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){∂t2u−Δu+∂tu=f(u),(t,x)∈[0,T(ε))×Rn,u(0,x)=εu0(x),x∈Rn,∂tu(0,x)=εu1(x),x∈Rn, 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 byT(ε)≤exp⁡(Cε−p) 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=pF.