Decay property of solutions for damped wave equations with space–time dependent damping term

We consider the Cauchy problem for the damped wave equation with space–time dependent potential b(t,x) and absorbing semilinear term |u|ρ−1u. Here, with b0>0, α,β⩾0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L2 decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N−α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L2-estimates of solutions for damped wave equations with space–time dependent damping term, J. Differential Equations 248 (2010) 403–422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,α,β) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.