Recently the critical constant of time-decaying damping in the scale invariant case is conjectured. Also it has been expected that the lifespan estimate is the same as associated semilinear heat equations if the constant is in “heat-like” domain. In this talk, I would like to point out that this is not true if the total integral of the initial position and speed vanish. In such a case, we have a new type of the lifespan estimates which is closely related to the non-damped case with a shifted dimension. This result is a joint work with M. Kato and K. Wakasa.

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