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This paper is concerned about the lifespan estimate to the Cauchy problem ofsemilinear damped wave equations with the Fujita critical exponent in highdimensions$(n\geq 4)$. We establish the sharp upper bound of the lifespan inthe following form \begin{equation}\nonumber\\ \begin{aligned}T(\varepsilon)\leq \exp(C\varepsilon^{-\frac 2n}), \end{aligned} \end{equation}by using the heat kernel as the test function.

The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions