be defined later
Observing the above results for problem ( )-( ), helpful in the potential well method which was introduced by Payne and Sattinger [ ], the global existence for the sup-critical case, i.e. E( ) >, is still not solved
The present paper solves this problem by introducing a new stable invariant set and, using the method of [ ], we are focusing on proving the global existence of the solution for problem ( )-( ) in the sup-critical case E( ) >
Summary
Introduction In the present paper, we mainly consider the initial boundary value problem for the nonlinear wave equation with combined power-type nonlinearities of different signs, utt – u = f (u), x ∈ , t ∈ [ , ∞), ( ) The authors in [ ] first considered problem ( )-( ) and obtained the global existence and blow up of solutions for the sub-critical case E( ) < d, where E( ) is the initial energy and d is the depth of the potential well or the mountain pass level, which will be defined later.
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