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.