Lifespan of solutions to the damped wave equation with a critical nonlinearity

In the present paper, we study a lifespan of solutions to the Cauchy problem for semilinear damped wave equations(DW) where n≥1, f(u)=±|u|^p-1u or |u|^p, p≥1, ε>0 is a small parameter, and (u₀, u₁) is a given initial data. The main purpose of this paper is to prove that if the nonlinear term is f(u)=|u|^p and the nonlinear power is the Fujita critical exponent p=pF=1+2n, then the upper estimate to the lifespan is estimated by for all ε∈(0, 1] and suitable data (u₀, u₁), without any restriction on the spatial dimension. Our proof is based on a test-function method utilized by Zhang [35]. We also prove a sharp lower estimate of the lifespan T(ε) to (DW) in the critical case p=p_F.