| Abstract: |
| In this talk, global existence for the Cauchy problem for a semilinear damped wave equation with positive initial displacement and negative initial velocity is investigated. In the classical theory, solutions are known to blow up if the sum of the mean values of the initial displacement and velocity is nonnegative, while global existence holds if the initial data satisfy a pointwise sign condition. However, it remains unclear whether solutions blow up in other cases. In this talk, a new condition on the initial data ensuring global existence is presented. Our approach is to show that the solution satisfies the classical global existence condition of Li and Zhou at some time. In particular, the solution is bounded by the initial displacement times a time-dependent amplification factor. This factor is shown to change sign at a certain time. The initial displacement and velocity are assumed to be given by a common polynomially decaying function multiplied by a positive constant and a negative constant, respectively. |
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