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This paper is concerned with the blowup phenomena for initial-boundary valueproblem for certain semi linear parabolic, dispersive and hyperbolic equationsin cone-like domain. The result proposes a unified treatment of estimates forlifespan of solutions to the problem by test function method. The Fujitaexponent p=1 + 2/N appears as a threshold of blowup phenomena for small datawhen $C_{{\Sigma}}=R^N$ , but the case of cone-like domain with boundary thethreshold changes and explicitly given via the first eigenvalue ofcorresponding Laplace-Beltrami operator with Dirichlet boundary condition as inLevine-Meier in 1989.

Upper bound for lifespan of solutions to certain semilinear parabolic, dispersive and hyperbolic equations via a unified test function method