We study estimates of lifespan and blow-up rates of solutions for the Cauchy problem of the wave equation with a time-dependent damping and a power-type nonlinearity. When the damping acts on the solutions effectively, and the nonlinearity belongs to the subcritical case, we show the sharp lifespan estimates and the blow-up rates of solutions. The upper estimates are proved by an ODE argument, and the lower estimates are given by a method of scaling variables.

Existence and nonexistence results on global solutions to the Cauchy problem for semirelativistic equations are shown by a simple compactness argument and a test function method, respectively. To obtain the nonexistence of global solutions, semirelativistic equations are transformed into a new equation without nonlocal operators in linear part but with a time derivative in nonlinear part, which is shown to be under control of special choice of test functions.

A lifespan estimate and a condition of the initial data for finite time blowup for the nonlinear Schrödinger equation are presented from a view point of ordinary differential equation (ODE) mechanism.

The nonexistence of global solutions for semirelativistic equations with nongauge invariant power type nonlinearity is revisited by a relatively direct way with a pointwise estimate of fractional derivative of some test functions.

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